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id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">文頭</div> </a> </li> <li id="toc-哲學背景" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#哲學背景"> <div class="vector-toc-text"> 1 哲學背景 </div> </a> <ul id="toc-哲學背景-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-基本程式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#基本程式"> <div class="vector-toc-text"> 2 基本程式 </div> </a> <ul id="toc-基本程式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-主要分類" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#主要分類"> <div class="vector-toc-text"> 3 主要分類 </div> </a> <button aria-controls="toc-主要分類-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 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class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 切換去各領域嘅應用細章節 </button> <ul id="toc-各領域嘅應用-sublist" class="vector-toc-list"> <li id="toc-物理學" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#物理學"> <div class="vector-toc-text"> 4.1 物理學 </div> </a> <ul id="toc-物理學-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-生物學" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#生物學"> <div class="vector-toc-text"> 4.2 生物學 </div> </a> <ul id="toc-生物學-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-認知科學" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#認知科學"> <div class="vector-toc-text"> 4.3 認知科學 </div> </a> <ul id="toc-認知科學-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-社會科學" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#社會科學"> <div class="vector-toc-text"> 4.4 社會科學 </div> </a> <ul id="toc-社會科學-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-工程學" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#工程學"> <div class="vector-toc-text"> 4.5 工程學 </div> </a> <ul id="toc-工程學-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-驗證同確認" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#驗證同確認"> <div class="vector-toc-text"> 5 驗證同確認 </div> </a> <ul id="toc-驗證同確認-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-創發嘅啟示" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#創發嘅啟示"> <div class="vector-toc-text"> 6 創發嘅啟示 </div> </a> <ul id="toc-創發嘅啟示-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-註釋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註釋"> <div class="vector-toc-text"> 7 註釋 </div> </a> <ul id="toc-註釋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-睇埋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#睇埋"> <div class="vector-toc-text"> 8 睇埋 </div> </a> <ul id="toc-睇埋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#文獻"> <div class="vector-toc-text"> 9 文獻 </div> </a> <ul id="toc-文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引述" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#引述"> <div class="vector-toc-text"> 10 引述 </div> </a> <ul id="toc-引述-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-拎" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#拎"> <div class="vector-toc-text"> 11 拎 </div> </a> <ul id="toc-拎-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目錄" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="開/收內容一覽" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > 開/收內容一覽 </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">電腦模擬</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="去睇另一種語文嘅文章。有38種語言版本。" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-38" aria-hidden="true" > 38種語言 </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AD%D8%A7%D9%83%D8%A7%D8%A9_%D8%A8%D8%A7%D9%84%D8%AD%D8%A7%D8%B3%D9%88%D8%A8" title="محاكاة بالحاسوب – 阿拉伯文" lang="ar" hreflang="ar" data-title="محاكاة بالحاسوب" data-language-autonym="العربية" data-language-local-name="阿拉伯文" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%8E%D1%82%D1%8A%D1%80%D0%BD%D0%B0_%D1%81%D0%B8%D0%BC%D1%83%D0%BB%D0%B0%D1%86%D0%B8%D1%8F" title="Компютърна симулация – 保加利亞文" lang="bg" hreflang="bg" data-title="Компютърна симулация" data-language-autonym="Български" data-language-local-name="保加利亞文" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Simulaci%C3%B3_d%27ordinador" title="Simulació d'ordinador – 加泰羅尼亞文" lang="ca" hreflang="ca" data-title="Simulació d'ordinador" data-language-autonym="Català" data-language-local-name="加泰羅尼亞文" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Po%C4%8D%C3%ADta%C4%8Dov%C3%A1_simulace" title="Počítačová simulace – 捷克文" lang="cs" hreflang="cs" data-title="Počítačová simulace" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Computersimulation" title="Computersimulation – 丹麥文" lang="da" hreflang="da" data-title="Computersimulation" data-language-autonym="Dansk" data-language-local-name="丹麥文" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Computersimulation" title="Computersimulation – 德文" lang="de" hreflang="de" data-title="Computersimulation" data-language-autonym="Deutsch" data-language-local-name="德文" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Computer_simulation" title="Computer simulation – 英文" lang="en" hreflang="en" data-title="Computer simulation" data-language-autonym="English" data-language-local-name="英文" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Komputilsimulado" title="Komputilsimulado – 世界文" lang="eo" hreflang="eo" data-title="Komputilsimulado" data-language-autonym="Esperanto" data-language-local-name="世界文" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Simulaci%C3%B3n_(inform%C3%A1tica)" title="Simulación (informática) – 西班牙文" lang="es" hreflang="es" data-title="Simulación (informática)" data-language-autonym="Español" data-language-local-name="西班牙文" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arvutisimulatsioon" title="Arvutisimulatsioon – 愛沙尼亞文" lang="et" hreflang="et" data-title="Arvutisimulatsioon" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%A8%DB%8C%D9%87%E2%80%8C%D8%B3%D8%A7%D8%B2%DB%8C_%DA%A9%D8%A7%D9%85%D9%BE%DB%8C%D9%88%D8%AA%D8%B1%DB%8C" title="شبیه‌سازی کامپیوتری – 波斯文" lang="fa" hreflang="fa" data-title="شبیه‌سازی کامپیوتری" data-language-autonym="فارسی" data-language-local-name="波斯文" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tietokonesimulointi" title="Tietokonesimulointi – 芬蘭文" lang="fi" hreflang="fi" data-title="Tietokonesimulointi" data-language-autonym="Suomi" data-language-local-name="芬蘭文" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Simulation_informatique" title="Simulation informatique – 法文" lang="fr" hreflang="fr" data-title="Simulation informatique" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9C%D7%A6%D7%99%D7%94_%D7%9E%D7%9E%D7%95%D7%97%D7%A9%D7%91%D7%AA" title="סימולציה ממוחשבת – 希伯來文" lang="he" hreflang="he" data-title="סימולציה ממוחשבת" data-language-autonym="עברית" data-language-local-name="希伯來文" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%95%E0%A4%B2%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A4%BE%E0%A4%B0" title="अभिकलित्र अनुकार – 北印度文" lang="hi" hreflang="hi" data-title="अभिकलित्र अनुकार" data-language-autonym="हिन्दी" data-language-local-name="北印度文" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Simulasi_komputer" title="Simulasi komputer – 印尼文" lang="id" hreflang="id" data-title="Simulasi komputer" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Komputerala_simulo" title="Komputerala simulo – 伊多文" lang="io" hreflang="io" data-title="Komputerala simulo" data-language-autonym="Ido" data-language-local-name="伊多文" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%88%D0%B8%D0%BD%D0%B0%D0%BB%D1%8B%D2%9B_%D2%B1%D2%9B%D1%81%D0%B0%D1%82%D1%82%D1%8B%D1%80%D1%83" title="Машиналық ұқсаттыру – 哈薩克文" lang="kk" hreflang="kk" data-title="Машиналық ұқсаттыру" data-language-autonym="Қазақша" data-language-local-name="哈薩克文" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BB%B4%ED%93%A8%ED%84%B0_%EC%8B%9C%EB%AE%AC%EB%A0%88%EC%9D%B4%EC%85%98" title="컴퓨터 시뮬레이션 – 韓文" lang="ko" hreflang="ko" data-title="컴퓨터 시뮬레이션" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Simulatio_computatralis" title="Simulatio computatralis – 拉丁文" lang="la" hreflang="la" data-title="Simulatio computatralis" data-language-autonym="Latina" data-language-local-name="拉丁文" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%8C%D1%8E%D1%82%D0%B5%D1%80%D0%B8%D0%B9%D0%BD_%D1%81%D0%B8%D0%BC%D1%83%D0%BB%D1%8F%D1%86%D0%B8" title="Компьютерийн симуляци – 蒙古文" lang="mn" hreflang="mn" data-title="Компьютерийн симуляци" data-language-autonym="Монгол" data-language-local-name="蒙古文" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Penyelakuan_komputer" title="Penyelakuan komputer – 馬來文" lang="ms" hreflang="ms" data-title="Penyelakuan komputer" data-language-autonym="Bahasa Melayu" data-language-local-name="馬來文" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Modelaige_cumputacional" title="Modelaige cumputacional – 米蘭德斯文" lang="mwl" hreflang="mwl" data-title="Modelaige cumputacional" data-language-autonym="Mirandés" data-language-local-name="米蘭德斯文" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Computersimulatie" title="Computersimulatie – 荷蘭文" lang="nl" hreflang="nl" data-title="Computersimulatie" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Datasimulering" title="Datasimulering – 巴克摩挪威文" lang="nb" hreflang="nb" data-title="Datasimulering" data-language-autonym="Norsk bokmål" data-language-local-name="巴克摩挪威文" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Symulacja_komputerowa" title="Symulacja komputerowa – 波蘭文" lang="pl" hreflang="pl" data-title="Symulacja komputerowa" data-language-autonym="Polski" data-language-local-name="波蘭文" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Modelagem_computacional" title="Modelagem computacional – 葡萄牙文" lang="pt" hreflang="pt" data-title="Modelagem computacional" data-language-autonym="Português" data-language-local-name="葡萄牙文" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%8C%D1%8E%D1%82%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5" title="Компьютерное моделирование – 俄文" lang="ru" hreflang="ru" data-title="Компьютерное моделирование" data-language-autonym="Русский" data-language-local-name="俄文" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Computer_model" title="Computer model – Simple English" lang="en-simple" hreflang="en-simple" data-title="Computer model" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Po%C4%8D%C3%ADta%C4%8Dov%C3%A1_simul%C3%A1cia" title="Počítačová simulácia – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Počítačová simulácia" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ra%C4%8Dunalni%C5%A1ka_simulacija" title="Računalniška simulacija – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Računalniška simulacija" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" 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</div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 第啲維基項目 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Computer_simulations" hreflang="en">維基同享</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q925667" title="連到接住資料儲存庫嘅項目[g]" accesskey="g">維基數據項</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="頁面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="閱讀設定"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">閱讀設定</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移去側欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">收埋</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">出自維基百科，自由嘅百科全書</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="yue" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1488945/mw-parser-output/.tmulti">.mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}</style><div class="thumb tmulti tright"><div class="thumbinner" style="width:304px;max-width:304px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><a href="/wiki/File:Typhoon_Mawar_2005_computer_simulation_thumbnail.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Typhoon_Mawar_2005_computer_simulation_thumbnail.gif/300px-Typhoon_Mawar_2005_computer_simulation_thumbnail.gif" decoding="async" width="300" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/8b/Typhoon_Mawar_2005_computer_simulation_thumbnail.gif 1.5x" data-file-width="450" data-file-height="403" /></a></div><div class="thumbcaption">用電腦模擬技術模擬一個<a href="/wiki/%E9%A2%B1%E9%A2%A8" title="颱風">颱風</a>出嘅圖像</div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Boids_OpenGL_example.webm/300px--Boids_OpenGL_example.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="300" height="300" data-durationhint="80" data-mwtitle="Boids_OpenGL_example.webm" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/43/Boids_OpenGL_example.webm/Boids_OpenGL_example.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="480" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/4/43/Boids_OpenGL_example.webm" type="video/webm; codecs="vp8"" data-width="500" data-height="500" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/43/Boids_OpenGL_example.webm/Boids_OpenGL_example.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="144" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/43/Boids_OpenGL_example.webm/Boids_OpenGL_example.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="240" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/43/Boids_OpenGL_example.webm/Boids_OpenGL_example.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="360" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/43/Boids_OpenGL_example.webm/Boids_OpenGL_example.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="360" data-height="360" /></video></div><div class="thumbcaption">用 <a href="/wiki/OpenGL" title="OpenGL">OpenGL</a> 整嘅群動模擬</div></div></div></div></div> 電腦模擬（<a href="/wiki/Wikipedia:%E7%B2%B5%E6%96%87%E6%8B%BC%E9%9F%B3" title="Wikipedia:粵文拼音">粵拼</a>：<a rel="nofollow" class="external text" href="http://yue.forvo.com/word/yue/電腦模擬/#yue">din6 nou5 mou4 ji5</a>；<a href="/wiki/%E8%8B%B1%E6%96%87" title="英文">英文</a>：computer simulation），又叫電腦仿真，係<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a>上嘅一種<a href="/wiki/%E7%A7%91%E5%AD%B8%E7%A0%94%E7%A9%B6" class="mw-redirect" title="科學研究">研究</a>方法。研究者會整個<a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a>出嚟<a href="/wiki/%E6%A8%A1%E6%93%AC" title="模擬">模擬</a>想研究嘅現象，個模型會講明個現象涉及邊啲<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">變數</a>同埋呢啲變數之間嘅<a href="/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式">方程式</a>係點，跟住個研究者就會將個模型寫成<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F" title="電腦程式">電腦程式</a>嚟行，睇吓「如果個數學模型正確，會發生乜嘢事」，做到按<a href="/wiki/%E7%A7%91%E5%AD%B8%E7%90%86%E8%AB%96" title="科學理論">理論</a>知識預測現實世界嘅現象。除此之外，電腦模擬通常會配合埋<a href="/wiki/%E7%A7%91%E5%AD%B8%E8%A6%96%E8%A6%BA%E5%8C%96" title="科學視覺化">科學視覺化</a>技術，將模擬結果以<a href="/wiki/%E5%9C%96%E5%83%8F" class="mw-redirect" title="圖像">圖像</a>形式呈現出嚟<a href="#cite_note-strogatz2007-1">[1]</a><a href="#cite_note-basicode-2">[2]</a>。 舉個簡單例子說明，<a href="/wiki/%E7%BE%A4%E5%8B%95" title="群動">群動</a>係電腦模擬嘅一個出名例子，想像家陣有一大柞個體一齊移動，途中佢哋會守三條規則：分隔－唔好撞到隔離嘅其他個體；順向－將自己方向對準周圍其他個體嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均</a>；凝聚－將自己置於周圍其他個體嘅平均嘅位置。用電腦模擬嘅話，個<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F" title="電腦程式">程式</a>會<a href="/wiki/%E8%99%9B%E6%93%AC%E7%A2%BC" title="虛擬碼">類似噉</a>： <pre> 設 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 件<a href="/wiki/%E7%89%A9%E4%BB%B6_(%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8)" title="物件 (電腦科學)">物件</a>；設呢 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 件物件嘅初始位置設 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">（時間）= 0； <a href="/wiki/While_%E8%BF%B4%E5%9C%88" title="While 迴圈">While</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 未到某個值， <a href="/wiki/Foreach" class="mw-redirect" title="Foreach">Foreach</a> 物件計「如果呢件物件跟嗰三條法則郁，佢嘅新位置會响邊」； <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="電腦圖像">顯示</a>出每件物件嘅新位置； <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 上升 1； </pre> 於是研究者就能夠知道「如果群動嘅數學模型正確，會發生乜嘢事（啲物件郁起上嚟望落會係點嘅樣）」。事實表明咗，群動模型可以準確噉預測某啲<a href="/wiki/%E8%87%AA%E7%84%B6%E7%8F%BE%E8%B1%A1" title="自然現象">自然現象</a>，例如係一大群<a href="/wiki/%E9%9B%80%E9%B3%A5" class="mw-redirect" title="雀鳥">雀鳥</a>一齊飛嗰陣成個雀群成嘅形態<a href="#cite_note-reynolds1987-3">[3]</a>。 電腦模擬係<a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a>不可或缺嘅一部份，喺<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a><a href="#cite_note-4">[4]</a>、<a href="/wiki/%E5%8C%96%E5%AD%B8" title="化學">化學</a><a href="#cite_note-5">[5]</a>同<a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8" title="生物學">生物學</a><a href="#cite_note-6">[6]</a>等嘅<a href="/wiki/%E8%87%AA%E7%84%B6%E7%A7%91%E5%AD%B8" title="自然科學">自然科學</a>上都會用到。除此之外，電腦模擬仲有俾人用落去<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>上，例如用<a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a>模擬<a href="/wiki/%E7%A4%BE%E6%9C%83" title="社會">社會</a>（想像「如果社會當中有咁多咁多<a href="/wiki/%E7%99%BE%E5%88%86%E6%AF%94" class="mw-redirect" title="百分比">百分比</a>嘅人跟噉樣噉樣嘅法則做<a href="/wiki/%E6%B1%BA%E7%AD%96" title="決策">決策</a>，個<a href="/wiki/%E7%B6%93%E6%BF%9F%E9%AB%94" class="mw-redirect" title="經濟體">經濟體</a>會變成噉樣噉樣」）<a href="#cite_note-7">[7]</a>，而<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>都有用到電腦模擬，例如模擬「如果部<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>用噉嘅設計，部機械嘅<a href="/wiki/%E6%A9%9F%E6%A2%B0%E6%95%88%E7%9B%8A" title="機械效益">效率</a>會係咁高咁高」<a href="#cite_note-8">[8]</a>。 以下嘅內容，假設讀者已經有齊嗮基本嘅<a href="/wiki/%E7%A8%8B%E5%BC%8F%E7%B7%A8%E5%AF%AB" class="mw-redirect" title="程式編寫">程式編寫</a>知識。 <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="哲學背景">哲學背景</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=1" title="編輯小節: 哲學背景">編輯</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:NewtonsLawOfUniversalGravitation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/300px-NewtonsLawOfUniversalGravitation.svg.png" decoding="async" width="300" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/450px-NewtonsLawOfUniversalGravitation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/600px-NewtonsLawOfUniversalGravitation.svg.png 2x" data-file-width="400" data-file-height="250" /></a><figcaption><a href="/wiki/%E8%90%AC%E6%9C%89%E5%BC%95%E5%8A%9B%E5%AE%9A%E5%BE%8B" title="萬有引力定律">萬有引力定律</a>以科學理論嚟講查實好<a href="/wiki/%E7%B0%A1%E5%96%AE" title="簡單">簡單</a>，成條式得嗰 4 個<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">變數</a><a href="#cite_note-9">[註 1]</a><a href="#cite_note-10">[9]</a>－ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1387d67417e45de53b554e8b775be7ad3cb236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.196ex; height:2.509ex;" alt="{\displaystyle F_{1}=F_{2}}"> 係兩件嘢之間嘅引力嘅數值； <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{m}_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{m}_{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301180698bd2ab5dbf739c179292409445c5005c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle {{m}_{1}}}"> 係第一件嘢嘅<a href="/wiki/%E8%B3%AA%E9%87%8F" title="質量">質量</a>；單位係<a href="/wiki/%E5%85%AC%E6%96%A4" class="mw-redirect" title="公斤">公斤</a>（kg） <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{m}_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{m}_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd08a1fe1bb494a015fcfc73471423b036665f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle {{m}_{2}}}"> 係第二件嘢嘅質量；單位係公斤（kg） <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> 係兩件嘢之間嘅<a href="/wiki/%E8%B7%9D%E9%9B%A2" title="距離">距離</a>；單位係<a href="/wiki/%E7%B1%B3_(%E9%95%B7%E5%BA%A6)" class="mw-redirect" title="米 (長度)">米</a>（m） 所以要用腦諗「如果 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{m}_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{m}_{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301180698bd2ab5dbf739c179292409445c5005c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle {{m}_{1}}}"> 同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{m}_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{m}_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd08a1fe1bb494a015fcfc73471423b036665f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle {{m}_{2}}}"> 係咁多咁多，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{1}}"> 會係幾多」並唔難。但唔係個個科學理論都咁簡單。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E7%A7%91%E5%AD%B8%E6%96%B9%E6%B3%95" title="科學方法">科學方法</a>、<a href="/wiki/%E7%A7%91%E5%AD%B8%E7%90%86%E8%AB%96" title="科學理論">科學理論</a>、<a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a>同<a href="/wiki/%E6%A8%A1%E6%93%AC" title="模擬">模擬</a></div> 電腦模擬係廿一世紀初嘅一種<a href="/wiki/%E7%A7%91%E7%A0%94" class="mw-redirect" title="科研">科研</a>方法。<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a>係用<a href="/wiki/%E7%A7%91%E5%AD%B8%E6%96%B9%E6%B3%95" title="科學方法">科學方法</a><a href="#cite_note-11">[e 1]</a>嚟追求<a href="/wiki/%E7%9F%A5%E8%AD%98" title="知識">知識</a>，而科學方法涉及以下嘅步驟<a href="#cite_note-12">[10]</a>： <ol><li>根據手上已知係真嘅<a href="/wiki/%E4%BA%8B%E5%AF%A6" title="事實">事實</a>，諗一啲<a href="/wiki/%E7%A7%91%E5%AD%B8%E7%90%86%E8%AB%96" title="科學理論">理論</a>出嚟（「基於我哋經已觀察到嘅呢啲現象，我推測啲現象背後嘅法則係噉噉噉，令呢種現象出現」），而為求精確，呢啲理論通常都想係用<a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a>嘅方式表達嘅；</li> <li>跟手就靠<a href="/wiki/%E9%82%8F%E8%BC%AF" title="邏輯">邏輯</a>嘅思考，去諗吓呢柞理論會做點樣嘅預測（「如果我提倡嘅呢個理論係真確嘅，理應會...」）；</li> <li>然後就做<a href="/wiki/%E5%AF%A6%E9%A9%97" title="實驗">實驗</a>同<a href="/wiki/%E8%A7%80%E5%AF%9F" title="觀察">觀察</a>嚟攞<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>，驗證吓個理論嘅預測係咪真確（「如果我呢個理論係真確，呢個實驗理應會得出某個某個結果，而個實際嘅實驗結果係...」）；</li> <li>如果數據顯示，個理論係做唔到準確嘅預測嘅話，就要一係根據攞到嘅數據睇吓個理論要點改先可以令佢做到準確啲嘅預測，一係就要諗新理論取代舊理論<a href="#cite_note-13">[11]</a>。</li></ol> 用電腦模擬做科研想要做嘅就係幫手預測<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>嘅行為：原則上，「一個理論會作出乜嘢預測」呢一點可以由個研究者齋靠諗嚟做得到，例子可以睇吓好似基本<a href="/wiki/%E7%89%9B%E9%A0%93%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="牛頓力學">牛頓力學</a>入面嗰啲相對<a href="/wiki/%E7%B0%A1%E5%96%AE" title="簡單">簡單</a>嘅<a href="/wiki/%E7%89%A9%E7%90%86%E5%AE%9A%E5%BE%8B" title="物理定律">物理定律</a>－呢啲定律吓吓都係一條式得嗰幾個<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">變數</a>，研究者可以輕易噉用人手計出「根據呢條定律，如果變數 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> 嘅數值係咁多咁多，變數 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 嘅數值應該會係咁多咁多」。不過當研究緊嗰樣現象有返咁上下<a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>嗰陣，「齋靠個研究者用<a href="/wiki/%E4%BA%BA%E8%85%A6" title="人腦">腦</a>諗」好多時就會難令人信服<a href="#cite_note-mcclelland2009-14">[12]</a>:p. 6，例如係<a href="/wiki/%E7%BE%A4%E5%8B%95" title="群動">群動</a>噉，好難說服人話「齋靠研究者用腦諗，就會準確計到群動三定律會出點嘅樣嘅移動」。於是研究者就將自己個理論（用數學模型嘅方式表示）<a href="/wiki/%E7%B7%A8%E7%A8%8B" class="mw-redirect" title="編程">編寫</a>成一個<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F" title="電腦程式">電腦程式</a>，然後再行呢個程式，睇吓個程式會引致乜嘢結果－呢啲結果就會係佢個理論會作出嘅預測<a href="#cite_note-15">[13]</a><a href="#cite_note-winsberg2003-16">[14]</a>。 電腦模擬係喺廿世紀嗰陣同<a href="/wiki/%E9%9B%BB%E8%85%A6" title="電腦">電腦</a>一齊冒起嘅：<a href="/wiki/%E7%A7%91%E5%AD%B8%E7%A0%94%E7%A9%B6" class="mw-redirect" title="科學研究">科學研究</a>呢家嘢喺<a href="/wiki/%E4%B8%AD%E4%B8%96%E7%B4%80" title="中世紀">中世紀</a>經已有，當時要計數淨係有得用人手計，而好似<a href="/wiki/%E8%90%AC%E6%9C%89%E5%BC%95%E5%8A%9B%E5%AE%9A%E5%BE%8B" title="萬有引力定律">萬有引力定律</a>等相對簡單嘅科學理論（條式通常得嗰 3 到 4 個變數）要用人手計並唔難，但一研究好似係<a href="/wiki/%E7%BE%A4%E5%8B%95" title="群動">群動</a>呢啲涉及閒閒地幾廿個個體隨時間分別噉變化嘅現象，用人手計就會好撈絞；不過廿世紀初出咗一樣好重要嘅發明－電腦，電腦係<a href="/wiki/%E9%81%8B%E7%AE%97%E6%A9%9F%E6%A2%B0" class="mw-redirect" title="運算機械">運算機械</a>，即係能夠用快過人手好多嘅速度做大量嘅<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>，所以電腦嘅出現令到人類能夠計一啲以前計唔到嘅<a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a>，引起咗<a href="/wiki/%E7%A7%91%E5%AD%B8%E5%AE%B6" title="科學家">科學家</a>用電腦模擬嚟做科研。到咗廿一世紀初，電腦模擬經已成為咗科研一個不可或缺嘅部份<a href="#cite_note-17">[15]</a>。 <div class="mw-heading mw-heading2"><h2 id="基本程式">基本程式</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=2" title="編輯小節: 基本程式">編輯</a>]</div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="電腦圖像">電腦圖像</a>同<a href="/wiki/%E7%A7%91%E5%AD%B8%E8%A6%96%E8%A6%BA%E5%8C%96" title="科學視覺化">科學視覺化</a></div> 喺最基本上，一個做電腦模擬嘅程式（或者最少係做<a href="/wiki/%E5%8B%95%E6%85%8B%E6%A8%A1%E6%93%AC" title="動態模擬">動態模擬</a>嘅程式）可以想像成以下噉嘅<a href="/wiki/%E8%99%9B%E6%93%AC%E7%A2%BC" title="虛擬碼">虛擬碼</a><a href="#cite_note-basicode-2">[2]</a>： <pre> <a href="/wiki/%E5%88%9D%E5%A7%8B%E5%8C%96" title="初始化">初始化</a>（講明個模型有乜<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%B7%A8%E7%A8%8B)" title="變數 (編程)">變數</a>，想像其中一個變數叫「<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">」）；設 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">（時間）= 0； <a href="/wiki/While_%E8%BF%B4%E5%9C%88" title="While 迴圈">While</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 未到某個值，按個理論嘅數學模型，計吓個現象跟住會變成點（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb3a22359174d9da835b238b25e2f89ce1cd2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.181ex; height:2.843ex;" alt="{\displaystyle y=f(t)}">）；<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> 係一個<a href="/wiki/%E5%87%BD%E6%95%B8" title="函數">函數</a>。 <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="電腦圖像">顯示</a>出個現象變成咗點嘅樣； <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 上升 1；<a href="#cite_note-18">[註 2]</a> </pre> 呢個過程會俾研究者視覺化噉睇到「如果嗰個數學模型成立，噉會見到啲乜嘢現象」呢一樣<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a><a href="#cite_note-19">[註 3]</a><a href="#cite_note-winsberg2003-16">[14]</a>。除此之外，一個電腦模擬仲可以將模擬途中嘅<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>記低，例如記住「變數 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 喺每個時間點嘅數值」（即係好似下面段碼噉），跟手再將呢啲數據畫成<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%9C%96" title="統計圖">圖表</a>，即係例如 <a href="/wiki/Y_%E8%BB%B8" class="mw-redirect" title="Y 軸">Y 軸</a>表示變數 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 嘅數值，<a href="/wiki/X_%E8%BB%B8" class="mw-redirect" title="X 軸">X 軸</a>表示時間，畫一條<a href="/wiki/%E7%B6%AB_(%E5%B9%BE%E4%BD%95)" title="綫 (幾何)">線</a>出嚟<a href="#cite_note-strogatz2007-1">[1]</a>。 <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>t=0: y = 0 t=1: y = 0 t=2: y = 10 t=3: y = 30 t=4: y = 60 t=5: y = 100 ... </pre></div> <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:TRUE_Procedural_Animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e7/TRUE_Procedural_Animation.gif" decoding="async" width="480" height="260" class="mw-file-element" data-file-width="480" data-file-height="260" /></a><figcaption>呢幅圖顯示用<a href="/wiki/%E5%8B%95%E6%85%8B%E6%A8%A1%E6%93%AC" title="動態模擬">動態模擬</a>模擬部<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>嘅郁動－「<a href="/wiki/%E6%B4%BB%E5%A1%9E" title="活塞">活塞</a>嘅位置」（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}">）會隨時間變化，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7ac17f7ad40d2acd962dede45ab8cf2ccbf0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.41ex; height:2.843ex;" alt="{\displaystyle x_{1}=f(t)}">，而個<a href="/wiki/%E8%BD%86" title="轆">轆</a>嘅轉動（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}">）係由 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"> 帶動嘅，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}=f(x_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190f57ce20bf29ab600a72dbd4795a7a4e09ff52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.954ex; height:2.843ex;" alt="{\displaystyle x_{2}=f(x_{1})}">，所以隨住 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 數值上升，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"> 同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"> 等嘅變數嘅數值會跟住變化。</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="主要分類">主要分類</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=3" title="編輯小節: 主要分類">編輯</a>]</div> 電腦模擬可以按多種準則分類，包括咗係<a href="#cite_note-20">[註 4]</a>： <div class="mw-heading mw-heading3"><h3 id="有冇隨機性">有冇隨機性</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=4" title="編輯小節: 有冇隨機性">編輯</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pi_30K.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Pi_30K.gif/300px-Pi_30K.gif" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Pi_30K.gif/450px-Pi_30K.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/84/Pi_30K.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>用蒙地卡羅方法估計 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> 數值嘅圖解；圖上高嘅字顯示計出嚟個 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> 數值同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 之間嘅關係－<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 愈大，計到出嚟嘅 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> 就愈近真嘅圓周率。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%9A%A8%E6%A9%9F%E6%95%B8%E7%94%9F%E6%88%90" title="隨機數生成">隨機數生成</a>同<a href="/wiki/%E8%92%99%E5%9C%B0%E5%8D%A1%E7%BE%85%E6%96%B9%E6%B3%95" title="蒙地卡羅方法">蒙地卡羅方法</a></div> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="隨機系統">隨機</a><a href="#cite_note-21">[e 2]</a>定<a href="/wiki/%E6%B1%BA%E5%AE%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="決定性系統">決定</a><a href="#cite_note-22">[e 3]</a>－前者指有<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>喺入面嘅電腦模擬，例如個程式其中一行碼係 <code>rand(1,10);</code>（<a href="/wiki/%E9%9A%A8%E6%A9%9F%E6%95%B8%E7%94%9F%E6%88%90" title="隨機數生成">隨機噉產生</a>一個數值响 1 同 10 之間嘅數值），將模擬緊嗰個現象想像成有隨機喺入面，而後者就指冇呢啲隨機嘅電腦模擬<a href="#cite_note-23">[16]</a>。 <a href="/wiki/%E8%92%99%E5%9C%B0%E5%8D%A1%E7%BE%85%E6%96%B9%E6%B3%95" title="蒙地卡羅方法">蒙地卡羅方法</a><a href="#cite_note-24">[e 4]</a>係一種出名嘅隨機模擬做法。蒙地卡羅方法係用隨機嘅做法應付<a href="/wiki/%E6%B1%BA%E5%AE%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="決定性系統">決定性系統</a>嘅<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>，源於 1940 年代：如果話一個<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>係「<a href="/wiki/%E6%B1%BA%E5%AE%9A%E6%80%A7" class="mw-redirect" title="決定性">決定咗</a>」嘅，意思係指個系統冇隨機喺裏面<a href="#cite_note-kroese2014-25">[17]</a>。蒙地卡羅方法最基本嘅流程如下<a href="#cite_note-kalos2008-26">[18]</a><a href="#cite_note-27">[19]</a>： <ol><li><a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a>個系統嘅<a href="/wiki/Input" class="mw-redirect" title="Input">輸入</a>嘅可能數值（<a href="/wiki/%E5%AE%9A%E7%BE%A9%E5%9F%9F" title="定義域">定義域</a><a href="#cite_note-28">[e 5]</a>）；</li> <li>用一個表示輸入數值、受定義域限制嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a><a href="#cite_note-29">[e 6]</a>隨機噉產生一個輸入；</li> <li>對個輸入做決定性嘅<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>；</li> <li>將步驟 2 同 3 重複 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 咁多次，得到 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 個<a href="/wiki/Output" class="mw-redirect" title="Output">輸出</a>，再<a href="/wiki/%E7%B5%90%E5%90%88%E6%95%B8%E6%93%9A" title="結合數據">結合數據</a><a href="#cite_note-30">[e 7]</a>－例如計嗰 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 個輸出嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均值</a>。</li></ol> 舉個簡單例子說明，蒙地卡羅方法可以用嚟估計<a href="/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率">圓周率</a>（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">）嘅數值（睇附圖）<a href="#cite_note-kalos2008-26">[18]</a>： <ol><li>畫一個<a href="/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形">正方形</a>，再喺個正方形入面畫個 90° 嘅<a href="/wiki/%E6%89%87%E5%BD%A2" title="扇形">扇形</a>，任何嘅輸入<a href="/wiki/%E5%9D%90%E6%A8%99" title="坐標">坐標</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> 都實會喺個正方形裏面（定義域）；</li> <li>用一個<a href="/wiki/%E5%9D%87%E5%8B%BB%E5%88%86%E4%BD%88" title="均勻分佈">均勻分佈</a><a href="#cite_note-31">[e 8]</a>產生一個輸入坐標數值，「均勻分佈」意思係指每個坐標可能數值出現嘅<a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a>都完全一樣；</li> <li>喺產生咗出嚟輸入坐標數值嘅位置嗰度畫一點（做運算）；</li> <li>將步驟 2 同 3 重複 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 咁多次，然後數吓有幾多點點係喺個扇形裏面（同<a href="/wiki/%E5%8E%9F%E9%BB%9E" title="原點">原點</a>距離細過 1）嘅，設呢個數值做 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">，如果 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 嘅數值大到<a href="/wiki/%E6%A5%B5%E9%99%90_(%E6%95%B8%E5%AD%B8)" title="極限 (數學)">接近</a><a href="/wiki/%E7%84%A1%E9%99%90%E5%A4%A7" class="mw-redirect" title="無限大">無限大</a>嘅話，以下呢樣嘢會成立： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m}{n}}={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}}={\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cb836d07e6eb065611f75f77033c808e553f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.143ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}}={\frac {\pi }{4}}}"></dd></dl></li></ol> 好似蒙地卡羅方法噉嘅隨機模擬帶出咗個諗頭：無論個系統係咪決定性嘅都好，用帶有隨機嘅方法都有可能俾得到有返咁上下準嘅答案－「有返咁上下準」意思係指，得出嘅答案同真實嘅答案差距唔係咁大，未至於大到能夠對解決問題嘅<a href="/wiki/%E6%95%88%E7%8E%87" title="效率">效率</a>造成明顯嘅負面影響。而事實表明咗，就算一個系統事實上冇隨機，好似蒙地卡羅方法噉嘅隨機模擬好多時都好使好用，例子可以睇吓<a href="/wiki/%E8%92%99%E5%9C%B0%E5%8D%A1%E7%BE%85%E6%A8%B9%E6%90%9C%E5%B0%8B" title="蒙地卡羅樹搜尋">蒙地卡羅樹搜尋</a>相關嘅嘢<a href="#cite_note-32">[20]</a><a href="#cite_note-33">[21]</a>。 <div class="mw-heading mw-heading3"><h3 id="方程定個體">方程定個體</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=5" title="編輯小節: 方程定個體">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="方程為本模型">方程為本模型</a>同<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本模型</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E7%89%A9%E4%BB%B6%E5%B0%8E%E5%90%91%E7%B7%A8%E7%A8%8B" title="物件導向編程">物件導向編程</a></div> 電腦模擬仲可以分做<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="方程為本模型">方程為本</a><a href="#cite_note-34">[e 9]</a>同<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本</a><a href="#cite_note-35">[e 10]</a>兩大種：方程為本模型常用於<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>同<a href="/wiki/%E5%8C%96%E5%AD%B8" title="化學">化學</a>等嘅<a href="/wiki/%E7%89%A9%E7%90%86%E7%A7%91%E5%AD%B8" title="物理科學">物理科學</a>，指成個模型靠住一大柞嘅<a href="/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式">方程式</a>－好多時會係<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>－嚟表達啲變數之間嘅啦掕，呢啲方程式當中通常會係涉及<a href="/wiki/%E6%99%82%E9%96%93" title="時間">時間</a>（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">），所以隨住時間演進（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 數值由 0 開始上升），個模型會係噉計算個系統嘅各變數响時間點每一刻數值會係幾多。例如以下呢段用 <a href="/wiki/C_%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" class="mw-redirect" title="C 程式語言">C 程式語言</a>寫嘅<a href="/wiki/%E6%BA%90%E7%A2%BC" class="mw-redirect" title="源碼">源碼</a>可以攞嚟模擬喺<a href="/wiki/%E7%89%9B%E9%A0%93%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" title="牛頓第二定律">牛頓第二定律</a>之下<a href="/wiki/%E9%83%81%E5%8B%95" class="mw-redirect" title="郁動">郁動</a>嘅<a href="/wiki/%E7%89%A9%E9%AB%94" title="物體">物體</a><a href="#cite_note-gaffer-36">[22]</a><a href="#cite_note-37">[註 5]</a>： <div style="clear: both;"></div> <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>double t = 0.0; float dt = 1.0f; float velocity = 0.0f; float position = 0.0f; float force = 10.0f; float mass = 1.0f; // 設一大柞變數，包括咗時間點（t）、時間間隔（dt）、速度（velocity）、位置（position）、件物體受嘅力（force）、同件物體嘅質量（mass）。 while ( t <= 10.0 ) // 重複噉計若干次，計到時間點係 10 為止。 { position = position + velocity * dt; velocity = velocity + ( force / mass ) * dt; // 用牛頓第二定律計吓件物體受嘅力同佢嘅質量會點影響佢嘅速度。 t += dt; } </pre></div> 相比之下，個體為本模型涉及模擬一大柞嘅個體－個體為本模型比較常用於<a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8" title="生物學">生物學</a>同<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>，因為呢啲領域研究嘅現象往往都涉及一大柞個體（<a href="/wiki/%E4%BA%BA" title="人">人</a>或者第啲<a href="/wiki/%E7%94%9F%E7%89%A9" title="生物">生物</a>）喺度個別噉做<a href="/wiki/%E6%B1%BA%E7%AD%96" title="決策">決策</a><a href="#cite_note-38">[23]</a>。基本上，行個體為本模型嘅程式做嘅嘢可以用以下呢幾個步驟總括<a href="#cite_note-39">[24]</a><a href="#cite_note-40">[25]</a>： <ol><li><a href="/wiki/%E5%88%9D%E5%A7%8B%E5%8C%96" title="初始化">初始化</a>，建構 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 個（<a href="/wiki/%E9%9B%A2%E6%95%A3%E8%AE%8A%E6%95%B8" class="mw-redirect" title="離散變數">離散</a>嘅）個體（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 通常閒閒地係幾百），每一個個體都掕住一啲數值表示佢嘅個體特性<a href="#cite_note-41">[註 6]</a>（睇<a href="/wiki/%E7%89%A9%E4%BB%B6%E5%B0%8E%E5%90%91%E7%B7%A8%E7%A8%8B" title="物件導向編程">物件導向編程</a>）；</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 設做 0；</li> <li>喺呢個時間點 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">，<a href="/wiki/Foreach" class="mw-redirect" title="Foreach">foreach</a> 個體，按呢個個體嘅行動法則同埋特性，計佢跟住落嚟嘅狀態會點變；</li> <li>如果個模型涉及一啲整體嘅變數，按啲個體嘅新狀態嚟計呢啲整體變數嘅新數值；</li> <li><code>t = t + 1</code>（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 數值上升 1）；</li> <li>一路係噉重複步驟 3 至 5，直至結束條件達到為止。</li></ol> 上面提咗嘅<a href="/wiki/%E7%BE%A4%E5%8B%95" title="群動">群動</a>就係個體為本模型嘅例子。 <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Swarming.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Swarming.jpg/480px-Swarming.jpg" decoding="async" width="480" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Swarming.jpg/720px-Swarming.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Swarming.jpg/960px-Swarming.jpg 2x" data-file-width="5134" data-file-height="3421" /></a><figcaption>一大群<a href="/wiki/%E8%9C%9C%E8%9C%82" title="蜜蜂">蜜蜂</a>；每隻蜜蜂都會按自己嘅行動法則行事，但佢哋個別行事可以<a href="/wiki/%E7%BE%A4%E9%AB%94%E6%99%BA%E8%83%BD" title="群體智能">令成個蜜蜂羣喺整體上產生一啲特別嘅規律</a>。</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="第啲分類">第啲分類</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=6" title="編輯小節: 第啲分類">編輯</a>]</div> 除咗以上嘅分類法之外，電腦模擬仲可以按以下呢啲特性嚟分： <ul><li><a href="/wiki/%E9%9B%A2%E6%95%A3%E4%BA%8B%E4%BB%B6%E6%A8%A1%E6%93%AC" title="離散事件模擬">離散事件模擬</a><a href="#cite_note-42">[e 11]</a>：指將模擬緊嗰個系統想像成喺<a href="/wiki/%E9%9B%A2%E6%95%A3%E6%99%82%E9%96%93" class="mw-redirect" title="離散時間">離散時間</a>（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},t_{2},t_{3},...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},t_{2},t_{3},...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b02f247752d8a8be2dd2921963ddc4790d70e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.498ex; height:2.343ex;" alt="{\displaystyle t_{1},t_{2},t_{3},...}">）當中存在嘅一連串事件；每件事件都會喺某個時間點發生，並且引起個系統嘅狀態 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"> 變化（某啲<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">變數</a>嘅數值變），而且呢啲模型通常都假設咗個系統喺事件之間嘅時間內唔會起乜嘢變化<a href="#cite_note-43">[註 7]</a>。<a href="/wiki/%E5%A4%A7%E8%87%AA%E7%84%B6" class="mw-redirect" title="大自然">大自然</a>當中嘅系統好少可會話「冇事件發生就企喺度唔郁」，不過離散事件模擬喺<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>同<a href="/wiki/%E5%95%86%E5%AD%B8" title="商學">商學</a>就頗為常用－呢啲領域成日都會分析一啲人造嘅系統，例如係用離散事件模擬一間<a href="/wiki/%E9%8A%80%E8%A1%8C" title="銀行">銀行</a>嘅變化，當中會影響間銀行嘅狀態嘅事件包括「客人 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"> 出現」同「客人 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"> 出現」呀噉<a href="#cite_note-44">[26]</a>。</li> <li><a href="/wiki/%E5%A4%9A%E9%87%8D%E8%A6%8F%E6%A8%A1%E6%A8%A1%E5%9E%8B" title="多重規模模型">多重規模模型</a><a href="#cite_note-45">[e 12]</a>：指個模型會同時模擬多過一個層面嘅現象；如果話一個模型有「多層」，即係例如每層都會有一柞變數，而每層嘅變數都會係由下層嗰啲變數結合而成， <ul><li><a href="/wiki/%E7%86%B1%E5%8A%9B%E5%AD%B8" title="熱力學">熱力學</a>例子：一嚿物體嘅<a href="/wiki/%E6%BA%AB%E5%BA%A6" title="溫度">溫度</a>（高層變數）取決於嚿嘢內部嘅<a href="/wiki/%E7%B2%92%E5%AD%90" title="粒子">粒子</a>嘅郁動（下層啲個體嘅變數）；</li> <li><a href="/wiki/%E9%80%B2%E5%8C%96%E8%AB%96" class="mw-redirect" title="進化論">進化論</a>例子：一個<a href="/wiki/%E7%89%A9%E7%A8%AE" title="物種">物種</a><a href="/wiki/%E7%B5%95%E7%A8%AE" title="絕種">絕種</a>嘅<a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a>（高層變數）取決於個物種啲個體嘅<a href="/wiki/%E7%B9%81%E6%AE%96" class="mw-redirect" title="繁殖">繁殖</a>能力（下層啲個體嘅變數）；</li> <li><a href="/wiki/%E7%B6%93%E6%BF%9F%E5%AD%B8" title="經濟學">經濟學</a>例子：一間<a href="/wiki/%E5%85%AC%E5%8F%B8" title="公司">公司</a>整體生產咗幾多件<a href="/wiki/%E7%94%A2%E5%93%81" title="產品">產品</a>（高層變數）取決於個別員工嘅<a href="/wiki/%E5%B7%A5%E4%BD%9C%E8%A1%A8%E7%8F%BE" title="工作表現">工作表現</a>（下層啲個體嘅變數）。</li> <li>一個多重規模模型會同時考慮多過一個層面嘅現象，而且會用一個層面嗰啲變數出嘅數值（例：嚿物體啲粒子嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均</a><a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>、個物種啲個體平均每個生咗幾多<a href="/wiki/%E5%BE%8C%E4%BB%A3" title="後代">後代</a>、間公司嘅員工平均每人生產咗幾多件產品...）嚟做另外一啲層面嘅模擬嘅<a href="/wiki/%E5%8F%83%E6%95%B8" title="參數">參數</a><a href="#cite_note-46">[27]</a>。</li></ul></li></ul> ... 等等。 <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="各領域嘅應用">各領域嘅應用</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=7" title="編輯小節: 各領域嘅應用">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a></div> <div class="mw-heading mw-heading3"><h3 id="物理學">物理學</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=8" title="編輯小節: 物理學">編輯</a>]</div> <figure typeof="mw:File/Thumb"><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/PhysicsEngine.ogv/300px--PhysicsEngine.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="300" height="225" data-durationhint="16" data-mwtitle="PhysicsEngine.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:PhysicsEngine.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/1/1c/PhysicsEngine.ogv" type="video/ogg; codecs="theora"" data-width="400" data-height="300" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/1c/PhysicsEngine.ogv/PhysicsEngine.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="400" data-height="300" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/1c/PhysicsEngine.ogv/PhysicsEngine.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/1c/PhysicsEngine.ogv/PhysicsEngine.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /></video><figcaption>四個<a href="/wiki/%E9%81%8A%E6%88%B2%E7%89%A9%E7%90%86" title="遊戲物理">遊戲物理</a>嘅例子： 1 冇任何物理法則。 2 有<a href="/wiki/%E9%87%8D%E5%8A%9B" title="重力">重力</a>，但冇<a href="/wiki/%E7%A2%B0%E6%92%9E%E6%8E%A2%E6%B8%AC" title="碰撞探測">碰撞探測</a>。 3 有重力同碰撞探測，但冇<a href="/wiki/%E5%89%9B%E9%AB%94%E5%8B%95%E5%8A%9B%E5%AD%B8" title="剛體動力學">剛體動力學</a>。 4 基本嘢冚唪唥齊嗮，最能夠創造迫真嘅<a href="/wiki/%E9%83%81%E5%8B%95" class="mw-redirect" title="郁動">郁動</a>。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E7%89%A9%E7%90%86%E5%BC%95%E6%93%8E" title="物理引擎">物理引擎</a>同<a href="/wiki/%E9%81%8B%E7%AE%97%E7%89%A9%E7%90%86%E5%AD%B8" title="運算物理學">運算物理學</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E9%81%8B%E7%AE%97%E5%8C%96%E5%AD%B8" title="運算化學">運算化學</a></div> <a href="/wiki/%E9%81%8B%E7%AE%97%E7%89%A9%E7%90%86%E5%AD%B8" title="運算物理學">運算物理學</a>涉及用<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>方法（包括電腦模擬）解決<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>上撞到嘅問題：物理學通常都會用一大柞<a href="/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式">方程式</a>（尤其係<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>）表達某啲變數之間嘅關係，用呢啲變數組成嘅<a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">模型</a>嚟描述研究緊嘅現象。做物理學相關工作嘅人成日都會用電腦模擬嚟處理一啲<a href="/wiki/%E8%A7%A3%E6%9E%90%E8%A7%A3" title="解析解">解析解</a>（簡單講即係齋靠<a href="/wiki/%E4%BB%A3%E6%95%B8" class="mw-disambig" title="代數">操弄</a>方程式）搞唔掂嘅問題－呢啲問題往往係要做「想像 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}">，個答案係噉噉噉，想像 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}">，個答案係噉噉噉...」噉嘅<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>方法嚟解（睇埋<a href="/wiki/%E6%95%B8%E5%80%BC%E5%88%86%E6%9E%90" title="數值分析">數值分析</a>），而當個冇得用解析解嚟解嘅問題涉及<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>嗰陣，變數嘅數量多得好交關，難以靠人手計嚟解決個問題，於是研究者就會用電腦模擬嘅方法，叫部電腦（例如）「計 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"> 嗰陣嘅答案，計 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}"> 嗰陣嘅答案... 再將呢啲答案畫一條線出嚟睇」<a href="#cite_note-47">[註 8]</a><a href="#cite_note-48">[28]</a>。 除此之外，用電腦嚟模擬<a href="/wiki/%E7%89%A9%E7%90%86" class="mw-redirect" title="物理">物理</a>現象仲可以幫到手製作更加迫真嘅<a href="/wiki/%E8%99%9B%E6%93%AC%E4%B8%96%E7%95%8C" title="虛擬世界">虛擬世界</a>－形成咗所謂嘅<a href="/wiki/%E7%89%A9%E7%90%86%E5%BC%95%E6%93%8E" title="物理引擎">物理引擎</a><a href="#cite_note-49">[e 13]</a>技術，呢種技術對<a href="/wiki/%E9%9B%BB%E5%AD%90%E9%81%8A%E6%88%B2" title="電子遊戲">電子遊戲</a>同<a href="/wiki/%E8%99%9B%E6%93%AC%E5%AF%A6%E5%A2%83" title="虛擬實境">虛擬實境</a>等嘅產品嚟講不可或缺<a href="#cite_note-50">[29]</a>。例如類似以下呢段 <a href="/wiki/C_%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" class="mw-redirect" title="C 程式語言">C</a> 碼噉嘅碼可以攞嚟模擬喺<a href="/wiki/%E7%89%9B%E9%A0%93%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="牛頓力學">牛頓力學</a><a href="#cite_note-51">[e 14]</a><a href="/wiki/%E4%BA%8C%E7%B6%AD%E7%A9%BA%E9%96%93" class="mw-redirect" title="二維空間">二維空間</a>下<a href="/wiki/%E9%83%81%E5%8B%95" class="mw-redirect" title="郁動">郁動</a>嘅<a href="/wiki/%E7%B2%92%E5%AD%90" title="粒子">粒子</a><a href="#cite_note-52">[註 9]</a><a href="#cite_note-53">[30]</a>，可以<a href="/wiki/%E9%81%8A%E6%88%B2%E7%89%A9%E7%90%86" title="遊戲物理">喺電子遊戲入面用嚟模擬物體嘅郁動</a>： <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>... while (currentTime < totalSimulationTime) { // 當 currentTime < totalSimulationTime，一路做... for (int i = 0; i < NUM_PARTICLES; ++i) { // 家陣模擬若干粒粒子，foreach 粒子，做... Particle *particle = &particles[i]; Vector2 force = ComputeForce(particle); // 有個子程序計吓呢粒粒子受緊幾多力。 Vector2 acceleration = (Vector2){force.x / particle->mass, force.y / particle->mass}; // 上面呢行碼係用牛頓第二定律計粒粒子跟住嘅加速度。 particle->velocity.x += acceleration.x * dt; // 用加速度 update X 軸速度... particle->velocity.y += acceleration.y * dt; // 用加速度 update Y 軸速度... particle->position.x += particle->velocity.x * dt; // 用速度 update X 軸位置... particle->position.y += particle->velocity.y * dt; // 用速度 update Y 軸位置... } PrintParticles(); // 按啲粒子嘅位置，更新個畫面嘅樣。 currentTime += dt; // currentTime 上升 1。 } ... </pre></div> <div class="mw-heading mw-heading3"><h3 id="生物學">生物學</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=9" title="編輯小節: 生物學">編輯</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wolf_with_Caribou_Hindquarter.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Wolf_with_Caribou_Hindquarter.jpg/240px-Wolf_with_Caribou_Hindquarter.jpg" decoding="async" width="240" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Wolf_with_Caribou_Hindquarter.jpg/360px-Wolf_with_Caribou_Hindquarter.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Wolf_with_Caribou_Hindquarter.jpg/480px-Wolf_with_Caribou_Hindquarter.jpg 2x" data-file-width="2400" data-file-height="3000" /></a><figcaption>喺<a href="/wiki/%E9%98%BF%E6%8B%89%E6%96%AF%E5%8A%A0" class="mw-redirect" title="阿拉斯加">阿拉斯加</a>影到嘅一隻<a href="/wiki/%E7%8B%BC" title="狼">狼</a>；佢擔住一隻<a href="/wiki/%E9%A6%B4%E9%B9%BF" title="馴鹿">馴鹿</a>嘅<a href="/wiki/%E9%AB%80" title="髀">髀</a>。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%81%8B%E7%AE%97%E7%94%9F%E7%89%A9%E5%AD%B8" title="運算生物學">運算生物學</a></div> <a href="/wiki/%E9%81%8B%E7%AE%97%E7%94%9F%E7%89%A9%E5%AD%B8" title="運算生物學">運算生物學</a>涉及用運算方法（包括電腦模擬）解決<a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8" title="生物學">生物學</a>上撞到嘅問題：同（例如）運算物理學比起嚟，運算生物學比較傾向用<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本模型</a>，例如係<a href="/wiki/%E9%80%B2%E5%8C%96%E8%AB%96" class="mw-redirect" title="進化論">進化論</a>同<a href="/wiki/%E7%94%9F%E6%85%8B%E5%AD%B8" title="生態學">生態學</a>等領域上嘅研究都成日要分析一大柞個別噉做<a href="/wiki/%E6%B1%BA%E7%AD%96" title="決策">決策</a>嘅個體（<a href="/wiki/%E5%8B%95%E7%89%A9" title="動物">動物</a>或者<a href="/wiki/%E6%A4%8D%E7%89%A9" title="植物">植物</a>個體呀噉）之間嘅互動，以及係呢啲互動出嘅結果，而由於<a href="/wiki/%E5%9F%BA%E5%9B%A0%E7%AA%81%E8%AE%8A" title="基因突變">基因突變</a>等嘅現象，<a href="/wiki/%E7%94%9F%E5%91%BD" title="生命">生命</a>呢家嘢本質上就實會有可能存在個體差異－因為噉，個體為本模型就啱嗮，例子可以睇吓對<a href="/wiki/%E7%89%A9%E7%AB%B6%E5%A4%A9%E6%93%87" title="物競天擇">物競天擇</a>等嘅現象嘅模擬<a href="#cite_note-54">[31]</a><a href="#cite_note-55">[32]</a>。 雖然話生命現象傾向似個體為本模型，但運算生物學當中都有用到<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="方程為本模型">方程為本模型</a>，例如<a href="/wiki/%E6%B4%9B%E5%8D%A1-%E4%BC%8F%E5%A1%94%E6%8B%89%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="洛卡-伏塔拉模型">洛卡-伏塔拉模型</a><a href="#cite_note-56">[e 15]</a>就係一個好出名嘅生物學方程為本模型，初頭呢個模型係俾人攞嚟模擬<a href="/wiki/%E6%8D%95%E9%A3%9F%E8%80%85" class="mw-redirect" title="捕食者">捕食者</a>同<a href="/wiki/%E7%8D%B5%E7%89%A9" class="mw-redirect" title="獵物">獵物</a>响<a href="/wiki/%E7%94%9F%E6%85%8B%E7%B3%BB%E7%B5%B1" title="生態系統">生態系統</a>當中嘅互動，打後呢個模型仲有俾人攞嚟模擬第啲生命現象。呢個模型涉及咗以下呢兩條<a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7" class="mw-redirect" title="非線性">非線性</a><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a><a href="#cite_note-brauer2012-57">[33]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dt}}=\alpha x-\beta xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>α</mi> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}=\alpha x-\beta xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088b7149be1adcc106e37463be680f2bfda5beae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.955ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}=\alpha x-\beta xy}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dt}}=\delta xy-\gamma y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>δ</mi> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>γ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dt}}=\delta xy-\gamma y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca9b4df2f4c0324a13613c05ce9c0771f34f792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.098ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dt}}=\delta xy-\gamma y}"></dd></dl> 當中 <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> 係<a href="/wiki/%E7%8D%B5%E7%89%A9" class="mw-redirect" title="獵物">獵物</a>（例：<a href="/wiki/%E5%85%94%E4%BB%94" class="mw-redirect" title="兔仔">兔仔</a>）嘅個體數量；</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 係<a href="/wiki/%E6%8D%95%E9%A3%9F%E8%80%85" class="mw-redirect" title="捕食者">捕食者</a>（例：<a href="/wiki/%E7%8B%BC" title="狼">狼</a>）嘅個體數量；</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 表示<a href="/wiki/%E6%99%82%E9%96%93" title="時間">時間</a>；</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd8f4577337ea1e3aa08b83b86f156fc04aff2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}}"> 表示獵物個體數量隨時間嘅<a href="/wiki/%E5%B0%8E%E6%95%B8" title="導數">導數</a>，即係獵物個體數量嘅<a href="/wiki/%E7%9E%AC%E9%96%93%E8%AE%8A%E7%8E%87" class="mw-redirect" title="瞬間變率">瞬間變率</a>（詳情可以睇<a href="/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分">微積分</a>）； <dl><dd>－呢個數值取決於 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">（獵物嘅數量會影響跟住落嚟有幾多隻新嘅獵物出世）同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}">（獵物同捕食者嘅數量會共同影響跟住落嚟有幾多隻獵物死亡）。</dd></dl></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21987ac7e66811e0ae301c442a2eeb258d54818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.207ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dt}}}"> 表示捕食者個體數量隨時間嘅導數； <dl><dd>－呢個數值取決於 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">（捕食者嘅數量會影響跟住落嚟有幾多隻新嘅捕食者出世）同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}">（獵物同捕食者嘅數量會共同影響跟住落嚟有幾多隻捕食者因為捉唔到獵物而餓死）。</dd></dl></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> 係一柞<a href="/wiki/%E5%AF%A6%E6%95%B8" title="實數">實數</a><a href="/wiki/%E5%8F%83%E6%95%B8" title="參數">參數</a>，反映獵物同捕食者「平均每隻生產幾多隻<a href="/wiki/%E5%BE%8C%E4%BB%A3" title="後代">後代</a>」等嘅因素；</li></ul> 如果將 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> 同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 畫做圖（打橫嗰條軸係<a href="/wiki/%E6%99%82%E9%96%93" title="時間">時間</a><a href="#cite_note-58">[註 10]</a>），會出好似以下呢兩條噉嘅線（藍線表示捕食者；紅線表示獵物）： <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Lotka_Volterra_equation_Maple_plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Lotka_Volterra_equation_Maple_plot.png/840px-Lotka_Volterra_equation_Maple_plot.png" decoding="async" width="840" height="294" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d7/Lotka_Volterra_equation_Maple_plot.png 1.5x" data-file-width="1142" data-file-height="400" /></a><figcaption></figcaption></figure> 跟住研究者仲有得調整吓啲參數，睇睇（例如）捕食者出生率突然勁跌會點樣影響個生態系統嘅情況<a href="#cite_note-brauer2012-57">[33]</a>。 <div class="mw-heading mw-heading3"><h3 id="認知科學">認知科學</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=10" title="編輯小節: 認知科學">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a>、<a href="/wiki/%E9%81%8B%E7%AE%97%E8%AA%8D%E7%9F%A5" title="運算認知">運算認知</a>同<a href="/wiki/%E9%81%8B%E7%AE%97%E7%A5%9E%E7%B6%93%E7%A7%91%E5%AD%B8" title="運算神經科學">運算神經科學</a></div> <a href="/wiki/%E8%AA%8D%E7%9F%A5%E7%A7%91%E5%AD%B8" title="認知科學">認知科學</a>上都有用到電腦模擬－電腦模擬可以攞嚟研究<a href="/wiki/%E8%AA%8D%E7%9F%A5" title="認知">認知</a>（<a href="/wiki/%E9%81%8B%E7%AE%97%E8%AA%8D%E7%9F%A5" title="運算認知">運算認知</a>）同<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B3%BB%E7%B5%B1" title="神經系統">神經系統</a>（<a href="/wiki/%E9%81%8B%E7%AE%97%E7%A5%9E%E7%B6%93%E7%A7%91%E5%AD%B8" title="運算神經科學">運算神經科學</a>），而<a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a>（AI）本質上就係嘗試緊用電腦方法嚟模擬<a href="/wiki/%E6%99%BA%E8%83%BD" title="智能">智能</a>，務求教電腦展現出好似有智能噉嘅行為<a href="#cite_note-59">[34]</a><a href="#cite_note-60">[35]</a>。例如<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="人工神經網絡">人工神經網絡</a><a href="#cite_note-61">[e 16]</a>就係一種好出名嘅<a href="/wiki/%E8%AA%8D%E7%9F%A5%E6%A8%A1%E5%9E%8B" title="認知模型">認知科學模型</a><a href="#cite_note-62">[36]</a>。人工神經網絡個諗頭係要用人工方法模仿動物神經細胞嘅功能：一粒<a href="/wiki/%E5%8B%95%E7%89%A9%E7%A5%9E%E7%B6%93%E7%B4%B0%E8%83%9E" class="mw-redirect" title="動物神經細胞">動物神經細胞</a>喺受到某啲特定刺激嗰陣，會用電或者化學方式向第啲神經細胞傳出<a href="/wiki/%E7%A5%9E%E7%B6%93%E8%A8%8A%E8%99%9F" class="mw-redirect" title="神經訊號">訊號</a>，然後同呢粒神經細胞相連嗰啲神經細胞會受到佢嘅訊號刺激，跟住又會各自噉射新嘅訊號－於是資訊就會喺成個<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B3%BB%E7%B5%B1" title="神經系統">神經系統</a>當中傳播開去<a href="#cite_note-63">[37]</a>；一個人工神經網絡由大量嘅<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B4%B0%E8%83%9E" title="人工神經細胞">人工神經細胞</a>組成，喺用電腦程式整神經網絡嗰陣，研究者會每粒人工神經細胞都同佢設個數值代表佢嘅啟動程度<a href="#cite_note-64">[e 17]</a><a href="#cite_note-65">[38]</a>，而每粒神經細胞嘅啟動程度嘅數值都有條式計，呢條式包括咗喺佢之前嗰啲神經細胞嘅啟動程度<a href="#cite_note-66">[39]</a>。即係例如個程式 <a href="/wiki/Foreach" class="mw-redirect" title="Foreach">foreach</a> 細胞有一條類似噉樣嘅算式： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=W_{1}A_{1}+W_{2}A_{2}...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=W_{1}A_{1}+W_{2}A_{2}...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb6665bdd84050b2886bfc2aee2210a7ed49b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.487ex; height:2.509ex;" alt="{\displaystyle A=W_{1}A_{1}+W_{2}A_{2}...}">；（<a href="/wiki/%E5%95%9F%E5%8B%95%E5%87%BD%E6%95%B8" title="啟動函數">啟動函數</a>）</dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> 代表粒神經細胞嘅啟動程度，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"> 代表前嗰排嘅神經細胞當中第 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 粒嘅啟動程度，而 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa0340872ef1d6511eaf27ed7c57f98589a693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.412ex; height:2.509ex;" alt="{\displaystyle W_{n}}"> 就係前嗰排嘅神經細胞當中第 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 粒嘅權重（指嗰粒細胞有幾影響到 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">）。所以當一粒人工神經細胞啟動嗰陣，會帶起佢後面啲細胞跟住佢啟動－似十足生物神經網絡入面嗰啲細胞噉。假如個神經網絡嘅程式令佢能夠自行按照經驗改變 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa0340872ef1d6511eaf27ed7c57f98589a693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.412ex; height:2.509ex;" alt="{\displaystyle W_{n}}"> 嘅數值嘅話，佢就會曉<a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">學習</a><a href="#cite_note-67">[40]</a>。 <div style="clear: both;"></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Artificial_neural_network.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Artificial_neural_network.svg/langyue-400px-Artificial_neural_network.svg.png" decoding="async" width="400" height="357" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Artificial_neural_network.svg/langyue-600px-Artificial_neural_network.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Artificial_neural_network.svg/langyue-800px-Artificial_neural_network.svg.png 2x" data-file-width="560" data-file-height="500" /></a><figcaption>一個<a href="/wiki/%E5%89%8D%E9%A5%8B%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="前饋神經網絡">前饋人工神經網絡</a>嘅抽象圖；每個圓圈代表咗一粒模擬嘅神經細胞，而每粒神經細胞嘅啟動程度由佢打前嗰排神經細胞嘅啟動程度話事<a href="#cite_note-68">[41]</a>。</figcaption></figure> 到咗廿一世紀初，人工神經網絡經已俾科學界廣泛噉用落去各種各樣嘅問題嗰度。喺應用上，事實證實咗人工神經網絡能夠處理<a href="/wiki/%E6%A9%9F%E6%A2%B0%E8%A6%96%E8%A6%BA" class="mw-redirect" title="機械視覺">機械視覺</a>同埋<a href="/wiki/%E8%AA%9E%E9%9F%B3%E8%AD%98%E5%88%A5" title="語音識別">語音識別</a>呢啲傳統嘅電腦程式好難解決到嘅問題<a href="#cite_note-69">[42]</a><a href="#cite_note-70">[43]</a>；而喺理論性質嘅<a href="/wiki/%E8%AA%8D%E7%9F%A5%E7%A7%91%E5%AD%B8" title="認知科學">認知科學</a>上亦有<a href="/wiki/%E8%81%AF%E7%B5%90%E4%B8%BB%E7%BE%A9" title="聯結主義">聯結主義</a><a href="#cite_note-71">[e 18]</a>理論，主張<a href="/wiki/%E5%BF%83%E9%9D%88" title="心靈">心靈</a>可以用人工神經網絡嚟模擬<a href="#cite_note-72">[44]</a>，例如有研究就成功噉用人工神經網絡令人工智能好似人噉識得睇一幅圖決定要將<a href="/wiki/%E6%B3%A8%E6%84%8F%E5%8A%9B" title="注意力">注意力</a>擺喺幅圖嘅邊一忽<a href="#cite_note-73">[45]</a>。 <div class="mw-heading mw-heading3"><h3 id="社會科學">社會科學</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=11" title="編輯小節: 社會科學">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%81%8B%E7%AE%97%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="運算社會科學">運算社會科學</a>、<a href="/wiki/%E9%81%8B%E7%AE%97%E7%B6%93%E6%BF%9F%E5%AD%B8" title="運算經濟學">運算經濟學</a>同<a href="/wiki/%E9%81%8B%E7%AE%97%E7%A4%BE%E6%9C%83%E5%AD%B8" title="運算社會學">運算社會學</a></div> <a href="/wiki/%E9%81%8B%E7%AE%97%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="運算社會科學">運算社會科學</a>係指用運算方法研究<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>，例如用電腦模擬解析<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%8F%BE%E8%B1%A1" title="社會現象">社會現象</a>。有學者指，電腦模擬呢種技術助長廿世紀起<a href="/wiki/%E7%B6%93%E6%BF%9F%E5%AD%B8" title="經濟學">經濟學</a>等嘅領域放棄唔真實嘅假設：廿世紀前嘅<a href="/wiki/%E5%8F%A4%E5%85%B8%E7%B6%93%E6%BF%9F%E5%AD%B8" title="古典經濟學">古典經濟學</a>成日會做一啲唔真實嘅假設，包括假設咗啲<a href="/wiki/%E7%B6%93%E6%BF%9F%E5%80%8B%E9%AB%94" title="經濟個體">經濟個體</a>喺做<a href="/wiki/%E7%B6%93%E6%BF%9F" title="經濟">經濟</a><a href="/wiki/%E6%B1%BA%E7%AD%96" title="決策">決策</a>嗰陣係<a href="/wiki/%E5%AE%8C%E5%85%A8%E8%B3%87%E8%A8%8A" title="完全資訊">乜都知嘅</a>（例如假設每個買家知道嗮<a href="/wiki/%E5%B8%82%E5%A0%B4" title="市場">市場</a>裏面啲<a href="/wiki/%E8%B2%A8" title="貨">貨</a>嘅<a href="/wiki/%E5%83%B9%E6%A0%BC" title="價格">價格</a>），呢個假設好明顯唔合符現實<a href="#cite_note-74">[46]</a>，但當時嗰啲研究者受到運算能力嘅局限－要計「如果每個個體响<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>上都有差異，個<a href="/wiki/%E7%B6%93%E6%BF%9F%E6%A8%A1%E5%9E%8B" title="經濟模型">模型</a>會得出乜結果」需要做嘅運算量大得滯，用人手根本做唔嚟（例：想像個模型模擬 1,000 個個體，佢哋個個<a href="/wiki/%E6%80%A7%E6%A0%BC" title="性格">性格</a>都唔同，研究者要人手噉<a href="/wiki/Foreach" class="mw-redirect" title="Foreach">逐個逐個</a>計佢哋嘅決策結果）；呢點搞到啲學者明知個假設唔真實，都焗住要照用。廿世紀電腦模擬技術有咗好似<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本模型</a>噉嘅架生，令到<a href="/wiki/%E7%B6%93%E6%BF%9F%E5%AD%B8%E5%AE%B6" title="經濟學家">經濟學家</a>喺建立佢哋嘅模型嗰時可以考慮埋個體差異，助長咗經濟學上嘅革新<a href="#cite_note-75">[47]</a>。 <a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a><a href="#cite_note-76">[e 19]</a>上嘅研究成日都會用到個體為本模型，當中<a href="/wiki/%E9%87%8D%E8%A4%87%E7%9B%A3%E7%8A%AF%E5%9B%B0%E5%A2%83" class="mw-redirect" title="重複監犯困境">重複監犯困境</a><a href="#cite_note-77">[e 20]</a>就係一個出名嘅例子：<a href="/wiki/%E7%9B%A3%E7%8A%AF%E5%9B%B0%E5%A2%83" title="監犯困境">監犯困境</a>係一種<a href="/wiki/%E5%8D%9A%E5%BC%88" class="mw-redirect" title="博弈">博弈</a>；喺一場監犯困境當中，每位博弈者有兩個選擇－合作同背叛；而一場兩人監犯困境嘅<a href="/wiki/%E5%A0%B1%E5%84%9F%E7%9F%A9%E9%99%A3" class="mw-redirect" title="報償矩陣">報償矩陣</a>如下，每個格嗰兩個數值表示兩個人嘅分別所得<a href="#cite_note-poundstone1993-78">[48]</a>： <div style="clear: both;"></div> <table border="2" cellpadding="4" cellspacing="0" style="background: #f9f9f9;border: 1px #aaa solid; border-collapse: collapse;"> <caption> </caption> <tbody><tr> <td> </td> <th scope="col" style="color: #900">紅色揀合作 </th> <th scope="col" style="color: #900">紅色揀背叛 </th></tr> <tr> <th scope="row" style="color: #009">藍色揀合作 </th> <td><center>R, R</center> </td> <td><center>S, T</center> </td></tr> <tr> <th scope="row" style="color: #009">藍色揀背叛 </th> <td><center>T, S</center> </td> <td><center>P, P</center> </td></tr></tbody></table> <a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a>上，喺監犯困境當中，以下呢條<a href="/wiki/%E4%B8%8D%E7%AD%89%E5%BC%8F" title="不等式">不等式</a>成立（假設報償值係愈<a href="/wiki/%E6%AD%A3%E6%95%B8" title="正數">正</a>愈理想）： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>R>P>S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>R</mi> <mo>></mo> <mi>P</mi> <mo>></mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>R>P>S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6fac90a83474d2b977ae4ff2075c146534cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.94ex; height:2.176ex;" alt="{\displaystyle T>R>P>S}"></dd></dl> 例：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0>-0.5>-5>-10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>></mo> <mo>−</mo> <mn>0.5</mn> <mo>></mo> <mo>−</mo> <mn>5</mn> <mo>></mo> <mo>−</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0>-0.5>-5>-10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104b1e5ca0a11a217f9abeea19946cba57510414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.341ex; height:2.343ex;" alt="{\displaystyle 0>-0.5>-5>-10}">；簡單講，監犯困境係指大家合作就可以得到 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> 咁多利益，一位博弈者單方面背叛可以攞到著數（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1fa60c9da906b73422a9f79364c30dda045601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.499ex; height:2.176ex;" alt="{\displaystyle T>R}">），但如果雙方都背叛就大家齊齊輸（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R>P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37afbcb755b88dab03bced457fbdbcd90d55a904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle R>P}">）<a href="#cite_note-poundstone1993-78">[48]</a>。重複監犯困境係指啲博弈者進入監犯困境嘅情境若干次<a href="#cite_note-79">[49]</a><a href="#cite_note-80">[50]</a>。 有研究者進行電腦模擬研究，行個體為本模型模擬重複監犯困境，喺最簡單嘅情況下即係<a href="#cite_note-81">[51]</a>： <ol><li><a href="/wiki/%E5%88%9D%E5%A7%8B%E5%8C%96" title="初始化">初始化</a>，將 <code>R</code>、<code>T</code>、<code>S</code> 同 <code>P</code> 設定做特定嘅數值；</li> <li>建立 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> 個模擬嘅個體，每個個體有兩個屬性－「採取嘅策略」 <code>strat</code> 同埋「累積咗嘅總得益」 <code>jik</code>；</li> <li>是但抽兩個個體，俾佢哋進行監犯困境博弈，每位模擬嘅博弈者會按佢嘅 <code>strat</code> 做<a href="/wiki/%E6%B1%BA%E7%AD%96" title="決策">決策</a>；</li> <li>按每個個體做嘅決定，計佢嘅總得益點變，例如兩個都揀咗合作嘅話，噉佢哋每個人都 <code>jik = jik + R</code>；</li> <li>重複步驟 3 同 4 若干次；</li></ol> 原則上，「最成功嘅策略」係指令到個體嘅總得益最大化嘅策略，所以研究者要做嘅嘢係喺步驟 3 同 4 重複咗若干次之後，<a href="/wiki/Foreach" class="mw-redirect" title="Foreach">foreach</a> 策略類型，（例如）計吓採取嗰種策略嘅個體嘅<a href="/wiki/%E5%B9%B3%E5%9D%87%E5%80%BC" title="平均值">平均</a> <code>jik</code>，而呢個數值最高嗰種策略就係「最成功嘅策略」。結果發現，「永遠淨係揀合作」係一種非常惡劣嘅策略，會搞到個個體係噉勁俾人背叛而最後輸（假設 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320976b4dd090d2df7a9e21c1ed07d63c6182cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.234ex; height:2.176ex;" alt="{\displaystyle T>S}">，而且 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T-S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>−</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T-S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c0bf1ad248aa6e574a4b733e8cb03479cff4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.976ex; height:2.343ex;" alt="{\displaystyle T-S}"> 數值有咁上下大）。啲研究者就靠呢個結果作出推論：重複監犯困境就好似人嘅現實生活噉－人係一種有<a href="/wiki/%E7%A4%BE%E6%9C%83%E6%80%A7" title="社會性">社會性</a>嘅動物，往往會一齊合作先可以做到最多嘢（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R>P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37afbcb755b88dab03bced457fbdbcd90d55a904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle R>P}">），但喺合作嘅情境當中，成日都會出現機會容許某啲參與者搵第啲參與者老襯（背叛係有可能發生嘅，而且 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>R>S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>R</mi> <mo>></mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>R>S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d00e8156d90998ca1fb135fb22edd3e8e5309d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.096ex; height:2.176ex;" alt="{\displaystyle T>R>S}">），而世界唔係<a href="/wiki/%E7%83%8F%E6%89%98%E9%82%A6" title="烏托邦">烏托邦</a>，完全<a href="/wiki/%E5%88%A9%E4%BB%96" class="mw-redirect" title="利他">利他</a>嘅人係缺乏靠自己生存嘅能力嘅－人實要識得喺俾人背叛嗰陣，用返背叛嚟應對，以防俾人搵老襯<a href="#cite_note-82">[52]</a>。 <div class="mw-heading mw-heading3"><h3 id="工程學">工程學</h3>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=12" title="編輯小節: 工程學">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%81%8B%E7%AE%97%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="運算工程學">運算工程學</a>同<a href="/wiki/%E6%9C%89%E9%99%90%E5%85%83%E5%88%86%E6%9E%90" title="有限元分析">有限元分析</a></div> <a href="/wiki/%E9%81%8B%E7%AE%97%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="運算工程學">運算工程學</a>涉及用電腦模擬等嘅運算方法分析<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>上應付嘅人造<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>（例如<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>同<a href="/wiki/%E5%BB%BA%E7%AF%89%E7%89%A9" class="mw-redirect" title="建築物">建築物</a>）。喺運算工程學上，電腦模擬通常會以<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>知識做基礎<a href="#cite_note-83">[註 11]</a>，即係「分析緊個系統，個系統嘅原理涉及咗呢柞呢柞<a href="/wiki/%E7%89%A9%E7%90%86%E5%AE%9A%E5%BE%8B" title="物理定律">物理定律</a>，將個系統涉及嗰啲物理定律編寫做程式，再行嚟睇，就做到模擬個系統行起上嚟會係點」<a href="#cite_note-84">[53]</a>。 例如廿一世紀初嘅<a href="/wiki/%E7%B5%90%E6%A7%8B%E5%88%86%E6%9E%90" title="結構分析">結構分析</a><a href="#cite_note-85">[e 21]</a>就時不時都會用到電腦模擬：結構分析常用於<a href="/wiki/%E5%9C%9F%E6%9C%A8%E5%B7%A5%E7%A8%8B" title="土木工程">土木工程</a>－包括咗<a href="/wiki/%E7%B5%90%E6%A7%8B%E5%B7%A5%E7%A8%8B" title="結構工程">結構工程</a>同<a href="/wiki/%E5%BB%BA%E7%AF%89%E5%B7%A5%E7%A8%8B" title="建築工程">建築工程</a>－指個<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%B8%AB" title="工程師">工程師</a>分析一個<a href="/wiki/%E7%B5%90%E6%A7%8B" title="結構">結構</a>同埋嗰個結構嘅各橛分別受到點樣嘅<a href="/wiki/%E7%B5%90%E6%A7%8B%E8%B2%A0%E8%8D%B7" title="結構負荷">負荷</a>，以及係分析呢啲負荷會對個結構造成乜嘢影響（例如係會唔會砸冧個結構），對於建設安全嘅結構嚟講不可或缺<a href="#cite_note-mcKenzie2013-86">[54]</a>，例如想像以下呢個噉嘅簡單結構，個結構係個<a href="/wiki/%E7%AD%89%E9%82%8A%E4%B8%89%E8%A7%92%E5%BD%A2" title="等邊三角形">等邊三角形</a>，個工程師打算用呢個結構嚟撐住啲嘢，個結構要承受一股由上面壓落嚟嘅<a href="/wiki/%E5%8A%9B" title="力">力</a>，股力嘅大細係 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10{\text{ N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> N</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10{\text{ N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20846e05bb1164bdf2b2e005715501f481cbb0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.649ex; height:2.176ex;" alt="{\displaystyle 10{\text{ N}}}"> 咁多： <dl><dd><figure class="mw-halign-center mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Full_Figure2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/Truss_Structure_Analysis%2C_Full_Figure2.jpg" decoding="async" width="690" height="487" class="mw-file-element" data-file-width="652" data-file-height="460" /></a><figcaption></figcaption></figure></dd></dl> 由<a href="/wiki/%E5%8A%9B%E5%AD%B8" title="力學">力學</a>上嘅知識（作為分析基礎嘅物理學知識）已知，呢股力會造成一股<a href="/wiki/%E5%8F%8D%E4%BD%9C%E7%94%A8%E5%8A%9B" title="反作用力">反作用力</a>。而假設組成個結構嗰啲<a href="/wiki/%E6%A8%91" title="樑">樑</a>夠嗮硬淨唔會變形（<a href="/wiki/%E5%89%9B%E9%AB%94" title="剛體">剛體</a>），而且啲<a href="/wiki/%E7%B5%90%E6%A7%8B%E7%AF%80%E9%BB%9E" title="結構節點">節點</a>又夠勁（啲樑唔會走位），噉啲反作用力（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{R}}_{A_{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{R}}_{A_{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f043b1781911e69ce42a239207bdc2e2ae84c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.103ex; height:2.676ex;" alt="{\displaystyle {\text{R}}_{A_{x}}}">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{R}}_{A_{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{R}}_{A_{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f61ca0375febe2ea9e2685859cee9c0c02066c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.003ex; height:3.009ex;" alt="{\displaystyle {\text{R}}_{A_{y}}}">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{R}}_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{R}}_{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf055cc68cdb94a4bca0b2e21ee13e7f56103735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.19ex; height:2.509ex;" alt="{\displaystyle {\text{R}}_{B}}">）會係噉嘅樣（想像啲節點有特殊構造）： <dl><dd><figure class="mw-halign-center mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_FBD2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/12/Truss_Structure_Analysis%2C_FBD2.jpg" decoding="async" width="480" height="462" class="mw-file-element" data-file-width="434" data-file-height="418" /></a><figcaption></figcaption></figure></dd></dl> 而按物理學上對力嘅知識，如果成個結構企到喺度唔郁（個結構會郁就表示可能會冧），噉啲力加埋嗮係 0，即係話喺 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"> 嗰點嘅力可以噉樣想像（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170e2b3a1ee797adf59f0a732b32eb0028e8437c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.544ex; height:2.843ex;" alt="{\displaystyle F_{y}}"> 係沿打戙嗰條軸嘅力，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0933c4941964a99c1aec59edb9e261468c4f9f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.667ex; height:2.509ex;" alt="{\displaystyle F_{x}}"> 係沿打橫嗰條軸嘅力，因為個結構係等邊三角形，所以每隻角嘅<a href="/wiki/%E8%A7%92%E5%BA%A6" title="角度">角度</a>係 60°）： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)={\frac {10}{2}}+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)={\frac {10}{2}}+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c57fac47c6a9769fe5c22a2ce4a12a20f2f908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:66.358ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)={\frac {10}{2}}+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ba7cea95e2ef40c429fa6a7415ae64ef2316d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:74.713ex; height:6.176ex;" alt="{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"></dd> <dd><figure class="mw-halign-center mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Method_of_Joints2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/af/Truss_Structure_Analysis%2C_Method_of_Joints2.png" decoding="async" width="480" height="490" class="mw-file-element" data-file-width="418" data-file-height="427" /></a><figcaption></figcaption></figure></dd></dl> ... 喺另外嗰幾點嘅力可以用同樣嘅道理嚟諗。如是者，個工程師就計到嗮「每條樑會受幾多力」等嘅重要資訊。上述呢個係一個簡化嘅例子：喺現實應用上，大些少嘅建築物會有成千上萬個類似噉嘅結構部件，唔可能人手計嗮。所以廿世紀末以來嘅工程師都會搵電腦模擬幫手，將結構唔同部份之間嘅互動定律組裝成<a href="/wiki/%E7%9F%A9%E9%99%A3" title="矩陣">矩陣</a>，然後解方程，得出結構每部分需要受幾多力同會郁幾多，知道成棟嘢會唔會因為噉而有機會冧，俾工程師做「成棟建築物噉樣起得唔得」嘅判斷<a href="#cite_note-mcKenzie2013-86">[54]</a>。 複雜啲嘅工程運算會涉及唔係<a href="/wiki/%E7%B7%9A%E6%80%A7" class="mw-redirect" title="線性">線性</a>嘅自然定律，呢啲定律加埋日常物件形狀上嘅不規則，工程運算原則上冇絕對準嘅答案，只能夠將現實處境拆做好多嚿細細個、會彼此互動嘅組件，然後用電腦模擬計個大約啱嘅答案出嚟。呢種做法就叫做<a href="/wiki/%E6%9C%89%E9%99%90%E5%85%83%E5%88%86%E6%9E%90" title="有限元分析">有限元分析</a><a href="#cite_note-87">[55]</a>。 <div class="mw-heading mw-heading2"><h2 id="驗證同確認">驗證同確認</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=13" title="編輯小節: 驗證同確認">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC%E6%A8%A1%E5%9E%8B%E9%A9%97%E8%AD%89%E5%90%8C%E7%A2%BA%E8%AA%8D" title="電腦模擬模型驗證同確認">電腦模擬模型驗證同確認</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E9%80%BC%E8%BF%91%E7%90%86%E8%AB%96" title="逼近理論">逼近理論</a>同<a href="/wiki/%E7%90%83%E7%8B%80%E4%B9%B3%E7%89%9B" title="球狀乳牛">球狀乳牛</a></div> 喺做咗一場電腦模擬之後，研究者仲要評估個模型掂唔掂：電腦模擬用嘅模型梗會涉及<a href="/wiki/%E6%8A%BD%E8%B1%A1%E5%8C%96" title="抽象化">抽象化</a>，即係有一啲特徵係現實嗰樣現象有、但個模型係冇嘅；例如<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>上用電腦模型模擬<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>或者<a href="/wiki/%E5%BB%BA%E7%AF%89%E7%89%A9" class="mw-redirect" title="建築物">建築物</a>等嘅結構嗰時，就好興假設組成個結構嗰啲<a href="/wiki/%E6%9F%B1" title="柱">柱</a>同<a href="/wiki/%E6%A8%91" title="樑">樑</a>係<a href="/wiki/%E5%89%9B%E9%AB%94" title="剛體">剛體</a>唔會變形－原則上，呢個假設係一種<a href="/wiki/%E7%90%86%E6%83%B3%E5%8C%96" title="理想化">理想化</a>，現實世界冇任何嘢係真正嘅剛體，不過如果啲<a href="/wiki/%E5%BB%BA%E7%AF%89%E6%9D%90%E6%96%99" title="建築材料">建築材料</a>真係造得好精良，硬淨得好交關，硬淨到要極大嘅力（例如有嚿<a href="/wiki/%E9%9A%95%E7%9F%B3" title="隕石">隕石</a>撞落去）先可以搞到佢變形，而呢啲「極大嘅力」出現嘅<a href="/wiki/%E6%A9%9F%E7%8E%87" class="mw-redirect" title="機率">機率</a>微乎其微（「有嚿隕石跌落嚟」係一件好少可會發生嘅事件）<a href="#cite_note-88">[56]</a>，噉「個結構係剛體」呢一點喺絕大多數時候都會成立，所以個假設雖然唔真實，但唔會對分析嘅準確度造成顯著嘅負面影響，可以照用<a href="#cite_note-89">[57]</a>。由呢個例子睇得出，一個電腦模擬模型實會有可能做唔到準確噉描述現實，問題只係「個模型失敗嘅機率有幾大」（一般係愈低愈好）同「個模型會喺幾多個情況下啱用」（一般係愈多愈好）。 評估一個電腦模擬模型主要有兩大過程－<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC%E6%A8%A1%E5%9E%8B%E9%A9%97%E8%AD%89" class="mw-redirect" title="電腦模擬模型驗證">驗證</a><a href="#cite_note-90">[e 22]</a>同<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC%E6%A8%A1%E5%9E%8B%E7%A2%BA%E8%AA%8D" class="mw-redirect" title="電腦模擬模型確認">確認</a><a href="#cite_note-91">[e 23]</a><a href="#cite_note-parker2013-92">[58]</a><a href="#cite_note-93">[59]</a>： <ul><li>驗證係指睇吓個模型嘅<a href="/wiki/Output" class="mw-redirect" title="Output">輸出</a>同原模型嗰啲<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>嘅解答接唔接近；呢個過程包括檢查個模型啲<a href="/wiki/%E6%BA%90%E7%A2%BC" class="mw-redirect" title="源碼">源碼</a>做嘅運算係咪同預想中嘅<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>相同，仲有係睇吓用個模型計出嚟嘅結果係咪合符用<a href="/wiki/%E8%A7%A3%E6%9E%90%E8%A7%A3" title="解析解">解析解</a>得出嘅結果（如果個模型模擬嗰樣現象係有可能用解析解計嘅話）<a href="#cite_note-94">[註 12]</a>。</li> <li>確認係指睇吓個模型「表示到現實嗰樣系統嘅程度」係咪足以達到場電腦模擬嘅目的；呢個過程通常會涉及個研究者行完場模擬之後走去做<a href="/wiki/%E5%AF%A6%E9%A9%97" title="實驗">實驗</a>攞<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>，睇吓啲數據係咪真係合乎個模型嘅預測，例如一個工程師做電腦模擬，模擬一個新設計嘅<a href="/wiki/%E6%91%A9%E6%89%93" title="摩打">摩打</a>行起上嚟係點，個模型預測個摩打嘅<a href="/wiki/%E6%95%88%E7%8E%87" title="效率">效率</a>「會係咁多咁多」，而個模型預測嘅效率數值非常好（根據佢個理論，個摩打效率非常咁高），令佢有信心郁手整個摩打嘅<a href="/wiki/%E5%8E%9F%E5%9E%8B" title="原型">原型</a>出嚟；响整好咗個原型之後，佢就開始試行吓個原型摩打，並且搵方法<a href="/wiki/%E9%87%8F%E5%BA%A6" title="量度">量度</a>個摩打嘅效率（攞到數據）－「啲數據係咪合符佢嘅原先預期」就反映佢個模型掂唔掂。 <ul><li>有好多做電腦模擬研究嘅科學家都覺得確認呢個工序好緊要－電腦模擬嘅方法曾經受唔少人抨擊，覺得呢啲模型傾向<a href="/wiki/%E6%8A%BD%E8%B1%A1%E5%8C%96" title="抽象化">抽象化</a>得滯，而成功噉確認一個電腦模擬模型<a href="#cite_note-95">[註 13]</a>反映嗰個模型能夠描述現實，令個模型唔淨只係流於理論<a href="#cite_note-96">[60]</a>。</li></ul></li></ul> <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="創發嘅啟示">創發嘅啟示</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=14" title="編輯小節: 創發嘅啟示">編輯</a>]</div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%89%B5%E7%99%BC" title="創發">創發</a>同<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E8%87%AA%E7%84%B6%E5%BD%A2%E6%85%8B%E8%A6%8F%E5%BE%8B" title="自然形態規律">自然形態規律</a></div> 有<a href="/wiki/%E5%93%B2%E5%AD%B8%E5%AE%B6" title="哲學家">哲學家</a>指出，電腦模擬對理解<a href="/wiki/%E5%89%B5%E7%99%BC" title="創發">創發</a><a href="#cite_note-97">[e 24]</a>嘅現象嚟講至關重要<a href="#cite_note-humphreys2004-98">[61]</a>。創發係一種現象，指一樣由好多唔同部份組成嘅事物具有一啲佢組成部份冇嘅特性。好似係<a href="/wiki/%E6%B0%B4" title="水">水</a>噉，水喺<a href="/wiki/%E5%8C%96%E5%AD%B8" title="化學">化學</a>上由<a href="/wiki/%E6%B0%AB" title="氫">氫</a>同<a href="/wiki/%E6%B0%A7" title="氧">氧</a>呢兩種<a href="/wiki/%E5%8C%96%E5%AD%B8%E5%85%83%E7%B4%A0" class="mw-redirect" title="化學元素">化學元素</a>組成嘅，水喺正常<a href="/wiki/%E6%BA%AB%E5%BA%A6" title="溫度">溫度</a>同<a href="/wiki/%E5%A3%93%E5%8A%9B" title="壓力">壓力</a>下處於<a href="/wiki/%E6%B6%B2%E9%AB%94" title="液體">液體</a>狀態，而呢種特性係氫同氧都冇嘅（氫同氧喺正常溫度壓力下會呈<a href="/wiki/%E6%B0%A3%E9%AB%94" title="氣體">氣體</a>狀態）－水具有最少一樣「氫同氧唔具有嘅特性」（創發出嚟嗰樣特性）；又或者好似係一個<a href="/wiki/%E4%BA%BA" title="人">人</a>噉，一個正常健康嘅人識<a href="/wiki/%E8%A1%8C%E8%B7%AF" title="行路">行路</a>識<a href="/wiki/%E6%80%9D%E8%80%83" class="mw-redirect" title="思考">思考</a>，而呢啲嘢都係組成人類身體嘅個別<a href="/wiki/%E7%B4%B0%E8%83%9E" title="細胞">細胞</a>唔能夠獨力做到嘅<a href="#cite_note-99">[62]</a>；創發仲可以分做兩種－<a href="/wiki/%E5%BC%B1%E5%89%B5%E7%99%BC" class="mw-redirect" title="弱創發">弱創發</a>同<a href="/wiki/%E5%BC%B7%E5%89%B5%E7%99%BC" class="mw-redirect" title="強創發">強創發</a>，弱創發係指創發出嚟嗰樣特性係可以由啲組成部份嘅特性解釋嘅，而強創發係指創發出嚟嗰樣特性係冇得由啲組成部份嘅特性嚟解釋<a href="#cite_note-100">[註 14]</a><a href="#cite_note-101">[63]</a>。 <div style="clear: both;"></div> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 435px"> <div class="thumb" style="width: 430px; height: 330px;"><a href="/wiki/File:ComputerHotline_-_Snow_crystals_(by).jpg" class="mw-file-description" title="水分子曉變成雪花，呢樣嘢氫同氧做唔到。"><img alt="水分子曉變成雪花，呢樣嘢氫同氧做唔到。" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/ComputerHotline_-_Snow_crystals_%28by%29.jpg/391px-ComputerHotline_-_Snow_crystals_%28by%29.jpg" decoding="async" width="391" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/ComputerHotline_-_Snow_crystals_%28by%29.jpg/587px-ComputerHotline_-_Snow_crystals_%28by%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/ComputerHotline_-_Snow_crystals_%28by%29.jpg/782px-ComputerHotline_-_Snow_crystals_%28by%29.jpg 2x" data-file-width="3181" data-file-height="2441" /></a></div> <div class="gallerytext"><center><a href="/wiki/%E6%B0%B4" title="水">水</a><a href="/wiki/%E5%88%86%E5%AD%90" title="分子">分子</a>曉變成<a href="/wiki/%E9%9B%AA%E8%8A%B1" title="雪花">雪花</a>，呢樣嘢氫同氧做唔到。</center></div> </li> </ul> 呢啲哲學家主張，電腦模擬嘅技術好幫到手理解弱創發嘅現象：研究者可以將一個預想中嘅系統寫做一個電腦模擬程式，個程式曉做（人手計唔嚟嘅）「<a href="/wiki/Foreach" class="mw-redirect" title="Foreach">foreach</a> 組成部份，做噉噉噉嘅運算」；而事實係，會用得電腦模擬嘅，好多時都係因為研究緊嗰樣現象有返咁上下<a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>，齋靠用個<a href="/wiki/%E4%BA%BA%E8%85%A6" title="人腦">人腦</a>諗唔能夠<a href="/wiki/%E6%83%B3%E5%83%8F" title="想像">想像</a>到個系統會出現乜嘢特性；不過假設個模型係掂嘅，個研究者跟住就有得靠住個模型，睇到個整體系統會出現啲乜嘢特性－包括係個系統有冇啲柞組成部份冇嘅特性，從而攞到一啲冇電腦模擬就攞唔到、有關弱創發嘅資訊<a href="#cite_note-102">[64]</a>。 <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="註釋">註釋</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=15" title="編輯小節: 註釋">編輯</a>]</div> <style data-mw-deduplicate="TemplateStyles:r2020813">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-3"> <ol class="references"> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> 係一個<a href="/wiki/%E5%B8%B8%E6%95%B8" title="常數">常數</a>，數值係 6.674×10−11 N • (m/kg)2。 </li> <li id="cite_note-18"><a href="#cite_ref-18">↑</a> 喺某啲電腦模擬當中，逐步升嘅唔係時間 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">，而係第個變數。 </li> <li id="cite_note-19"><a href="#cite_ref-19">↑</a> 技術化啲講，電腦模擬嘅過程會涉及<a href="/wiki/%E9%9B%A2%E6%95%A3%E5%8C%96" title="離散化">離散化</a>－假想時間係<a href="/wiki/%E9%80%A3%E7%BA%8C%E8%AE%8A%E6%95%B8" class="mw-redirect" title="連續變數">連續</a>嘅，好似上述噉嘅電腦模擬實會將 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 想像成一個<a href="/wiki/%E9%9B%A2%E6%95%A3%E8%AE%8A%E6%95%B8" class="mw-redirect" title="離散變數">離散變數</a>。 </li> <li id="cite_note-20"><a href="#cite_ref-20">↑</a> 以下呢啲類型有好多都唔係<a href="/wiki/%E4%BA%92%E6%96%A5" title="互斥">互斥</a>嘅，一個電腦模擬程式有可能同時屬於多過一個類別。例如<a href="/wiki/%E9%9A%A8%E6%A9%9F%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="隨機系統">隨機</a>同<a href="/wiki/%E6%B1%BA%E5%AE%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="決定性系統">決定</a>之間<a href="/wiki/%E5%AE%9A%E7%BE%A9" title="定義">定義</a>上就係互斥嘅，但隨機同<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本</a>呢兩個類別唔互斥。 </li> <li id="cite_note-37"><a href="#cite_ref-37">↑</a> 呢個模型冇隨機，所以屬決定性模型。 </li> <li id="cite_note-41"><a href="#cite_ref-41">↑</a> 呢點令佢哋同物理學上研究嘅<a href="/wiki/%E7%B2%92%E5%AD%90" title="粒子">粒子</a>唔同－同一個種類嘅粒子粒粒都一個板，但同一個<a href="/wiki/%E7%89%A9%E7%A8%AE" title="物種">物種</a>嘅唔同個體可以有唔同<a href="/wiki/%E6%80%A7%E6%A0%BC" title="性格">性格</a>。 </li> <li id="cite_note-43"><a href="#cite_ref-43">↑</a> 即係話設 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} _{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed04541609bb507fa23518d151dfd7ab25990f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.882ex; height:2.009ex;" alt="{\displaystyle \mathbf {s} _{t}}"> 表示「時間點 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> 嘅 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }">」，如果 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"> 同 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749fee708b41e7079eabd50d61c8bf3e965db16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}"> 之間冇事件發生，噉 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} _{t_{1}}=\mathbf {s} _{t_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} _{t_{1}}=\mathbf {s} _{t_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52875409f033ac273e4d5044dc0d81a08f4c7b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.525ex; height:2.343ex;" alt="{\displaystyle \mathbf {s} _{t_{1}}=\mathbf {s} _{t_{2}}}">。 </li> <li id="cite_note-47"><a href="#cite_ref-47">↑</a> 會輸出一個列表，列嗮啲數值出嚟。 </li> <li id="cite_note-52"><a href="#cite_ref-52">↑</a> <a href="/wiki/%E5%88%9D%E5%A7%8B%E5%8C%96" title="初始化">初始化</a>嗰部份省略咗。 </li> <li id="cite_note-58"><a href="#cite_ref-58">↑</a> 會涉及「<a href="/wiki/Foreach" class="mw-redirect" title="Foreach">foreach</a> 時間點」噉嘅運算，用人手計會好撈絞。 </li> <li id="cite_note-83"><a href="#cite_ref-83">↑</a> 不過工程學都有分好多門，例如<a href="/wiki/%E5%8C%96%E5%AD%B8%E5%B7%A5%E7%A8%8B" title="化學工程">化學工程</a>就會用多啲<a href="/wiki/%E5%8C%96%E5%AD%B8" title="化學">化學</a>知識。 </li> <li id="cite_note-94"><a href="#cite_ref-94">↑</a> 有人因此而批評，話「驗證」嘅概念根本冇用－研究者想用電腦模擬通常就正正係因為個問題用解析解搞唔掂。 </li> <li id="cite_note-95"><a href="#cite_ref-95">↑</a> 再配合埋<a href="/wiki/%E5%AF%A6%E9%A9%97" title="實驗">實驗</a>攞到嘅<a href="/wiki/%E6%95%B8%E6%93%9A" title="數據">數據</a>。 </li> <li id="cite_note-100"><a href="#cite_ref-100">↑</a> 不過都有學者覺得呢種分法係多餘。 </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="睇埋">睇埋</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=16" title="編輯小節: 睇埋">編輯</a>]</div> <div class="div-col columns column-count column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; column-count:3"> <ul><li><a href="/wiki/Foreach" class="mw-redirect" title="Foreach">Foreach</a></li> <li><a href="/wiki/%E6%A8%A1%E6%93%AC%E5%99%A8" title="模擬器">模擬器</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E5%8F%AF%E8%A6%96%E5%8C%96" title="資訊可視化">資訊可視化</a></li> <li><a href="/wiki/%E6%A0%BC%E4%BB%94%E8%87%AA%E5%8B%95%E6%A9%9F" title="格仔自動機">格仔自動機</a></li> <li><a href="/wiki/%E9%9B%A2%E6%95%A3%E5%8C%96" title="離散化">離散化</a></li> <li><a href="/wiki/%E6%99%82%E9%96%93%E5%BA%8F%E5%88%97" title="時間序列">時間序列</a></li> <li><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a></li> <li><a href="/wiki/%E7%B5%B1%E8%A8%88%E6%A8%A1%E5%9E%8B" title="統計模型">統計模型</a></li> <li><a href="/wiki/%E6%80%9D%E6%83%B3%E5%AF%A6%E9%A9%97" title="思想實驗">思想實驗</a></li> <li><a href="/wiki/%E6%83%85%E6%99%AF%E8%A8%88%E5%8A%83" title="情景計劃">情景計劃</a></li> <li><a href="/wiki/%E9%81%8A%E6%88%B2%E7%89%A9%E7%90%86" title="遊戲物理">遊戲物理</a></li> <li><a href="/wiki/%E9%9B%BB%E5%AD%90%E9%81%8A%E6%88%B2" title="電子遊戲">電子遊戲</a></li> <li><a href="/wiki/%E6%A8%A1%E6%93%AC%E9%81%8A%E6%88%B2" title="模擬遊戲">模擬遊戲</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="文獻">文獻</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=17" title="編輯小節: 文獻">編輯</a>]</div> <div class="div-col columns column-count column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; font-size:90%; column-count:3"> <ul><li>Desa, W. L. H. M., Kamaruddin, S., & Nawawi, M. K. M. (2012). <a rel="nofollow" class="external text" href="https://www.scientific.net/AMR.591-593.557.pdf">Modeling of aircraft composite parts using simulation</a> (PDF). In Advanced Materials Research (Vol. 591, pp. 557-560). Trans Tech Publications Ltd.</li> <li>Hartmann, S. (1996). <a rel="nofollow" class="external text" href="https://sas-space.sas.ac.uk/1087/1/S_Hartmann_Simulations.pdf">The world as a process</a> (PDF). In Modelling and simulation in the social sciences from the philosophy of science point of view (pp. 77-100). Springer, Dordrecht.</li> <li>Humphreys, P. (2004). Extending ourselves: Computational science, empiricism, and scientific method. Oxford University Press.</li> <li>Kaizer, J. S., Heller, A. K., & Oberkampf, W. L. (2015). <a rel="nofollow" class="external text" href="https://reader.elsevier.com/reader/sd/pii/S0951832015000307?token=199F2FEDBF6AE2B9F0602DB9944715F866AA9961FFFAAB1D692E89684F0ADC5A6BC6DB54DA2E386D48B298772306893A">Scientific computer simulation review</a> (PDF). Reliability Engineering & System Safety, 138, 210-218.</li> <li>Macal, C. M., & North, M. J. (2008, December). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4736060">Agent-based modeling and simulation: ABMS examples</a> (PDF). In 2008 Winter Simulation Conference (pp. 101-112). IEEE.</li> <li>McClelland, J. L. (2009). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/pdfdirect/10.1111/j.1756-8765.2008.01003.x">The place of modeling in cognitive science</a> (PDF). Topics in Cognitive Science, 1(1), 11-38.</li> <li>Nutaro, J. J. (2011). <a rel="nofollow" class="external text" href="http://www.r-5.org/files/books/computers/algo-list/test-dev/James_J_Nutaro-Building_Software_for_Simulation-EN.pdf">Building software for simulation: Theory and algorithms, with application in C++</a> (PDF). John Wiley & Sons Ltd.</li> <li>Schelling, T. C., (1971). "<a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/pdf/10.1080/0022250X.1971.9989794">Dynamic Models of Segregation</a> (PDF)," Journal of Mathematical Sociology, 1: 143–186.</li> <li>Winsberg, E. (2003). <a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/pdf/10.1086/367872">Simulated experiments: Methodology for a virtual world</a> (PDF). Philosophy of science, 70(1), 105-125.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="引述">引述</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=18" title="編輯小節: 引述">編輯</a>]</div> 篇文用咗嘅<a href="/wiki/%E8%A1%8C%E8%A9%B1" title="行話">行話</a>，<a href="/wiki/%E8%8B%B1%E6%96%87" title="英文">英文</a>名如下： <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2020813"><div class="reflist reflist-columns references-column-width reflist-columns-3"> <ol class="references"> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> scientific method </li> <li id="cite_note-21"><a href="#cite_ref-21">↑</a> stochastic </li> <li id="cite_note-22"><a href="#cite_ref-22">↑</a> deterministic </li> <li id="cite_note-24"><a href="#cite_ref-24">↑</a> Monte Carlo method </li> <li id="cite_note-28"><a href="#cite_ref-28">↑</a> domain </li> <li id="cite_note-29"><a href="#cite_ref-29">↑</a> probability distribution </li> <li id="cite_note-30"><a href="#cite_ref-30">↑</a> aggregate data </li> <li id="cite_note-31"><a href="#cite_ref-31">↑</a> uniform distribution </li> <li id="cite_note-34"><a href="#cite_ref-34">↑</a> equation-based </li> <li id="cite_note-35"><a href="#cite_ref-35">↑</a> agent-based </li> <li id="cite_note-42"><a href="#cite_ref-42">↑</a> discrete event simulation，DES </li> <li id="cite_note-45"><a href="#cite_ref-45">↑</a> multiscale modeling </li> <li id="cite_note-49"><a href="#cite_ref-49">↑</a> physics engine </li> <li id="cite_note-51"><a href="#cite_ref-51">↑</a> Newtonian mechanics </li> <li id="cite_note-56"><a href="#cite_ref-56">↑</a> Lotka-Volterra model </li> <li id="cite_note-61"><a href="#cite_ref-61">↑</a> artificial neural network </li> <li id="cite_note-64"><a href="#cite_ref-64">↑</a> activation level </li> <li id="cite_note-71"><a href="#cite_ref-71">↑</a> connectionism </li> <li id="cite_note-76"><a href="#cite_ref-76">↑</a> game theory </li> <li id="cite_note-77"><a href="#cite_ref-77">↑</a> iterated prisoner's dilemma </li> <li id="cite_note-85"><a href="#cite_ref-85">↑</a> structural analysis </li> <li id="cite_note-90"><a href="#cite_ref-90">↑</a> verification </li> <li id="cite_note-91"><a href="#cite_ref-91">↑</a> validation </li> <li id="cite_note-97"><a href="#cite_ref-97">↑</a> emergence </li> </ol></div> 篇文引用咗以下呢啲<a href="/wiki/%E6%96%87%E7%8D%BB" title="文獻">文獻</a>同<a href="/wiki/%E7%B6%B2%E9%A0%81" title="網頁">網頁</a>： <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2020813"><div class="reflist reflist-columns references-column-width reflist-columns-3"> <ol class="references"> <li id="cite_note-strogatz2007-1">↑ <a href="#cite_ref-strogatz2007_1-0">1.0</a> <a href="#cite_ref-strogatz2007_1-1">1.1</a> Strogatz, Steven (2007). "The End of Insight". In Brockman, John (ed.). What is your dangerous idea?. HarperCollins. </li> <li id="cite_note-basicode-2">↑ <a href="#cite_ref-basicode_2-0">2.0</a> <a href="#cite_ref-basicode_2-1">2.1</a> Donald Craig (1996). <a rel="nofollow" class="external text" href="http://www.cs.mun.ca/~donald/msc/node69.html">The Simulation Algorithm</a>. </li> <li id="cite_note-reynolds1987-3"><a href="#cite_ref-reynolds1987_3-0">↑</a> Reynolds, Craig W. (1987). "Flocks, herds and schools: A distributed behavioral model.". ACM SIGGRAPH Computer Graphics. 21. pp. 25–34. </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> Thijssen, J. (2007). Computational physics. Cambridge university press. </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Lewars, E. (2003). Computational chemistry. Introduction to the theory and applications of molecular and quantum mechanics, 318. </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> Banzhaf, W., & Eckman, F. H. (1995). 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M.; Zdravković, L. (1999). Finite element analysis in geotechnical engineering. Thomas Telford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9780727727534" title="Special:書本來源/9780727727534"><bdi>9780727727534</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+element+analysis+in+geotechnical+engineering&rft.pub=Thomas+Telford&rft.date=1999&rft.isbn=9780727727534&rft.aulast=Potts&rft.aufirst=D.+M.&rft.au=Zdravkovi%C4%87%2C+L.&rfr_id=info%3Asid%2Fzh-yue.wikipedia.org%3A%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC" class="Z3988"> </li> <li id="cite_note-88"><a href="#cite_ref-88">↑</a> Nester, C. J., Liu, A. M., Ward, E., Howard, D., Cocheba, J., & Derrick, T. (2010). Error in the description of foot kinematics due to violation of rigid body assumptions. Journal of biomechanics, 43(4), 666-672. </li> <li id="cite_note-89"><a href="#cite_ref-89">↑</a> Timan, A. F. (2014). Theory of approximation of functions of a real variable. Elsevier. </li> <li id="cite_note-parker2013-92"><a href="#cite_ref-parker2013_92-0">↑</a> Parker, W. (2013). Computer Simulation, in S. Psillos and M. Curd (eds.), The Routledge Companion to Philosophy of Science, 2nd Edition, London: Routledge. </li> <li id="cite_note-93"><a href="#cite_ref-93">↑</a> Oberkampf, W. and C. Roy, (2010). Verification and Validation in Scientific Computing, Cambridge: Cambridge University Press. </li> <li id="cite_note-96"><a href="#cite_ref-96">↑</a> Heckbert, S., Baynes, T., & Reeson, A. (2010). <a rel="nofollow" class="external text" href="https://d1wqtxts1xzle7.cloudfront.net/44161975/Agent-based_modeling_in_ecological_econo20160328-28027-wu6auw-with-cover-page-v2.pdf?Expires=1638771424&Signature=bB0xwlUDTT7CryMe4kKVjLwR7jBCYKaD3ffyrT1RMY89TzvZLfPd4ELnC9IAEmUCElah~ycMWga4Zl6yzOViRGcxMHUt4Ey8CSo1kTATYy6Gg-zspECz6N-b86731t-ReMdYZP5989vruSUIpF0t0Y7n2yAlXvdVpamyZu~oqk3p-nyWl0IfjH4QPpn7iSah15ky8YhoipA~OzXBQc0Tr27grSgkZBTLlhK0qIhnW6qV5Ny1vHCmUNpRL5cRFE0elhwJY0FxSnWn7KJe3TubmwsGbnz40R8rcvoLn2-6Okicc~VsBHreSRfNegCs63htPV2THnlmdQgOVnjwljWx1A__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA">Agent‐based modeling in ecological economics</a> <a href="/wiki/Wayback_Machine" title="Wayback Machine">互聯網檔案館</a>嘅<a rel="nofollow" class="external text" href="https://web.archive.org/web/20211206051808/https://d1wqtxts1xzle7.cloudfront.net/44161975/Agent-based_modeling_in_ecological_econo20160328-28027-wu6auw-with-cover-page-v2.pdf?Expires=1638771424&Signature=bB0xwlUDTT7CryMe4kKVjLwR7jBCYKaD3ffyrT1RMY89TzvZLfPd4ELnC9IAEmUCElah~ycMWga4Zl6yzOViRGcxMHUt4Ey8CSo1kTATYy6Gg-zspECz6N-b86731t-ReMdYZP5989vruSUIpF0t0Y7n2yAlXvdVpamyZu~oqk3p-nyWl0IfjH4QPpn7iSah15ky8YhoipA~OzXBQc0Tr27grSgkZBTLlhK0qIhnW6qV5Ny1vHCmUNpRL5cRFE0elhwJY0FxSnWn7KJe3TubmwsGbnz40R8rcvoLn2-6Okicc~VsBHreSRfNegCs63htPV2THnlmdQgOVnjwljWx1A__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA">歸檔</a>，歸檔日期2021年12月6號，. (PDF). Annals of the New York Academy of Sciences, 1185(1), 39-53. </li> <li id="cite_note-humphreys2004-98"><a href="#cite_ref-humphreys2004_98-0">↑</a> Humphreys, P. (2004). Extending ourselves: Computational science, empiricism, and scientific method. Oxford University Press. </li> <li id="cite_note-99"><a href="#cite_ref-99">↑</a> O'Connor, Timothy; Wong, Hong Yu (February 28, 2012). Edward N. Zalta (ed.). "Emergent Properties". The Stanford Encyclopedia of Philosophy (Spring 2012 Edition). </li> <li id="cite_note-101"><a href="#cite_ref-101">↑</a> Bedau, Mark A. (1997), <a rel="nofollow" class="external text" href="http://people.reed.edu/~mab/papers/weak.emergence.pdf">Weak Emergence</a> (PDF). </li> <li id="cite_note-102"><a href="#cite_ref-102">↑</a> Bedau, M. A. (2013). Weak emergence and computer simulation. In Models, simulations, and representations (pp. 109-132). Routledge. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="拎">拎</h2>[<a href="/w/index.php?title=%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC&action=edit&section=19" title="編輯小節: 拎">編輯</a>]</div> <style data-mw-deduplicate="TemplateStyles:r2018312">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r2018311">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist"><a href="https://commons.wikimedia.org/wiki/%E9%A0%AD%E7%89%88" class="extiw" title="commons:頭版">維基同享</a>有多媒體嘅嘢： <a href="https://commons.wikimedia.org/wiki/Category:Computer_simulations" class="extiw" title="commons:Category:Computer simulations">電腦模擬</a></div></div> </div> <ul><li>（英文） <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/simulations-science/">Computer Simulations in Science</a>，<a href="/wiki/%E5%8F%B2%E4%B8%B9%E7%A6%8F%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹福哲學百科全書">史丹福哲學百科全書</a></li> <li>（英文） <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/models-science/">Models in Science</a>，史丹福哲學百科全書</li> <li>（英文） <a rel="nofollow" class="external text" href="https://www.lib.ncsu.edu/findingaids/mc00488">Guide to the Computer Simulation Oral History Archive 2003-2018</a>.</li></ul> <div role="navigation" class="navbox" aria-labelledby="電腦科學" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Template:電腦科學"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8&action=edit&redlink=1" class="new" title="Template talk:電腦科學 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="電腦科學" style="font-size:114%;margin:0 4em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="電腦科學">電腦科學</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="研究運算嘅科學工程領域">研究<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>嘅<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a><a href="/wiki/%E5%B7%A5%E7%A8%8B" class="mw-redirect" title="工程">工程</a>領域</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要概念</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a> · <a href="/wiki/I/O" title="I/O">I/O</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6" title="電腦">電腦</a> · <a href="/wiki/%E9%81%8B%E7%AE%97%E9%82%8F%E8%BC%AF" title="運算邏輯">運算邏輯</a> · <a href="/wiki/%E6%95%B8%E8%AB%96" title="數論">數論</a> · <a href="/wiki/%E4%BA%8C%E9%80%B2%E5%88%B6" title="二進制">二進制</a> · <a href="/wiki/%E6%95%B8%E5%AD%B8%E9%82%8F%E8%BC%AF" title="數學邏輯">數學邏輯</a> · <a href="/wiki/%E9%9B%A2%E6%95%A3%E6%95%B8%E5%AD%B8" title="離散數學">離散數學</a>（<a href="/wiki/%E5%9C%96%E8%AB%96" title="圖論">圖論</a>） · <a href="/wiki/%E6%8A%BD%E8%B1%A1%E5%8C%96_(%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8)" title="抽象化 (電腦科學)">抽象化</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8%E8%A9%9E%E5%BD%99%E8%A1%A8" title="電腦科學詞彙表">電腦科學詞彙表</a></div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><a href="/wiki/File:Binario_cropped.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binario_cropped.png/150px-Binario_cropped.png" decoding="async" width="150" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binario_cropped.png/225px-Binario_cropped.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/17/Binario_cropped.png 2x" data-file-width="254" data-file-height="218" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%90%86%E8%AB%96%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="理論電腦科學">電腦理論</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="運算理論" scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%87%AA%E5%8B%95%E6%A9%9F%E7%90%86%E8%AB%96" title="自動機理論">自動機理論</a> · <a href="/wiki/%E5%8F%AF%E9%81%8B%E7%AE%97%E5%BA%A6%E7%90%86%E8%AB%96" title="可運算度理論">可運算度理論</a> · <a href="/wiki/%E9%81%8B%E7%AE%97%E8%A4%87%E9%9B%9C%E5%BA%A6%E7%90%86%E8%AB%96" title="運算複雜度理論">運算複雜度理論</a> · <a href="/wiki/%E5%9C%96%E9%9D%88%E6%A9%9F" title="圖靈機">圖靈機</a> · <a href="/wiki/%E9%87%8F%E5%AD%90%E9%81%8B%E7%AE%97" title="量子運算">量子運算</a></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a>（<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a> · <a href="/wiki/%E8%A8%8A%E8%99%9F" title="訊號">訊號</a> · <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%86%B5" title="資訊熵">資訊熵</a> · <a href="/wiki/%E7%B7%A8%E7%A2%BC%E7%90%86%E8%AB%96" title="編碼理論">編碼理論</a>） · <a href="/wiki/%E8%B3%87%E6%96%99%E9%A1%9E%E5%9E%8B" title="資料類型">資料類型‎</a>同<a href="/wiki/%E6%95%B8%E6%93%9A%E7%B5%90%E6%A7%8B" title="數據結構">數據結構</a> · <a href="/wiki/%E7%B3%BB%E7%B5%B1%E8%B3%87%E6%BA%90" title="系統資源">系統資源</a> · <a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>（<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E5%88%86%E6%9E%90" title="演算法分析">分析</a>） · <a href="/wiki/%E5%BD%A2%E5%BC%8F%E5%8C%96%E6%96%B9%E6%B3%95" title="形式化方法">形式化方法</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦系統</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E6%9E%B6%E6%A7%8B" title="電腦架構">電腦架構</a>（<a href="/wiki/%E9%82%8F%E8%BC%AF%E9%96%80" title="邏輯門">邏輯門</a> · <a href="/wiki/%E6%8C%87%E4%BB%A4%E9%9B%86%E6%9E%B6%E6%A7%8B" title="指令集架構">指令集架構</a> · <a href="/wiki/%E5%BE%AE%E6%9E%B6%E6%A7%8B" title="微架構">微架構</a> · <a href="/wiki/%E5%A4%9A%E5%85%83%E8%99%95%E7%90%86" title="多元處理">多元處理</a> · <a href="/wiki/%E4%BD%9C%E6%A5%AD%E7%B3%BB%E7%B5%B1" title="作業系統">作業系統</a>） · <a href="/wiki/%E4%B8%A6%E8%A1%8C%E9%81%8B%E7%AE%97" title="並行運算">並行</a>、<a href="/wiki/%E5%B9%B3%E8%A1%8C%E9%81%8B%E7%AE%97" title="平行運算">平行</a>同<a href="/wiki/%E5%88%86%E6%95%A3%E9%81%8B%E7%AE%97" title="分散運算">分散</a>運算 · <a href="/wiki/%E9%9B%BB%E8%85%A6%E6%80%A7%E8%83%BD" title="電腦性能">性能</a> · <a href="/wiki/%E8%B6%85%E7%B4%9A%E9%9B%BB%E8%85%A6" title="超級電腦">超級電腦</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦通訊</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%B6%B2%E7%B5%A1" title="電腦網絡">電腦網絡</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E5%82%B3%E8%BC%B8%E5%8D%94%E8%AD%B0" title="網絡傳輸協議">網絡傳輸協議</a> · <a href="/wiki/%E7%B6%B2%E7%B5%A1%E6%8B%93%E6%A8%B8" title="網絡拓樸">網絡拓樸</a> · <a href="/wiki/%E6%95%B8%E6%93%9A%E5%82%B3%E8%BC%B8" title="數據傳輸">數據傳輸</a> · <a href="/wiki/%E4%BA%92%E8%81%AF%E7%B6%B2" title="互聯網">互聯網</a> · <a href="/wiki/%E7%B6%B2%E9%A0%81%E8%A8%AD%E8%A8%88" title="網頁設計">網頁設計</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BF%9D%E5%AE%89" title="電腦保安">電腦保安</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E4%BF%9D%E5%AE%89" title="網絡保安">網絡保安</a> · <a href="/wiki/%E6%BC%8F%E6%B4%9E" title="漏洞">漏洞</a> · <a href="/wiki/%E5%85%AC%E5%85%B1%E6%BC%8F%E6%B4%9E%E5%90%8C%E6%9A%B4%E9%9C%B2" title="公共漏洞同暴露">公共漏洞同暴露</a> · <a href="/wiki/%E7%B6%B2%E7%B5%A1%E6%94%BB%E6%93%8A" title="網絡攻擊">網絡攻擊</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E7%97%85%E6%AF%92" title="電腦病毒">電腦病毒</a> · <a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F%E7%B7%A8%E5%AF%AB" title="電腦程式編寫">程式編寫</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="程式語言">程式語言</a>（<a href="/wiki/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80%E7%90%86%E8%AB%96" title="程式語言理論">程式語言理論</a> · <a href="/wiki/%E4%BD%8E%E7%B4%9A%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="低級程式語言">低級</a>同<a href="/wiki/%E9%AB%98%E7%B4%9A%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="高級程式語言">高級</a>程式語言 · <a href="/wiki/%E6%8E%A7%E5%88%B6%E6%B5%81%E7%A8%8B" title="控制流程">控制流程</a>） · <a href="/wiki/%E7%A8%8B%E5%BC%8F%E5%9F%B7%E8%A1%8C" title="程式執行">程式執行</a> · <a href="/wiki/%E7%B7%A8%E8%AD%AF%E5%99%A8" title="編譯器">編譯器</a> · <a href="/wiki/%E7%A8%8B%E5%BC%8F%E7%B7%A8%E5%AF%AB%E7%AF%84%E5%BC%8F" title="程式編寫範式">程式編寫範式</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦應用</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E6%95%B8%E7%A2%BC%E8%97%9D%E8%A1%93" title="數碼藝術">數碼藝術</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="電腦圖像">電腦圖像</a>（<a href="/wiki/%E4%BA%8C%E7%B6%AD%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" class="mw-redirect" title="二維電腦圖像">2D</a> · <a href="/wiki/%E4%B8%89%E7%B6%AD%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" class="mw-redirect" title="三維電腦圖像">3D</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%8B%95%E7%95%AB" title="電腦動畫">電腦動畫</a> · <a href="/wiki/%E8%B3%87%E8%A8%8A%E5%8F%AF%E8%A6%96%E5%8C%96" title="資訊可視化">資訊可視化</a> · <a href="/wiki/%E8%99%9B%E6%93%AC%E5%AF%A6%E5%A2%83" title="虛擬實境">VR</a> · <a href="/wiki/%E6%93%B4%E5%BC%B5%E5%AF%A6%E5%A2%83" title="擴張實境">AR</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F%E5%AD%B8" title="電腦圖像學">電腦圖像學</a>） · <a href="/wiki/%E9%9B%BB%E8%85%A6%E9%9F%B3%E6%A8%82" title="電腦音樂">電腦音樂</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E8%BB%9F%E4%BB%B6%E5%B7%A5%E7%A8%8B" title="軟件工程">軟件工程</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E8%BB%9F%E4%BB%B6" title="電腦軟件">電腦軟件</a> · <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%96%8B%E7%99%BC" title="軟件開發">軟件開發</a> · <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%9C%80%E6%B1%82" title="軟件需求">軟件需求</a> · <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%A0%85%E7%9B%AE%E7%AE%A1%E7%90%86" title="軟件項目管理">軟件項目管理</a> · <a href="/wiki/%E9%9B%BB%E5%AD%90%E9%81%8A%E6%88%B2%E8%A3%BD%E4%BD%9C" title="電子遊戲製作">電子遊戲製作</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E4%BA%BA%E6%A9%9F%E4%BA%92%E5%8B%95" title="人機互動">HCI</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%94%A8%E5%AE%B6" title="用家">用家</a> · <a href="/wiki/%E6%98%93%E7%94%A8%E5%BA%A6" title="易用度">易用度</a> · <a href="/wiki/%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2" title="用家介面">用家介面</a>（<a href="/wiki/%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2%E8%A8%AD%E8%A8%88" title="用家介面設計">設計</a> · <a href="/wiki/%E5%9C%96%E5%83%8F%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2" title="圖像用家介面">圖像</a> · <a href="/wiki/%E5%91%BD%E4%BB%A4%E8%A1%8C%E4%BB%8B%E9%9D%A2" title="命令行介面">命令行</a> · <a href="/wiki/%E8%85%A6%E6%A9%9F%E4%BB%8B%E9%9D%A2" title="腦機介面">腦機</a>）</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">其他應用</div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%B3%BB%E7%B5%B1" title="資訊系統">資訊系統</a>（<a href="/wiki/%E6%95%B8%E6%93%9A%E5%BA%AB" title="數據庫">數據庫</a>） · <a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a>（<a class="mw-selflink selflink">電腦模擬</a>） · <a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E8%AB%96" title="運算數論">運算數論</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BB%A3%E6%95%B8" title="電腦代數">電腦代數</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">AI</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD%E5%93%B2%E5%AD%B8" title="人工智能哲學">AI 哲學</a> · <a href="/wiki/%E7%9F%A5%E8%AD%98%E8%A1%A8%E7%A4%BA" title="知識表示">知識表示</a> · <a href="/wiki/%E8%87%AA%E5%8B%95%E8%A8%88%E5%8A%83" title="自動計劃">自動計劃</a> · <a href="/wiki/%E9%9B%BB%E8%85%A6%E8%A6%96%E8%A6%BA" title="電腦視覺">電腦視覺</a> · <a href="/wiki/%E8%87%AA%E7%84%B6%E8%AA%9E%E8%A8%80%E8%99%95%E7%90%86" title="自然語言處理">NLP</a> · <a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>（<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="人工神經網絡">ANN</a> · <a href="/wiki/%E7%9B%A3%E7%9D%A3%E5%BC%8F%E5%AD%B8%E7%BF%92" title="監督式學習">監督式學習</a> · <a href="/wiki/%E9%9D%9E%E7%9B%A3%E7%9D%A3%E5%BC%8F%E5%AD%B8%E7%BF%92" title="非監督式學習">非監督式學習</a> · <a href="/wiki/%E5%BC%B7%E5%8C%96%E5%AD%B8%E7%BF%92" title="強化學習">強化學習</a>）</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">相關領域</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="_工程學" scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">　<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E5%AD%90%E5%B7%A5%E7%A8%8B" title="電子工程">電子工程</a>（<a href="/wiki/%E9%9B%BB%E8%A8%8A" title="電訊">電訊</a>） · <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%B7%A5%E7%A8%8B" title="電腦工程">電腦工程</a> · <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%A7%91%E6%8A%80" title="資訊科技">資訊科技</a> · <a href="/wiki/%E6%A9%9F%E6%A2%B0%E4%BA%BA%E5%AD%B8" title="機械人學">機械人學</a> · <a href="/wiki/%E6%A9%9F%E9%9B%BB%E5%B7%A5%E7%A8%8B" class="mw-redirect" title="機電工程">機電工程</a></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%82%8F%E8%BC%AF%E5%AD%B8" class="mw-redirect" title="邏輯學">邏輯學</a> · <a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>（<a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a> · <a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a>） · <a href="/wiki/%E8%AA%8D%E7%9F%A5%E7%A7%91%E5%AD%B8" title="認知科學">認知科學</a> · <a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a> · <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%A7%91%E5%AD%B8" title="資訊科學">資訊科學</a> · <a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%A7%91%E5%AD%B8" title="系統科學">系統科學</a></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><a href="/wiki/Category:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Category:電腦科學">電腦科學類</a></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="複雜系統" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="Template:複雜系統"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1&action=edit&redlink=1" class="new" title="Template talk:複雜系統 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="複雜系統" style="font-size:114%;margin:0 4em"><a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="顧名思義係指複雜嘅系統，變化難以預測">顧名思義係指<a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>嘅<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>，變化難以預測</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要概念</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>同<a href="/wiki/%E7%B0%A1%E5%96%AE" title="簡單">簡單</a></li> <li><a href="/wiki/%E7%B5%84%E5%90%88%E7%88%86%E7%99%BC" title="組合爆發">組合爆發</a></li> <li><a href="/wiki/%E5%89%B5%E7%99%BC" title="創發">創發</a>同<a href="/wiki/%E8%87%AA%E6%88%91%E7%B5%84%E7%B9%94" title="自我組織">自我組織</a></li> <li><a href="/wiki/%E5%8B%95%E6%85%8B%E7%B3%BB%E7%B5%B1" title="動態系統">動態系統</a></li> <li><a href="/wiki/%E8%A4%87%E9%9B%9C%E9%81%A9%E6%87%89%E7%B3%BB%E7%B5%B1" title="複雜適應系統">適應系統</a></li> <li><a href="/wiki/%E5%8F%AF%E9%80%B2%E5%8C%96%E5%BA%A6" title="可進化度">可進化</a></li> <li><a href="/wiki/%E6%95%B4%E9%AB%94%E4%B8%BB%E7%BE%A9" title="整體主義">整體主義</a>同<a href="/wiki/%E9%82%84%E5%8E%9F%E8%AB%96" title="還原論">還原論</a></li> <li><a href="/wiki/%E8%87%AA%E7%84%B6%E5%BD%A2%E6%85%8B%E8%A6%8F%E5%BE%8B" title="自然形態規律">自然形態規律</a></li></ul> </div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><a href="/wiki/File:Lorenz_attractor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/150px-Lorenz_attractor.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/225px-Lorenz_attractor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/300px-Lorenz_attractor.svg.png 2x" data-file-width="750" data-file-height="750" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要例子</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%94%9F%E5%91%BD" title="生命">生命</a>同<a href="/wiki/%E7%94%9F%E7%89%A9" title="生物">生物</a></li> <li><a href="/wiki/%E7%B4%B0%E8%83%9E" title="細胞">細胞</a></li> <li>神經<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B4%B0%E8%83%9E" title="神經細胞">細胞</a>同<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B3%BB%E7%B5%B1" title="神經系統">系統</a>－包括<a href="/wiki/%E8%85%A6" title="腦">腦</a>（尤其係<a href="/wiki/%E4%BA%BA%E8%85%A6" title="人腦">人腦</a>）</li> <li><a href="/wiki/%E7%A8%AE%E7%BE%A4" title="種群">種群</a></li> <li><a href="/wiki/%E7%94%9F%E6%85%8B%E7%B3%BB%E7%B5%B1" title="生態系統">生態系統</a>（<a href="/wiki/%E9%97%9C%E9%8D%B5%E7%89%A9%E7%A8%AE" title="關鍵物種">關鍵物種</a>）</li> <li><a href="/wiki/%E8%87%AA%E7%84%B6" title="自然">自然</a></li> <li><a href="/wiki/%E6%B0%A3%E5%80%99" title="氣候">氣候</a></li></ul> <ul><li><a href="/wiki/%E7%B5%84%E7%B9%94" title="組織">組織</a></li> <li><a href="/wiki/%E7%A4%BE%E6%9C%83" title="社會">社會</a></li> <li><a href="/wiki/%E7%B6%93%E6%BF%9F%E9%AB%94" class="mw-redirect" title="經濟體">經濟體</a>（<a href="/wiki/%E9%87%91%E8%9E%8D%E5%B8%82%E5%A0%B4" title="金融市場">金融市場</a> · <a href="/wiki/%E5%9C%8B%E9%9A%9B%E7%B6%93%E6%BF%9F" class="mw-redirect" title="國際經濟">國際經濟</a>）</li> <li><a href="/wiki/%E5%9F%8E%E5%B8%82" title="城市">城市</a></li> <li>現代<a href="/wiki/%E5%9F%BA%E7%A4%8E%E5%BB%BA%E8%A8%AD" class="mw-redirect" title="基礎建設">基礎建設</a></li> <li>進階<a href="/wiki/%E9%9B%BB%E8%85%A6%E8%BB%9F%E4%BB%B6" title="電腦軟件">電腦軟件</a></li> <li><a href="/wiki/%E5%9C%B0%E7%90%83" title="地球">地球</a></li> <li><a href="/wiki/%E6%98%9F%E7%B3%BB" title="星系">星系</a></li> <li><a href="/wiki/%E5%AE%87%E5%AE%99" title="宇宙">宇宙</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" title="非線性系統">非線系統</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>（<a href="/wiki/%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="常微分方程">常</a>同<a href="/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程">偏</a>）</li> <li><a href="/wiki/%E6%99%82%E9%96%93%E5%BA%8F%E5%88%97" title="時間序列">時間序列</a></li> <li><a href="/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96" title="混沌理論">混沌理論</a>（<a href="/wiki/%E8%87%A8%E7%95%8C%E9%81%8E%E6%B8%A1" title="臨界過渡">臨界過渡</a> · <a href="/wiki/%E5%90%B8%E5%BC%95%E5%AD%90" title="吸引子">吸引子</a>）</li> <li><a href="/wiki/%E5%88%86%E5%B2%94%E7%90%86%E8%AB%96" title="分岔理論">分岔理論</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E7%90%86%E8%AB%96" title="網絡理論">網絡理論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E7%A7%91%E5%AD%B8" title="網絡科學">網絡科學</a></li> <li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E4%B8%AD%E5%BF%83%E5%BA%A6" title="網絡中心度">中心度</a></li> <li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E8%A6%8F%E5%BE%8B" title="網絡規律">網絡規律</a></li> <li><a href="/wiki/%E7%B4%B0%E4%B8%96%E7%95%8C%E7%B6%B2%E7%B5%A1" title="細世界網絡">細世界網絡</a></li> <li><a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="神經網絡">神經網絡</a>、<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%B6%B2%E7%B5%A1" title="社會網絡">社會網絡</a>同<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%B6%B2%E7%B5%A1" title="電腦網絡">電腦網絡</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E9%AB%94%E8%A1%8C%E7%82%BA" title="集體行為">集體行為</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%A4%BE%E6%9C%83%E5%8B%95%E5%8A%9B%E5%AD%B8" title="社會動力學">社會動力學</a></li> <li><a href="/wiki/%E8%B6%85%E7%94%9F%E7%89%A9" title="超生物">超生物</a></li> <li><a href="/wiki/%E5%A4%A7%E7%BE%A4%E8%A1%8C%E7%82%BA" title="大群行為">大群行為</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%99%BA%E8%83%BD" title="集體智能">集體智能</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E8%A1%8C%E5%8B%95" title="集體行動">集體行動</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%84%8F%E8%AD%98" title="集體意識">集體意識</a></li> <li><a href="/wiki/%E7%BE%8A%E7%BE%A4%E5%BF%83%E7%90%86" class="mw-redirect" title="羊群心理">羊群心理</a></li> <li><a href="/wiki/%E5%90%8C%E6%AD%A5" title="同步">同步</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%84%8F%E6%9C%83" title="集體意會">集體意會</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%90%86%E8%AB%96" title="系統理論">系統理論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Feedback" title="Feedback">Feedback</a></li> <li><a href="/wiki/%E8%87%AA%E6%8C%87" title="自指">自指</a></li> <li><a href="/wiki/%E9%A0%91%E5%81%A5%E5%BA%A6" title="頑健度">頑健</a>同<a href="/wiki/%E5%AE%B9%E9%8C%AF%E8%83%BD%E5%8A%9B" title="容錯能力">容錯</a></li> <li><a href="/wiki/%E5%8F%AF%E6%93%B4%E7%B8%AE%E5%BA%A6" title="可擴縮度">可擴縮度</a></li> <li><a href="/wiki/%E9%AB%94%E5%86%85%E5%B9%B3%E8%A1%A1" title="體内平衡">體内平衡</a></li> <li><a href="/wiki/%E8%87%AA%E7%94%9F%E7%B3%BB%E7%B5%B1" title="自生系統">自生系統</a>同<a href="/wiki/%E8%87%AA%E6%88%91%E8%A4%87%E8%A3%BD" title="自我複製">自我複製</a></li> <li><a href="/wiki/%E5%A4%9A%E7%A9%A9%E6%85%8B" title="多穩態">多穩態</a></li> <li><a href="/wiki/%E7%B3%BB%E7%B5%B1%E5%8B%95%E5%8A%9B%E5%AD%B8" title="系統動力學">系統動力學</a></li> <li><a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>相關</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%81%8B%E7%AE%97%E8%A4%87%E9%9B%9C%E5%BA%A6" class="mw-redirect" title="運算複雜度">運算複雜度</a></li> <li><a class="mw-selflink selflink">電腦模擬</a>（<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本模型</a>）</li> <li><a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>（<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="人工神經網絡">人工神經網絡</a>）</li> <li><a href="/wiki/%E9%80%B2%E5%8C%96%E9%81%8B%E7%AE%97" title="進化運算">進化運算</a>（<a href="/wiki/%E9%81%BA%E5%82%B3%E6%BC%94%E7%AE%97%E6%B3%95" title="遺傳演算法">遺傳演算法</a>）</li></ul> <ul><li><a href="/wiki/%E6%9C%80%E4%BD%B3%E5%8C%96" title="最佳化">最佳化</a>（<a href="/wiki/%E5%85%A8%E5%B1%80%E6%9C%80%E4%BD%B3%E5%8C%96" title="全局最佳化">全局最佳化</a> · <a href="/wiki/%E7%B2%92%E5%AD%90%E7%BE%A4%E6%9C%80%E4%BD%B3%E5%8C%96" title="粒子群最佳化">粒子群最佳化</a> · <a href="/wiki/%E8%9F%BB%E7%BE%A4%E6%9C%80%E4%BD%B3%E5%8C%96%E6%BC%94%E7%AE%97%E6%B3%95" title="蟻群最佳化演算法">蟻群最佳化演算法</a>）</li> <li><a href="/wiki/%E6%A0%BC%E4%BB%94%E8%87%AA%E5%8B%95%E6%A9%9F" title="格仔自動機">格仔自動機</a>（<a href="/wiki/%E7%94%9F%E5%91%BD%E6%A3%8B" title="生命棋">生命棋</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">相關領域</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%A7%91%E5%AD%B8" title="系統科學">系統科學</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></li> <li><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a></li> <li><a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a></li> <li><a href="/wiki/%E5%9C%B0%E7%90%83%E7%A7%91%E5%AD%B8" title="地球科學">地球科學</a>（<a href="/wiki/%E5%9C%B0%E5%BD%A2%E5%AD%B8" title="地形學">地形學</a>）</li> <li><a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8" title="生物學">生物學</a>（<a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%94%9F%E7%89%A9%E5%AD%B8" title="系統生物學">系統生物學</a> · <a href="/wiki/%E9%80%B2%E5%8C%96%E7%99%BC%E8%82%B2%E7%94%9F%E7%89%A9%E5%AD%B8" title="進化發育生物學">進化發育生物學</a>）</li></ul> <ul><li><a href="/wiki/%E9%80%B2%E5%8C%96%E5%8D%9A%E5%BC%88%E8%AB%96" title="進化博弈論">進化博弈論</a></li> <li><a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>（<a href="/wiki/%E7%90%86%E6%80%A7%E9%81%B8%E6%93%87%E7%90%86%E8%AB%96" title="理性選擇理論">理性選擇理論</a>）</li> <li><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a></li> <li><a href="/wiki/%E6%8E%A7%E5%88%B6%E7%90%86%E8%AB%96" title="控制理論">控制理論</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a></li> <li><a href="/wiki/%E9%80%B2%E5%8C%96%E6%A9%9F%E6%A2%B0%E4%BA%BA%E5%AD%B8" title="進化機械人學">進化機械人學</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%94%9F%E5%91%BD" title="人工生命">人工生命</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>拉雜相關 <a href="/wiki/%E5%A5%A7%E5%9D%8E%E5%89%83%E5%88%80" title="奧坎剃刀">奧坎剃刀</a>、<a href="/wiki/%E8%AA%BF%E7%AF%80%E6%95%88%E6%87%89" class="mw-redirect" title="調節效應">調節效應</a>同<a href="/wiki/%E4%B8%8D%E5%85%85%E5%88%86%E6%B1%BA%E5%AE%9A%E8%AB%96" title="不充分決定論">不充分決定論</a> · <a href="/wiki/%E7%A2%8E%E5%BD%A2" title="碎形">碎形</a> · <a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a> · <a href="/wiki/%E7%86%B5" title="熵">熵</a> · <a href="/wiki/%E8%80%97%E6%95%A3%E7%B3%BB%E7%B5%B1" title="耗散系統">耗散系統</a> · <a href="/wiki/%E6%BB%B2%E6%B5%81" title="滲流">滲流</a> · <a href="/wiki/%E6%9C%89%E9%99%90%E7%90%86%E6%80%A7" title="有限理性">有限理性</a> · <a href="/wiki/%E7%AA%AE%E8%88%89%E6%90%9C%E5%B0%8B" title="窮舉搜尋">窮舉搜尋</a> · <a href="/wiki/%E7%94%9F%E6%88%90%E8%97%9D%E8%A1%93" title="生成藝術">生成藝術</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: 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