Document Zbl 1473.65349

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Comput. Phys.</a> <a href="/?q=in%3A471737" title="Articles in this Issue">28, No. 5, 2042-2074 (2020)</a>. </div> <div class="abstract">Summary: Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.<br class="zbmathjax-paragraph">As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7419159">Cited in <strong>99</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N99" title="MSC2020">65N99</a> </td> <td class="space"> Numerical methods for partial differential equations, boundary value problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M99" title="MSC2020">65M99</a> </td> <td class="space"> Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M12" title="MSC2020">65M12</a> </td> <td class="space"> Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N12" title="MSC2020">65N12</a> </td> <td class="space"> Stability and convergence of numerical methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J25" title="MSC2020">35J25</a> </td> <td class="space"> Boundary value problems for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K20" title="MSC2020">35K20</a> </td> <td class="space"> Initial-boundary value problems for second-order parabolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T07" title="MSC2020">68T07</a> </td> <td class="space"> Artificial neural networks and deep learning </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aphysics+informed+neural+networks">physics informed neural networks</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3AH%C3%B6lder+regularization">Hölder regularization</a>; <a href="/?q=ut%3Aelliptic+and+parabolic+PDEs">elliptic and parabolic PDEs</a>; <a href="/?q=ut%3ASchauder+approach">Schauder approach</a></div> <div class="software"> <h3>Software:</h3><a href="/software/16033">DiffSharp</a>; <a href="/software/22205">Adam</a>; <a href="/software/3229">L-BFGS</a>; <a href="/software/32456">DeepXDE</a>; <a href="/software/39282">DGM</a>; <a href="/software/40570">FPINNs</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1473.65349" data-ciurl="/ci/07419159" data-biburl="/bibtex/07419159.bib" data-amsurl="/amsrefs/07419159.bib" data-xmlurl="/xml/07419159.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07419159.pdf" title="Zbl 1473.65349 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.4208/cicp.OA-2020-0193" aria-label="DOI for “On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs”" title="10.4208/cicp.OA-2020-0193">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2004.01806"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">H. Attouch, G. Buttazzo, and G. Michaille. Variational analysis in Sobolev and BV spaces: appli-cations to PDEs and optimization. SIAM, 2014. · <a href="/1311.49001" class="nowrap">Zbl 1311.49001</a></td> </tr><tr> <td>[2]</td> <td class="space">N. Baker, F. Alexander, T. Bremer, A. Hagberg, Y. Kevrekidis, H. Najm, M. Parashar, A. Patra, J. Sethian, S. Wild, and ET AL. Workshop report on basic research needs for scientific ma-chine learning: Core technologies for artificial intelligence. Technical report, USDOE Office of Science (SC), Washington, DC (United States), 2019.</td> </tr><tr> <td>[3]</td> <td class="space">A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind. Automatic differentiation in machine learning: a survey. The Journal of Machine Learning Research, 18(1):5595-5637, 2017. · <a href="/06982909" class="nowrap">Zbl 06982909</a></td> </tr><tr> <td>[4]</td> <td class="space">J. Berg and K. Nyström. A unified deep artificial neural network approach to partial differ-ential equations in complex geometries. Neurocomputing, 317:28-41, 2018.</td> </tr><tr> <td>[5]</td> <td class="space">L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Advances in Neural Information Processing Systems, pages 161-168, 2008.</td> </tr><tr> <td>[6]</td> <td class="space">H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, 2010. · <a href="/1218.46002" class="nowrap">Zbl 1218.46002</a></td> </tr><tr> <td>[7]</td> <td class="space">J. Calder. Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data. SIAM Journal on Mathematics of Data Science, 1(4):780-812, 2019. · <a href="/1499.35598" class="nowrap">Zbl 1499.35598</a></td> </tr><tr> <td>[8]</td> <td class="space">J. Darbon, G. P. Langlois, and T. Meng. Overcoming the curse of dimensionality for some hamilton-jacobi partial differential equations via neural network architectures. arXiv preprint arXiv:1910.09045, 2019. · <a href="/1445.35119" class="nowrap">Zbl 1445.35119</a></td> </tr><tr> <td>[9]</td> <td class="space">J. Darbon and T. Meng. On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations. arXiv preprint arXiv:2002.09750, 2020.</td> </tr><tr> <td>[10]</td> <td class="space">M. Dissanayake and N. Phan-Thien. Neural-network-based approximations for solving par-tial differential equations. Communications in Numerical Methods in Engineering, 10(3):195-201, 1994. · <a href="/0802.65102" class="nowrap">Zbl 0802.65102</a></td> </tr><tr> <td>[11]</td> <td class="space">W. E and B. Yu. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12, 2018. · <a href="/1392.35306" class="nowrap">Zbl 1392.35306</a></td> </tr><tr> <td>[12]</td> <td class="space">L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. CRC press, 2015. · <a href="/1310.28001" class="nowrap">Zbl 1310.28001</a></td> </tr><tr> <td>[13]</td> <td class="space">C. Finlay, J. Calder, B. Abbasi, and A. Oberman. Lipschitz regularized deep neural networks generalize and are adversarially robust. arXiv preprint arXiv:1808.09540, 2018.</td> </tr><tr> <td>[14]</td> <td class="space">A. Friedman. Partial differential equations of parabolic type. Courier Dover Publications, 2008.</td> </tr><tr> <td>[15]</td> <td class="space">D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Springer, 2015. · <a href="/1042.35002" class="nowrap">Zbl 1042.35002</a></td> </tr><tr> <td>[16]</td> <td class="space">X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 249-256, 2010.</td> </tr><tr> <td>[17]</td> <td class="space">P. Grohs, F. Hornung, A. Jentzen, and P. Von Wurstemberger. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of black-scholes partial differential equations. arXiv preprint arXiv:1809.02362, 2018.</td> </tr><tr> <td>[18]</td> <td class="space">J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505-8510, 2018. · <a href="/1416.35137" class="nowrap">Zbl 1416.35137</a></td> </tr><tr> <td>[19]</td> <td class="space">B. Houska and B. Chachuat. Global optimization in hilbert space. Mathematical programming, 173(1-2):221-249, 2019. · <a href="/1408.49027" class="nowrap">Zbl 1408.49027</a></td> </tr><tr> <td>[20]</td> <td class="space">A. D. Jagtap, K. Kawaguchi, and G. E. Karniadakis. Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks. arXiv preprint arXiv:1909.12228, 2019.</td> </tr><tr> <td>[21]</td> <td class="space">D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.</td> </tr><tr> <td>[22]</td> <td class="space">I. E. Lagaris, A. Likas, and D. I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on Neural Networks, 9(5):987-1000, 1998.</td> </tr><tr> <td>[23]</td> <td class="space">I. E. Lagaris, A. C. Likas, and G. D. Papageorgiou. Neural-network methods for boundary value problems with irregular boundaries. IEEE Transactions on Neural Networks, 11(5):1041-1049, 2000.</td> </tr><tr> <td>[24]</td> <td class="space">Y. LeCun, Y. Bengio, and G. Hinton. Deep learning. Nature, 521(7553):436-444, 2015.</td> </tr><tr> <td>[25]</td> <td class="space">D. C. Liu and J. Nocedal. On the limited memory bfgs method for large scale optimization. Mathematical programming, 45(1-3):503-528, 1989. · <a href="/0696.90048" class="nowrap">Zbl 0696.90048</a></td> </tr><tr> <td>[26]</td> <td class="space">L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis. Deepxde: A deep learning library for solving differential equations. arXiv preprint arXiv:1907.04502, 2019. · <a href="/1459.65002" class="nowrap">Zbl 1459.65002</a></td> </tr><tr> <td>[27]</td> <td class="space">Z. Mao, A. D. Jagtap, and G. E. Karniadakis. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360:112789, 2020. · <a href="/1442.76092" class="nowrap">Zbl 1442.76092</a></td> </tr><tr> <td>[28]</td> <td class="space">M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of machine learning. MIT press, 2018. · <a href="/1407.68007" class="nowrap">Zbl 1407.68007</a></td> </tr><tr> <td>[29]</td> <td class="space">P. Niyogi and F. Girosi. Generalization bounds for function approximation from scattered noisy data. Advances in Computational Mathematics, 10(1):51-80, 1999. · <a href="/1053.65506" class="nowrap">Zbl 1053.65506</a></td> </tr><tr> <td>[30]</td> <td class="space">G. Pang, L. Lu, and G. E. Karniadakis. fPINNs: Fractional physics-informed neural net-works. SIAM Journal on Scientific Computing, 41(4):A2603-A2626, 2019. · <a href="/1420.35459" class="nowrap">Zbl 1420.35459</a></td> </tr><tr> <td>[31]</td> <td class="space">A. Pinkus. Approximation theory of the mlp model in neural networks. Acta numerica, 8:143-195, 1999. · <a href="/0959.68109" class="nowrap">Zbl 0959.68109</a></td> </tr><tr> <td>[32]</td> <td class="space">M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019. · <a href="/1415.68175" class="nowrap">Zbl 1415.68175</a></td> </tr><tr> <td>[33]</td> <td class="space">M. Raissi, A. Yazdani, and G. E. Karniadakis. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 367(6481):1026-1030, 2020. · <a href="/1478.76057" class="nowrap">Zbl 1478.76057</a></td> </tr><tr> <td>[34]</td> <td class="space">S. Ruder. An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747, 2016.</td> </tr><tr> <td>[35]</td> <td class="space">J. Sirignano and K. Spiliopoulos. DGM: A deep learning algorithm for solving partial differ-ential equations. Journal of Computational Physics, 375:1339-1364, 2018. · <a href="/1416.65394" class="nowrap">Zbl 1416.65394</a></td> </tr><tr> <td>[36]</td> <td class="space">F. Song, G. Pange, C. Meneveau, and G. E. Karniadakis. Fractional physical-inform neural networks (fPINNs) for turbulent flows. Bulletin of the American Physical Society, 2019.</td> </tr><tr> <td>[37]</td> <td class="space">S. Wang, Y. Teng, and P. Perdikaris. Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536, 2020.</td> </tr><tr> <td>[38]</td> <td class="space">K. Yosida. Functional analysis, volume 123. springer, 1988. · <a href="/0126.11504" class="nowrap">Zbl 0126.11504</a></td> </tr><tr> <td>[39]</td> <td class="space">D. Zhang, L. Guo, and G. E. Karniadakis. Learning in modal space: Solving time-dependent stochastic pdes using physics-informed neural networks. SIAM Journal on Scientific Comput-ing, 42(2):A639-A665, 2020 · <a href="/1440.60067" class="nowrap">Zbl 1440.60067</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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