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On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs

On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
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Year:    2020

Author:    Yeonjong Shin, Jérôme Darbon, George Em Karniadakis

Communications in Computational Physics, Vol. 28 (2020), Iss. 5 : pp. 2042–2074

Abstract

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.
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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0193

Communications in Computational Physics, Vol. 28 (2020), Iss. 5 : pp. 2042–2074

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    33

Keywords:    Physics informed neural networks convergence Hölder regularization elliptic and parabolic PDEs Schauder approach.

Author Details

Yeonjong Shin

Jérôme Darbon

George Em Karniadakis

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    Bulletin of Mathematical Biology, Vol. 86 (2024), Iss. 11

    https://doi.org/10.1007/s11538-024-01357-2 [Citations: 0]

  78. Generalization of PINNs for elliptic interface problems

    Jiang, Xuelian | Wang, Ziming | Bao, Wei | Xu, Yingxiang

    Applied Mathematics Letters, Vol. 157 (2024), Iss. P.109175

    https://doi.org/10.1016/j.aml.2024.109175 [Citations: 0]

  79. A metalearning approach for Physics-Informed Neural Networks (PINNs): Application to parameterized PDEs

    Penwarden, Michael | Zhe, Shandian | Narayan, Akil | Kirby, Robert M.

    Journal of Computational Physics, Vol. 477 (2023), Iss. P.111912

    https://doi.org/10.1016/j.jcp.2023.111912 [Citations: 20]

  80. Self-adaptive loss balanced Physics-informed neural networks

    Xiang, Zixue | Peng, Wei | Liu, Xu | Yao, Wen

    Neurocomputing, Vol. 496 (2022), Iss. P.11

    https://doi.org/10.1016/j.neucom.2022.05.015 [Citations: 163]

  81. Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

    Mouratidou, Aliki D. | Drosopoulos, Georgios A. | Stavroulakis, Georgios E.

    Acta Mechanica, Vol. 235 (2024), Iss. 11 P.6703

    https://doi.org/10.1007/s00707-024-04053-3 [Citations: 0]

  82. Physics-Informed Neural Networks With Weighted Losses by Uncertainty Evaluation for Accurate and Stable Prediction of Manufacturing Systems

    Hua, Jiaqi | Li, Yingguang | Liu, Changqing | Wan, Peng | Liu, Xu

    IEEE Transactions on Neural Networks and Learning Systems, Vol. 35 (2024), Iss. 8 P.11064

    https://doi.org/10.1109/TNNLS.2023.3247163 [Citations: 6]

  83. Physics-informed deep learning of rate-and-state fault friction

    Rucker, Cody | Erickson, Brittany A.

    Computer Methods in Applied Mechanics and Engineering, Vol. 430 (2024), Iss. P.117211

    https://doi.org/10.1016/j.cma.2024.117211 [Citations: 2]

  84. Enhanced physics-informed neural networks with Augmented Lagrangian relaxation method (AL-PINNs)

    Son, Hwijae | Cho, Sung Woong | Hwang, Hyung Ju

    Neurocomputing, Vol. 548 (2023), Iss. P.126424

    https://doi.org/10.1016/j.neucom.2023.126424 [Citations: 16]

  85. A Gentle Introduction to Physics-Informed Neural Networks, with Applications in Static Rod and Beam Problems

    Katsikis, Dimitrios | Muradova, Aliki D. | Stavroulakis, Georgios E.

    Journal of Advances in Applied & Computational Mathematics, Vol. 9 (2022), Iss. P.103

    https://doi.org/10.15377/2409-5761.2022.09.8 [Citations: 9]

  86. Physics-Informed neural network solver for numerical analysis in geoengineering

    Chen, Xiao-Xuan | Zhang, Pin | Yin, Zhen-Yu

    Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, Vol. 18 (2024), Iss. 1 P.33

    https://doi.org/10.1080/17499518.2024.2315301 [Citations: 1]

  87. An artificial neural network approach to bifurcating phenomena in computational fluid dynamics

    Pichi, Federico | Ballarin, Francesco | Rozza, Gianluigi | Hesthaven, Jan S.

    Computers & Fluids, Vol. 254 (2023), Iss. P.105813

    https://doi.org/10.1016/j.compfluid.2023.105813 [Citations: 32]

  88. Boundary-safe PINNs extension: Application to non-linear parabolic PDEs in counterparty credit risk

    P. Villarino, Joel | Leitao, Álvaro | García Rodríguez, J.A.

    Journal of Computational and Applied Mathematics, Vol. 425 (2023), Iss. P.115041

    https://doi.org/10.1016/j.cam.2022.115041 [Citations: 0]

  89. A novel physics-informed deep learning strategy with local time-updating discrete scheme for multi-dimensional forward and inverse consolidation problems

    Guo, Hongwei | Yin, Zhen-Yu

    Computer Methods in Applied Mechanics and Engineering, Vol. 421 (2024), Iss. P.116819

    https://doi.org/10.1016/j.cma.2024.116819 [Citations: 6]

  90. Physical informed neural network for thermo-hydral analysis of fire-loaded concrete

    Gao, Zhiran | Fu, Zhuojia | Wen, Minjie | Guo, Yuan | Zhang, Yiming

    Engineering Analysis with Boundary Elements, Vol. 158 (2024), Iss. P.252

    https://doi.org/10.1016/j.enganabound.2023.10.027 [Citations: 3]

  91. The Deep Learning Galerkin Method for the General Stokes Equations

    Li, Jian | Yue, Jing | Zhang, Wen | Duan, Wansuo

    Journal of Scientific Computing, Vol. 93 (2022), Iss. 1

    https://doi.org/10.1007/s10915-022-01930-8 [Citations: 9]

  92. Physics-informed neural networks for data-driven simulation: Advantages, limitations, and opportunities

    Fernández de la Mata, Félix | Gijón, Alfonso | Molina-Solana, Miguel | Gómez-Romero, Juan

    Physica A: Statistical Mechanics and its Applications, Vol. 610 (2023), Iss. P.128415

    https://doi.org/10.1016/j.physa.2022.128415 [Citations: 10]
  93. New Trends in the Applications of Differential Equations in Sciences

    Physics Informed Cellular Neural Networks for Solving Partial Differential Equations

    Slavova, Angela | Litsyn, Elena

    2024

    https://doi.org/10.1007/978-3-031-53212-2_3 [Citations: 0]

  94. Error Analysis Based on Inverse Modified Differential Equations for Discovery of Dynamics Using Linear Multistep Methods and Deep Learning

    Zhu, Aiqing | Wu, Sidi | Tang, Yifa

    SIAM Journal on Numerical Analysis, Vol. 62 (2024), Iss. 5 P.2087

    https://doi.org/10.1137/22M152373X [Citations: 0]

  95. Neural network expression rates and applications of the deep parametric PDE method in counterparty credit risk

    Glau, Kathrin | Wunderlich, Linus

    Annals of Operations Research, Vol. 336 (2024), Iss. 1-2 P.331

    https://doi.org/10.1007/s10479-023-05315-4 [Citations: 1]

  96. Modeling the dynamics of Covid-19 in Japan: employing data-driven deep learning approach

    Nelson, S. Patrick | Raja, R. | Eswaran, P. | Alzabut, J. | Rajchakit, G.

    International Journal of Machine Learning and Cybernetics, Vol. (2024), Iss.

    https://doi.org/10.1007/s13042-024-02301-5 [Citations: 0]

  97. Variational temporal convolutional networks for I-FENN thermoelasticity

    Abueidda, Diab W. | Mobasher, Mostafa E.

    Computer Methods in Applied Mechanics and Engineering, Vol. 429 (2024), Iss. P.117122

    https://doi.org/10.1016/j.cma.2024.117122 [Citations: 1]

  98. Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

    Falini, Antonella | D’Inverno, Giuseppe Alessio | Sampoli, Maria Lucia | Mazzia, Francesca

    Mathematics, Vol. 11 (2023), Iss. 10 P.2406

    https://doi.org/10.3390/math11102406 [Citations: 3]

  99. Physics-Informed Neural Networks (PINNs) for Parameterized PDEs: A Metalearning Approach

    Penwarden, Michael | Zhe, Shandian | Narayan, Akil | Kirby, Robert M.

    SSRN Electronic Journal , Vol. (2021), Iss.

    https://doi.org/10.2139/ssrn.3965238 [Citations: 4]

  100. M‐PINN: A mesh‐based physics‐informed neural network for linear elastic problems in solid mechanics

    Wang, Lu | Liu, Guangyan | Wang, Guanglun | Zhang, Kai

    International Journal for Numerical Methods in Engineering, Vol. 125 (2024), Iss. 9

    https://doi.org/10.1002/nme.7444 [Citations: 4]

  101. Tunable complexity benchmarks for evaluating physics-informed neural networks on coupled ordinary differential equations

    New, Alexander | Eng, Benjamin | Timm, Andrea C. | Gearhart, Andrew S.

    2023 57th Annual Conference on Information Sciences and Systems (CISS), (2023), P.1

    https://doi.org/10.1109/CISS56502.2023.10089728 [Citations: 1]

  102. INN: Interfaced neural networks as an accessible meshless approach for solving interface PDE problems

    Wu, Sidi | Lu, Benzhuo

    Journal of Computational Physics, Vol. 470 (2022), Iss. P.111588

    https://doi.org/10.1016/j.jcp.2022.111588 [Citations: 13]

  103. A block-coordinate approach of multi-level optimization with an application to physics-informed neural networks

    Gratton, Serge | Mercier, Valentin | Riccietti, Elisa | Toint, Philippe L.

    Computational Optimization and Applications, Vol. 89 (2024), Iss. 2 P.385

    https://doi.org/10.1007/s10589-024-00597-1 [Citations: 0]

  104. A deep First-Order System Least Squares method for solving elliptic PDEs

    Bersetche, Francisco M. | Borthagaray, Juan Pablo

    Computers & Mathematics with Applications, Vol. 129 (2023), Iss. P.136

    https://doi.org/10.1016/j.camwa.2022.11.014 [Citations: 5]

  105. Polynomial-Spline Networks with Exact Integrals and Convergence Rates

    Actor, Jonas A. | Huang, Andy | Trask, Nat

    2022 IEEE Symposium Series on Computational Intelligence (SSCI), (2022), P.1156

    https://doi.org/10.1109/SSCI51031.2022.10022123 [Citations: 0]
  106. Computational Science – ICCS 2023

    Fixed-Budget Online Adaptive Learning for Physics-Informed Neural Networks. Towards Parameterized Problem Inference

    Nguyen, Thi Nguyen Khoa | Dairay, Thibault | Meunier, Raphaël | Millet, Christophe | Mougeot, Mathilde

    2023

    https://doi.org/10.1007/978-3-031-36027-5_36 [Citations: 0]
  107. Advances in Computational Modeling and Simulation

    Efficient Physics Informed Neural Networks Coupled with Domain Decomposition Methods for Solving Coupled Multi-physics Problems

    Nguyen, Long | Raissi, Maziar | Seshaiyer, Padmanabhan

    2022

    https://doi.org/10.1007/978-981-16-7857-8_4 [Citations: 4]

  108. Solving nonlinear soliton equations using improved physics-informed neural networks with adaptive mechanisms

    Guo, Yanan | Cao, Xiaoqun | Peng, Kecheng

    Communications in Theoretical Physics, Vol. 75 (2023), Iss. 9 P.095003

    https://doi.org/10.1088/1572-9494/accb8d [Citations: 8]

  109. On Physics-Informed Neural Networks training for coupled hydro-poromechanical problems

    Millevoi, Caterina | Spiezia, Nicolò | Ferronato, Massimiliano

    Journal of Computational Physics, Vol. 516 (2024), Iss. P.113299

    https://doi.org/10.1016/j.jcp.2024.113299 [Citations: 1]

  110. Acceleration of Numerical Modeling of Uranium In Situ Leaching: Application of IDW Interpolation and Neural Networks for Solving the Hydraulic Head Equation

    Kurmanseiit, Maksat B. | Tungatarova, Madina S. | Abdullayeva, Banu Z. | Aizhulov, Daniar Y. | Shayakhmetov, Nurlan M.

    Minerals, Vol. 14 (2024), Iss. 10 P.1043

    https://doi.org/10.3390/min14101043 [Citations: 0]

  111. Physics-informed neural networks for learning fluid flows with symmetry

    Kim, Younghyeon | Kwak, Hyungyeol | Nam, Jaewook

    Korean Journal of Chemical Engineering, Vol. 40 (2023), Iss. 9 P.2119

    https://doi.org/10.1007/s11814-023-1420-4 [Citations: 3]

  112. Approximating Fracture Paths in Random Heterogeneous Materials: A Probabilistic Learning Perspective

    Quek, Ariana | Yi Yong, Jin | Guilleminot, Johann

    Journal of Engineering Mechanics, Vol. 150 (2024), Iss. 8

    https://doi.org/10.1061/JENMDT.EMENG-7617 [Citations: 1]

  113. A mechanism informed neural network for predicting machining deformation of annular parts

    Ni, Yang | Li, Yingguang | Liu, Changqing | Liu, Xu

    Advanced Engineering Informatics, Vol. 53 (2022), Iss. P.101661

    https://doi.org/10.1016/j.aei.2022.101661 [Citations: 5]

  114. Deep learning methods for partial differential equations and related parameter identification problems

    Nganyu Tanyu, Derick | Ning, Jianfeng | Freudenberg, Tom | Heilenkötter, Nick | Rademacher, Andreas | Iben, Uwe | Maass, Peter

    Inverse Problems, Vol. 39 (2023), Iss. 10 P.103001

    https://doi.org/10.1088/1361-6420/ace9d4 [Citations: 9]

  115. Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

    Demir, Kubilay Timur | Logemann, Kai | Greenberg, David S.

    Mathematics, Vol. 12 (2024), Iss. 21 P.3315

    https://doi.org/10.3390/math12213315 [Citations: 0]

  116. Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations

    Goraya, Shoaib | Sobh, Nahil | Masud, Arif

    Computational Mechanics, Vol. 72 (2023), Iss. 2 P.267

    https://doi.org/10.1007/s00466-023-02334-7 [Citations: 5]

  117. Physics-Informed Neural Network (PINN) Evolution and Beyond: A Systematic Literature Review and Bibliometric Analysis

    Lawal, Zaharaddeen Karami | Yassin, Hayati | Lai, Daphne Teck Ching | Che Idris, Azam

    Big Data and Cognitive Computing, Vol. 6 (2022), Iss. 4 P.140

    https://doi.org/10.3390/bdcc6040140 [Citations: 40]

  118. ERROR ESTIMATES OF RESIDUAL MINIMIZATION USING NEURAL NETWORKS FOR LINEAR PDES

    Shin, Yeonjong | Zhang, Zhongqiang | Karniadakis, George Em

    Journal of Machine Learning for Modeling and Computing, Vol. 4 (2023), Iss. 4 P.73

    https://doi.org/10.1615/JMachLearnModelComput.2023050411 [Citations: 13]

  119. Isogeometric neural networks: A new deep learning approach for solving parameterized partial differential equations

    Gasick, Joshua | Qian, Xiaoping

    Computer Methods in Applied Mechanics and Engineering, Vol. 405 (2023), Iss. P.115839

    https://doi.org/10.1016/j.cma.2022.115839 [Citations: 7]

  120. A Deep Fourier Residual method for solving PDEs using Neural Networks

    Taylor, Jamie M. | Pardo, David | Muga, Ignacio

    Computer Methods in Applied Mechanics and Engineering, Vol. 405 (2023), Iss. P.115850

    https://doi.org/10.1016/j.cma.2022.115850 [Citations: 12]

  121. Learning thermoacoustic interactions in combustors using a physics-informed neural network

    Mariappan, Sathesh | Nath, Kamaljyoti | Em Karniadakis, George

    Engineering Applications of Artificial Intelligence, Vol. 138 (2024), Iss. P.109388

    https://doi.org/10.1016/j.engappai.2024.109388 [Citations: 1]

  122. Improved multi-scale fusion network for solving non-smooth elliptic interface problems with applications

    Ying, Jinyong | Li, Jiao | Liu, Qiong | Chen, Yinghao

    Applied Mathematical Modelling, Vol. 132 (2024), Iss. P.274

    https://doi.org/10.1016/j.apm.2024.04.039 [Citations: 0]

  123. CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method

    Chiu, Pao-Hsiung | Wong, Jian Cheng | Ooi, Chinchun | Dao, My Ha | Ong, Yew-Soon

    Computer Methods in Applied Mechanics and Engineering, Vol. 395 (2022), Iss. P.114909

    https://doi.org/10.1016/j.cma.2022.114909 [Citations: 108]

  124. Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

    Fan, Chen | Ali, Muhammad Aamir | Zhang, Zhiyue

    Computer Physics Communications, Vol. 303 (2024), Iss. P.109275

    https://doi.org/10.1016/j.cpc.2024.109275 [Citations: 0]

  125. Novel prediction of fluid forces on obstacle in a periodic flow regime using hybrid FEM-ANN simulations

    Mahmood, Rashid | Majeed, Afraz Hussain | Shahzad, Hasan | Khan, Ilyas

    The European Physical Journal Plus, Vol. 138 (2023), Iss. 8

    https://doi.org/10.1140/epjp/s13360-023-04225-5 [Citations: 6]

  126. Phase space approach to solving higher order differential equations with artificial neural networks

    Tori, Floriano | Ginis, Vincent

    Physical Review Research, Vol. 4 (2022), Iss. 4

    https://doi.org/10.1103/PhysRevResearch.4.043090 [Citations: 1]

  127. A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations

    Fatone, L. | Funaro, D. | Manzini, G.

    Journal of Scientific Computing, Vol. 92 (2022), Iss. 1

    https://doi.org/10.1007/s10915-022-01853-4 [Citations: 1]

  128. A framework based on symbolic regression coupled with eXtended Physics-Informed Neural Networks for gray-box learning of equations of motion from data

    Kiyani, Elham | Shukla, Khemraj | Karniadakis, George Em | Karttunen, Mikko

    Computer Methods in Applied Mechanics and Engineering, Vol. 415 (2023), Iss. P.116258

    https://doi.org/10.1016/j.cma.2023.116258 [Citations: 9]

  129. Optimization of Physics-Informed Neural Networks for Solving the Nolinear Schrödinger Equation

    Chuprov, I. | Gao, Jiexing | Efremenko, D. | Kazakov, E. | Buzaev, F. | Zemlyakov, V.

    Doklady Mathematics, Vol. 108 (2023), Iss. S2 P.S186

    https://doi.org/10.1134/S1064562423701120 [Citations: 0]

  130. A mutually embedded perception model for solar corona

    Zhao, Jingmin | Feng, Xueshang | Xiang, Changqing | Jiang, Chaowei

    Monthly Notices of the Royal Astronomical Society, Vol. 523 (2023), Iss. 1 P.1577

    https://doi.org/10.1093/mnras/stad1516 [Citations: 4]

  131. Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs

    Mojgani, Rambod | Balajewicz, Maciej | Hassanzadeh, Pedram

    Computer Methods in Applied Mechanics and Engineering, Vol. 404 (2023), Iss. P.115810

    https://doi.org/10.1016/j.cma.2022.115810 [Citations: 21]
  132. Materials Data Science

    Advanced Deep Learning Architectures and Techniques

    Sandfeld, Stefan

    2024

    https://doi.org/10.1007/978-3-031-46565-9_19 [Citations: 0]

  133. Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

    Yoo, Jihahm | Lee, Haesung

    AIMS Mathematics, Vol. 9 (2024), Iss. 10 P.27000

    https://doi.org/10.3934/math.20241314 [Citations: 0]

  134. Weight initialization algorithm for physics-informed neural networks using finite differences

    Tarbiyati, Homayoon | Nemati Saray, Behzad

    Engineering with Computers, Vol. 40 (2024), Iss. 3 P.1603

    https://doi.org/10.1007/s00366-023-01883-y [Citations: 4]