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

Journal Article · · Communications in Computational Physics
 [1];  [2];  [2]
  1. Brown Univ., Providence, RI (United States); Brown University
  2. Brown Univ., Providence, RI (United States)
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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 C0. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes H1. 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.
Research Organization:
Brown Univ., Providence, RI (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
SC0019453
OSTI ID:
2281993
Journal Information:
Communications in Computational Physics, Journal Name: Communications in Computational Physics Journal Issue: 5 Vol. 28; ISSN 1815-2406
Publisher:
Global Science PressCopyright Statement
Country of Publication:
United States
Language:
English

References (1)

Approximation theory of the MLP model in neural networks journal January 1999

Cited By (4)

Error estimates of residual minimization using neural networks for linear PDEs preprint January 2020
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks text January 2020
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations preprint January 2021
On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networks text January 2022

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Related Subjects

97 MATHEMATICS AND COMPUTING
Hölder regularization
Schauder approach
convergence
elliptic and parabolic PDEs
physics informed neural networks