Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

Abstract

The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time → ∞.

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BibTeX RIS

BibTeX

@misc{sirignano2026global,
  title = {{Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs}},
  author = {Justin Sirignano and Konstantinos Spiliopoulos and Samuel Cohen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.24726},
  eprint = {2607.24726},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24726},
}

RIS

TY  - UNPB
TI  - Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
AU  - Justin Sirignano
AU  - Konstantinos Spiliopoulos
AU  - Samuel Cohen
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2607.24726
UR  - https://arxiv.org/abs/2607.24726
AB  - The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time → ∞.
ER  -

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