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Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

arXiv 2607.24726 ↗First posted 27 Jul 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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 rightarrow ∞.

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References · 10

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  3. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  4. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  5. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  6. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  7. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
  8. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  9. 2022Neural Q-learning for solving PDEscited by 8
  10. 2024Deep Backward and Galerkin Methods for the Finite State Master Equationcited by 6

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