Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

2026-07-27

AI Summary

This work addresses the challenge that deep Galerkin methods (DGM) and physics-informed neural networks (PINNs) often converge to spurious local minima when solving nonlinear partial differential equations (PDEs). Focusing on a class of semilinear PDEs, the authors develop a residual minimization framework based on deep neural networks and stochastic gradient descent. Through rigorous theoretical analysis, they establish, for the first time, that under suitable conditions this framework guarantees global convergence to the true solution. This result provides the first global convergence guarantee for DGM and PINN approaches in non-convex optimization settings for nonlinear PDEs, significantly strengthening the mathematical foundation and reliability of scientific machine learning methodologies.

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. For a class of semi-linear PDEs (nonlinear in the solution and its first derivative), we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution.

Problem

• Global Convergence
• Nonlinear PDEs
• Deep Galerkin Method
• Physics Informed Neural Networks
• Non-convex Optimization

Innovation

• global convergence
• Deep Galerkin Method
• Physics Informed Neural Networks
• nonlinear PDEs
• gradient descent