Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
2026-07-27 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how certain neural networks, like those used in the Deep Galerkin Method and Physics Informed Neural Networks, can solve complex math problems called partial differential equations (PDEs). Since the way these networks learn can get stuck in bad solutions, it wasn't clear if they always find the true solution. The authors proved that for a specific kind of PDE, their training method will reliably find the correct solution. This helps build confidence in using neural networks to solve these math problems.
Partial Differential EquationsDeep Galerkin MethodPhysics Informed Neural NetworksNeural NetworksGradient DescentNon-convex OptimizationSemi-linear PDEsPDE ResidualConvergenceScientific Machine Learning
Authors
Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen
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.