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This research establishes a critical mathematical foundation for Scientific Machine Learning by proving the global convergence of Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs). The study resolves longstanding concerns about local minima in non-convex optimization, confirming that gradient descent reliably finds true solutions for semi-linear partial differential equations. This advancement strengthens confidence in using neural networks as rigorous solvers for complex physical systems without relying on traditional mesh-based methods.
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
arXiv:2607.24726v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2607.24726 [cs.LG]
(or arXiv:2607.24726v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.24726 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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