# Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations

arXiv: https://arxiv.org/abs/2305.06000 · HTML: https://mlsanity.com/paper/2305.06000/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 18 references that are in the map.
- First posted on arXiv 10 May 2023.

## Builds on (most cited first, 6 of 6)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Asymptotics of Reinforcement Learning with Neural Networks](https://mlsanity.com/paper/1911.07304/) (2019)

## Built on by (most cited first, 10 of 10)

- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression](https://mlsanity.com/paper/2410.17796/) (2024)
- [Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies](https://mlsanity.com/paper/2507.06428/) (2025)
- [Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems](https://mlsanity.com/paper/2405.13346/) (2024)
- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Enforcing Dirichlet Boundary Conditions in Operator Learning](https://mlsanity.com/paper/2608.27256/) (2026)
- [Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions](https://mlsanity.com/paper/2407.20741/) (2024)
- [Error Analysis of Deep PDE Solvers for Option Pricing](https://mlsanity.com/paper/2505.05121/) (2025)
- [Convergence of the generalization error for deep gradient flow methods for PDEs](https://mlsanity.com/paper/2512.25017/) (2025)
- [Man, Machine, and Mathematics](https://mlsanity.com/paper/2604.27052/) (2026)

## Abstract (arXiv)

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A variety of deep learning methods have been recently developed to try and solve high-dimensional PDEs by approximating the solution using a neural network. In this paper, we prove global convergence for one of the commonly-used deep learning algorithms for solving PDEs, the Deep Galerkin Method (DGM). DGM trains a neural network approximator to solve the PDE using stochastic gradient descent. We prove that, as the number of hidden units in the single-layer network goes to infinity (i.e., in the ``wide network limit"), the trained neural network converges to the solution of an infinite-dimensional linear ordinary differential equation (ODE). The PDE residual of the limiting approximator converges to zero as the training time rightarrow ∞. Under mild assumptions, this convergence also implies that the neural network approximator converges to the solution of the PDE. A closely related class of deep learning methods for PDEs is Physics Informed Neural Networks (PINNs). Using the same mathematical techniques, we can prove a similar global convergence result for the PINN neural network approximators. Both proofs require analyzing a kernel function in the limit ODE governing the evolution of the limit neural network approximator. A key technical challenge is that the kernel function, which is a composition of the PDE operator and the neural tangent kernel (NTK) operator, lacks a spectral gap, therefore requiring a careful analysis of its properties.