Global convergence of Deep Galerkin and PINNs methods for solving partial differential equations

Journal article

Numerically solving high-dimensional partial differential equations (PDEs) is a ma5 jor challenge. Conventional methods, such as finite difference methods, are unable to solve high6 dimensional PDEs due to the curse-of-dimensionality. This is in particular a fundamental challenge for the solution of financial models, which are often inherently high-dimensional. Option pricing, hedging, mean-field financial models, order book models, and dynamic portfolio investment can all require the solution of high-dimensional PDEs. 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. These deep learning methods have been widely applied to high-dimensional PDEs in financial engineering. 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 → ∞. 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), which has been widely-used in a variety of fields (including financial mathematics but also physics and engineering). 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.

Authors: Jiang, D; Sirignano, J; Cohen, SN

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Financial Mathematics
• Volume: 17
• Issue: 2
• Pages: 620-645
• Publication date: 2026-05-29
• Acceptance date: 2025-11-19
• DOI: 10.1137/24M1701502
• EISSN: 1945-497X
• Language: English
• Keywords: neural network; partial differential equation; neural tangent kernel; machine learning; gradient flow