Deep Neural networks for solving high-dimensional parabolic partial differential equations

TL;DR

This paper reviews neural network-based methods for solving high-dimensional parabolic PDEs, highlighting three main paradigms, their mathematical foundations, and practical applications to problems with up to 1000 dimensions.

Contribution

It provides a comprehensive tutorial and comparison of three unifying neural network paradigms for high-dimensional PDEs, emphasizing their strengths and limitations.

Findings

Neural network methods can solve PDEs in up to 1000 dimensions.

Different paradigms show varying effectiveness depending on the problem.

Benchmark problems demonstrate the scalability and accuracy of these methods.

Abstract

The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh free alternative, enabling the approximation of PDE solutions in tens to thousands of dimensions. This review provides a tutorial--oriented introduction to neural--network--based methods for solving high dimensional parabolic PDEs, emphasizing conceptual clarity and methodological connections. We organize the literature around three unifying paradigms: (i) PDE residual--based approaches, including physicsinformed neural networks and their high dimensional variants; (ii) stochastic methods derived from Feynman--Kac and backward stochastic differential equation formulations; and (iii) hybrid derivative--free random difference approaches designed to alleviate the computational cost of derivatives in high dimensions. For each paradigm, we outline the underlying mathematical formulation, algorithmic implementation, and practical strengths and limitations. Representative benchmark problems--including Hamilton--Jacobi--Bellman and Black--Scholes equations in up to 1000 dimensions --illustrate the scalability, effectiveness, and accuracy of the methods. The paper concludes with a discussion of open challenges and future directions for reliable and scalable solvers of high dimensional PDEs.