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

TL;DR

This survey analyzes neural-network approaches for solving high-dimensional parabolic PDEs, organizing methods into three paradigms: PDE-residual PINNs, SDE-based probabilistic representations, and random-difference derivative-free schemes. It details mathematical foundations (e.g., Feynman–Kac and Pardoux–Peng FBSDEs), algorithmic implementations, and practical trade-offs, and showcases benchmark results up to $d=1000$ for Hamilton–Jacobi–Bellman and Black–Scholes equations to illustrate scalability. Key contributions include architecture- and loss-function designs (PINN variants, Deep BSDE, DeepMartNet, DRDM/Shotgun) and variance-reduction techniques (SDGD, HTE, STDE) that address the curse of dimensionality. The article highlights open challenges in sampling strategies, boundary conditions, and computational efficiency, and outlines future directions toward robust, scalable solvers for high-dimensional PDEs with broad impact in physics, finance, and data science.

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.