Deep Neural networks for solving high-dimensional parabolic partial differential equations
It addresses the curse of dimensionality in PDEs for researchers and practitioners in computational mathematics and machine learning, but it is incremental as it reviews existing methods rather than proposing new ones.
This review tackles the problem of solving high-dimensional parabolic partial differential equations, which are impractical with classical methods due to the curse of dimensionality, by introducing neural-network-based methods that can handle tens to thousands of dimensions, as demonstrated with benchmark problems in up to 1000 dimensions.
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.