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.

翻译：高维偏微分方程的数值求解深受维度灾难的制约，使得基于网格的经典方法在维度略高时即不具实用性。近年来，深度神经网络作为一种无网格的替代方案崭露头角，能够近似求解数十至数千维的偏微分方程。本综述以教程为导向，介绍了基于神经网络求解高维抛物型偏微分方程的方法，强调概念清晰性与方法论关联。我们围绕三个统一范式组织文献：(i) 基于PDE残差的方法，包括物理信息神经网络及其高维变体；(ii) 源自Feynman-Kac公式与倒向随机微分方程表述的随机方法；(iii) 旨在缓解高维导数计算成本的、免导数的混合随机差分方法。针对每种范式，我们概述了其基本数学表述、算法实现以及实际优势与局限。代表性基准问题——包括维度高达1000的Hamilton-Jacobi-Bellman方程与Black-Scholes方程——展示了这些方法的可扩展性、有效性与准确性。本文最后讨论了高维偏微分方程可靠且可扩展求解器所面临的开放性挑战与未来发展方向。