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

2026-01-19

AI Summary

This work addresses the long-standing challenge of the curse of dimensionality in numerically solving high-dimensional parabolic partial differential equations (PDEs), where traditional grid-based methods fail to scale. The study systematically categorizes existing deep learning approaches into three distinct paradigms: physics-informed neural networks (PINNs) based on PDE residuals, stochastic methods rooted in the Feynman–Kac formula and backward stochastic differential equations (BSDEs), and derivative-free hybrid stochastic finite difference schemes. By establishing a unified framework, the paper clarifies the conceptual relationships and applicability boundaries among these paradigms and provides a comprehensive evaluation of their respective strengths and limitations. The scalability, accuracy, and practicality of the reviewed methods are rigorously demonstrated on benchmark problems—including Hamilton–Jacobi–Bellman and Black–Scholes equations—up to 1,000 dimensions.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

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.

Problem

• high-dimensional PDEs
• curse of dimensionality
• parabolic partial differential equations
• numerical solution

Innovation

• deep neural networks
• high-dimensional PDEs
• physics-informed neural networks
• Feynman–Kac formula
• curse of dimensionality