A Deep Learning Framework for High-Dimensional Partial Differential Equations

Abstract

Partial Differential Equations (PDEs) are fundamental to modeling various physical, biological, and financial systems. However, solving high-dimensional PDEs remains a significant challenge due to the curse of dimensionality. Traditional numerical methods, such as finite difference and finite element methods, struggle with scalability and computational efficiency in high dimensions. This paper presents a novel deep learning framework for solving high-dimensional PDEs, leveraging the expressive power of neural networks to approximate solutions efficiently. Our approach combines deep neural networks with stochastic gradient descent to minimize a loss function derived from the PDE and its boundary conditions. We demonstrate the effectiveness of our framework through several high-dimensional PDE examples, including the Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Fokker-Planck equation. The results show that our method outperforms traditional techniques in terms of accuracy and computational efficiency, offering a promising direction for solving complex high-dimensional PDEs in various scientific and engineering domains.