Physics-Informed Neural Networks for High-Dimensional Stochastic Differential Equations

Abstract

This paper presents a comprehensive framework for solving high-dimensional stochastic differential equations (SDEs) using Physics-Informed Neural Networks (PINNs). Traditional numerical methods for SDEs, such as Monte Carlo and finite difference schemes, suffer from the curse of dimensionality, rendering them computationally intractable for problems involving a large number of stochastic factors. PINNs offer a promising alternative by leveraging the expressive power of deep neural networks to approximate the solution of the associated Kolmogorov partial differential equation (PDE) in a mesh-free manner. The core idea is to embed the differential operator into the loss function of the neural network, thereby constraining the model to satisfy the underlying physical laws described by the SDE. We formulate the problem by transforming the SDE into a high-dimensional parabolic PDE and design a PINN architecture that effectively learns the solution landscape. The network is trained by minimizing a composite loss function that includes the PDE residual, initial conditions, and boundary conditions, sampled at a set of collocation points. We demonstrate the efficacy of this approach on several benchmark problems, including high-dimensional Black-Scholes equations and nonlinear SDEs arising in physics. Our results show that PINNs can accurately and efficiently solve SDEs in dimensions far beyond the reach of conventional methods, providing a powerful tool for uncertainty quantification, financial modeling, and computational science.