Adaptive Numerical Solution Algorithm for High-Dimensional SPDEs Based on the Feynman-Kac Formula and Neural Networks

Abstract

The Feynman-Kac formula establishes a connection between stochastic processes and partial differential equations (PDEs), providing a novel approach for the numerical solution of PDEs by expressing the solution of a PDE as the expected value of a random variable. Moreover, this formula can be used to solve for the expected solution of stochastic partial differential equations (SPDEs). However, in practical applications, the formula is often constrained by the range of values that the random variable can take. When the values of the random variable are too large or too small, the numerical stability of the formula is reduced, and it may even become unusable. To address this issue, this paper proposes an adaptive algorithm that dynamically adjusts relevant parameters, enabling the formula to be applied to a broader range of situations. Furthermore, this adaptive algorithm is applied to high-dimensional stochastic partial differential equations, and a neural network algorithm is used to fit the expected solution of the SPDE. Experimental results show that, compared to traditional polynomial regression methods, this approach demonstrates higher precision and stability in high-dimensional problems. The adaptive algorithm proposed in this paper provides a novel approach for solving high-dimensional stochastic partial differential equations, with significant theoretical and practical value.