Numerical Solution Method of PDE Based on Neural Network and Feynman-Kac Formula

Abstract

Aiming at the computational difficulty of partial differential equations (PDEs) in high-dimensional problems, this paper proposes a new numerical solution to the uncertainty quantification problem based on the Feynman-Kac formula and probability statistics to effectively deal with the computational difficulty of traditional methods in high-dimensional problems. The curse of dimensionality in the scene and the accuracy limitations in low-dimensional scenes. Specifically, this paper innovatively designs a sampling strategy, which improves the representativeness and statistical reliability of the sample set by increasing the number of sampling points and reducing the number of single-point simulations. At the same time, the polynomial regression model is used to replace the traditional interpolation method to avoid the overfitting problem, and the neural network is combined to deal with the complexity of high-dimensional nonlinear problems. Numerical experiments show that the improved method significantly reduces the error in both low-dimensional and high-dimensional cases, and the relative error of the neural network in high-dimensional problems is kept below 2%, which is better than the polynomial regression model. This study provides new ideas for the numerical solution and uncertainty quantification of high-dimensional partial differential equations and demonstrates its significant advantages in low-dimensional high precision and high-dimensional high efficiency.