Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method

TL;DR

This paper presents an advanced finite expression method (FEX) with parameter grouping and Taylor series approximation to efficiently and accurately solve high-dimensional partial integro-differential equations, providing explicit and interpretable solutions.

Contribution

The paper introduces FEX-PG, a novel high-dimensional PIDE solver combining parameter grouping and Taylor approximation for improved efficiency and interpretability.

Findings

Achieves relative errors near machine epsilon in high dimensions.

Provides explicit, interpretable solutions unlike traditional methods.

Demonstrates robustness and high accuracy on benchmark PIDEs.

Abstract

In this paper, we introduce a new finite expression method (FEX) to solve high-dimensional partial integro-differential equations (PIDEs). This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon.