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

2024-10-01

arXiv.org

AI Summary

Solving high-dimensional partial integro-differential equations (PIDEs) numerically remains challenging due to computational intractability and poor interpretability. To address this, we propose FEX-PG, a finite-expression method featuring a novel parameter grouping (PG) strategy that drastically reduces the number of trainable coefficients required for high-dimensional function approximation. FEX-PG explicitly approximates nonlocal integral terms via truncated Taylor series, balancing computational efficiency with enhanced accuracy. The method yields compact, physically meaningful explicit analytical solutions. Evaluated on multiple high-dimensional benchmark PIDEs, FEX-PG achieves relative errors on the order of single-precision machine epsilon (~1×10⁻⁷), substantially outperforming conventional finite element and finite difference methods as well as state-of-the-art deep learning approaches. Thus, FEX-PG simultaneously delivers high accuracy, strong interpretability, and computational feasibility.

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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.

Problem

• Solving high-dimensional partial integro-differential equations efficiently
• Reducing coefficients via novel parameter grouping method
• Improving integral term evaluation with Taylor series approximation

Innovation

• Parameter grouping reduces high-dimensional coefficients
• Taylor series enhances integral term efficiency
• FEX-PG provides interpretable explicit equation solutions