新方法高效求解高维积分微分方程，结果可解释且精度接近极限。
Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method
- 通过参数分组减少高维函数近似中的系数数量。
- 用泰勒展开加速积分项计算，误差逼近单精度机器精度。
- 适合需要高精度与可解释解的金融、物理建模场景。
本文提出一种新的有限表达方法（FEX），用于求解高维偏积分微分方程（PIDEs）。该方法在原始FEX基础上引入两项改进：1）提出新型参数分组策略，显著降低高维函数近似中的系数数量；2）采用泰勒级数近似法，大幅提升PIDE中积分项的计算效率与精度。改进后的算法称为FEX-PG，能提供高精度且可解释的数值解，输出为显式方程，便于理解解的内在结构。传统方法如有限元法（FEM）、有限差分法及基于深度学习的方法通常缺乏可解释性。在文献中的基准测试中，FEX-PG在高维场景下表现稳健，相对误差达到单精度机器精度量级。
原文摘要 · Abstract (English)
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.
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