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.

翻译：本文提出了一种新的有限表达式方法（FEX）用于求解高维偏积分微分方程（PIDEs）。该方法基于原始FEX及其固有优势，并引入了新的进展：1）提出了一种新颖的参数分组方法，以减少高维函数逼近中的系数数量；2）采用泰勒级数逼近方法，显著提升了PIDE积分项计算的计算效率与精度。这种基于新FEX的方法，记为FEX-PG以表明算法中加入了参数分组（PG）步骤，既能提供高精度的数值解，又能保证解的可解释性，其输出结果为显式方程，便于直观理解底层解的结构。这些特性在传统方法（如有限元法（FEM）和有限差分法）以及基于深度学习的方法中通常难以实现。为了将我们的方法与最新进展进行基准测试，我们将新的FEX-PG应用于求解文献中的基准PIDE问题。在高维设置下，FEX-PG展现出强大而稳健的性能，其相对误差达到单精度机器epsilon量级。