Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in designing numerical schemes that scale in dimension. This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution.

翻译：设计高效且精确的高维偏微分方程数值求解器仍是计算科学与工程领域具有挑战性的重要课题，其主要困难在于设计随维度扩展的数值格式时面临的"维度灾难"。本文提出一种新方法，在具有有限解析表达式的函数空间中寻找偏微分方程的近似解，故将该方法命名为有限表达式方法（FEX）。逼近理论证明，FEX可避免维度灾难。作为概念验证，本文提出深度强化学习方法实现FEX，针对不同维度的多种高维偏微分方程，该方法在多项式量级的内存复杂度与适宜的时间复杂度下，实现了高精度乃至机器精度。具有有限解析表达式的近似解还能为真实偏微分方程解提供可解释的见解，进而有助于深化对物理系统的理解，并为精化解设计后处理技术。