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：在计算科学和工程中,设计高维部分差异方程式(PDEs)的高效和准确的数字解析器仍然是一个具有挑战性和重要性的专题,这主要是因为在设计规模尺度的数值方法时“维度的极限”,本文件采用了一种新的方法,以有限的多种分析表达方式在函数空间寻求大致的PDE解决方案,因此,这种方法被称为有限表达法(FEX),近似理论证明FEX可以避免维度的诅咒。作为概念的证明,建议了一种深强化学习方法,为不同维度的不同PDEs实施FEX,实现高甚至机器的精度,其内存复杂度为多元性和易变时间复杂度。一个有有限分析表达方式的近似解决方案也提供了可解释的地面真理PDE解决方案,这可以进一步帮助增进对物理系统的了解,并设计精细的解决方案的后处理技术。