论文标题
解决高维偏微分方程的有限表达方法
Finite Expression Method for Solving High-Dimensional Partial Differential Equations
论文作者
论文摘要
在计算科学和工程学中，设计高维偏微分方程（PDE）的高效数值求解器（PDE）仍然是一个具有挑战性且重要的主题，这主要是由于设计数字方案的“维度诅咒”，这些方案的规模规模扩展。本文介绍了一种新方法，该方法在具有有限的分析表达式的功能空间中寻求近似PDE解决方案，因此，该方法被命名为有限表达方法（FEX）。在近似理论中证明，FEX可以避免维数的诅咒。作为概念的证明，提出了一种深入的增强学习方法，以在不同维度的各种高维PDE中实现FEX，从而在维度和可观的时间复杂性方面具有内存复杂性多项式，从而实现高甚至机器的精度。具有有限分析表达式的近似解决方案还为地面真相PDE解决方案提供了可解释的见解，这可以进一步帮助促进对物理系统的理解，并为精制解决方案设计后处理技术。
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.
