标题： 一种求解高维分界问题的有限表达式方法 显示英文标题

标题： A Finite Expression Method for Solving High-Dimensional Committor Problems

摘要： 过渡路径理论（TPT）是一个数学框架，用于量化在一对选定的稳定状态$A$和$B$之间的稀有过渡事件。 TPT 的核心是传递函数，它描述了从相空间中的任何给定起始点出发，在到达稳定状态$A$之前到达稳定状态$B$的概率。 一旦计算出传递函数，就可以轻松找到过渡通道和过渡速率。 传递函数是带有适当边界条件的后向柯尔莫哥洛夫方程的解。 然而，由于需要对环境空间的整个区域进行网格划分，因此在高维情况下求解是一项具有挑战性的任务。 在本工作中，我们探索了有限表达式方法（FEX，Liang 和 Yang（2022））作为一种计算传递函数的工具。 FEX 通过一个涉及固定有限数量的非线性函数和二进制算术运算的代数表达式来近似传递函数。 表达式模板中的最优非线性函数、二进制运算以及数值系数是通过强化学习找到的。 基于 FEX 的传递函数求解器在多个高维基准问题上进行了测试。 它的结果与基于神经网络的求解器相当或更好。 最重要的是，FEX 能够正确识别解的代数结构，这使得可以将传递函数问题简化为低维问题，并以任何所需的精度找到传递函数。 显示英文摘要

摘要： Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states $A$ and $B$. Central to TPT is the committor function, which describes the probability to hit the metastable state $B$ prior to $A$ from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.