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.

翻译：过渡路径理论（TPT）是一个用于量化选定亚稳态$A$与$B$之间稀有跃迁事件的数学框架。TPT的核心是承诺子函数，该函数描述了从相空间中任意给定起点出发，在抵达亚稳态$A$之前首次到达亚稳态$B$的概率。一旦计算出承诺子，跃迁通道与跃迁速率便可轻松得出。承诺子是满足特定边界条件的后向柯尔莫哥洛夫方程的解。然而，在高维情形下求解该方程极具挑战性，因为需要对整个环境空间区域进行网格剖分。本文探讨了有限表达式方法（FEX，Liang与Yang（2022））作为计算承诺子的工具。FEX通过包含固定有限个非线性函数与二元算术运算的代数表达式来逼近承诺子。表达式模板中的最优非线性函数、二元运算及数值系数通过强化学习进行寻优。基于FEX的承诺子求解器在多个高维基准问题上进行了测试，其结果与基于神经网络的求解器相当或更优。最重要的是，FEX能够正确识别解的代数结构，从而可将承诺子问题降至低维，并以任意所需精度求解承诺子。