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）是一个数学框架，用于量化选定亚稳态$A$与$B$之间的罕见过渡事件。TPT的核心是提交函数，它描述了从相空间任意起点出发，在到达$A$之前到达亚稳态$B$的概率。一旦计算出提交函数，过渡通道和过渡速率即可轻松求得。提交函数是带有适当边界条件的向后科尔莫戈罗夫方程的解。然而，在高维空间中求解该方程极具挑战性，因为需要对整个周围区域进行网格划分。在本工作中，我们探索了有限表达方法（FEX，Liang and Yang (2022)）作为计算提交函数的工具。FEX通过包含固定有限个非线性函数和二元算术运算的代数表达式来逼近提交函数。最优的非线性函数、二元运算以及表达式模板中的数值系数通过强化学习确定。基于FEX的提交函数求解器在多个高维基准问题上进行了测试。与基于神经网络的求解器相比，它给出了相当或更优的结果。最重要的是，FEX能够正确识别解的代数结构，从而将提交函数问题降维至低维问题，并以任意期望的精度求解提交函数。