A Finite Expression Method for Solving High-Dimensional Committor
  Problems
Zezheng Song Maria K. Cameron Haizhao Yang
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.