A Finite Expression Method for Solving High-Dimensional Committor Problems
This work addresses a computational bottleneck in analyzing rare transition events for researchers in computational physics and chemistry, though it is incremental as it builds on an existing method (FEX).
The paper tackles the challenge of solving high-dimensional committor problems in transition path theory by applying the finite expression method (FEX), which approximates the committor with algebraic expressions optimized via reinforcement learning. It achieves comparable or better results than neural network-based solvers on benchmark problems and can identify the solution's algebraic structure to reduce dimensionality and achieve any desired accuracy.
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.