A Finite Expression Method for Solving High-Dimensional Committor Problems

TL;DR

The paper tackles the high-dimensional committor problem from transition path theory by adapting the finite expression method (FEX) to learn compact algebraic expressions that approximate the solution to the backward Kolmogorov equation $\beta^{-1} \Delta q - \nabla V \cdot \nabla q = 0$ with appropriate boundary conditions. By representing the committor as a finite, structured combination of nonlinear operators arranged in binary trees and optimizing both the operator sequence and the parameters through a mixed combinatorial-continuous framework, the approach not only achieves competitive accuracy with neural-network solvers but also exposes the algebraic structure of the solution. This structure enables automatic dimensionality reduction to low-dimensional subproblems and facilitates subsequent high-precision solutions via Chebyshev spectral methods or finite element methods. The results on benchmark problems, including high-dimensional double-well potentials, concentric spheres, rugged Mueller’s potential, and butane, demonstrate FEX’s ability to identify the governing variables and achieve machine-precision-type solutions when reduced to lower dimensions, highlighting its practical impact for scalable and interpretable high-dimensional committor computations.

Abstract

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.

Figures (9)

• Figure 1: Representation of the components of our FEX implementation. (a) The searching loop for the symbolic solution encompasses multiple stages, namely expression generation, score computation, controller update, and candidate optimization. (b) Illustration of the expression generation with a binary tree and a controller $\bm{\chi}$. The controller produces probability mass functions for each node of the tree, enabling the sampling of node values. Furthermore, we incorporate learnable scaling and bias parameters to generate expressions based on the predefined tree structure and the sampled node values.
• Figure 2: Computational rule of a binary tree. Each node within the binary tree holds either a unary or a binary operator. Initially, we outline the computation flow of a depth-1 tree comprising a solitary operator. Subsequently, for binary trees extending beyond a single layer, the computation process is recursively executed.
• Figure 3: The representation of the computation flow of solving for \ref{['eqn:regularized_variational_form']}. The committor function $q$ is represented by the summation of three "FEX trees", two of which are weighted with $\frac{1}{|\mathbf{x}|^{d-2}}$ type singularities.
• Figure 4: The committor function for the double-well potential along $x_1$ dimension when $\beta^{-1}=0.2$ for an arbitrary $(x_2,\cdots,x_d)$ with $d=10$.
• Figure 5: Committor function for the double-well potential with sublevel sets boundary. As $\beta$ increases, the problem transforms from a 2D problem to a 1D problem, and FEX can capture such behavior of the committor function.

Theorems & Definitions (3)

• Definition 2.1: Mathematical expression liang2022finite
• Definition 2.2: $k$-finite expression liang2022finite
• Definition 2.3: Finite expression method liang2022finite