# A Finite Expression Method for Solving High-Dimensional Committor Problems

arXiv: https://arxiv.org/abs/2306.12268 · HTML: https://mlsanity.com/paper/2306.12268/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 21 Jun 2023.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Computing Committor Functions for the Study of Rare Events Using Deep Learning](https://mlsanity.com/paper/1906.06285/) (2019)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)
- [Active Importance Sampling for Variational Objectives Dominated by Rare Events: Consequences for Optimization and Generalization](https://mlsanity.com/paper/2008.06334/) (2020)
- [Data-driven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions](https://mlsanity.com/paper/2005.12787/) (2020)

## Built on by (most cited first, 9 of 9)

- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Sharp error estimates for target measure diffusion maps with applications to the committor problem](https://mlsanity.com/paper/2312.14418/) (2023)
- [From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs](https://mlsanity.com/paper/2503.09986/) (2025)
- [On the Stochastic (Variance-Reduced) Proximal Gradient Method for Regularized Expected Reward Optimization](https://mlsanity.com/paper/2401.12508/) (2024)
- [Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs](https://mlsanity.com/paper/2410.22371/) (2024)
- [An exact multiple-time-step variational formulation for the committor and the transition rate](https://mlsanity.com/paper/2509.03539/) (2025)
- [Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains](https://mlsanity.com/paper/2510.22497/) (2025)
- [The finite expression method for turbulent dynamics with high-order moment recovery](https://mlsanity.com/paper/2605.10687/) (2026)

## Abstract (arXiv)

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.