# Solving for high dimensional committor functions using artificial neural networks

arXiv: https://arxiv.org/abs/1802.10275 · HTML: https://mlsanity.com/paper/1802.10275/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 30 of 34)

- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
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- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations](https://mlsanity.com/paper/2101.01708/) (2021)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [Reinforcement learning of rare diffusive dynamics](https://mlsanity.com/paper/2105.04321/) (2021)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction](https://mlsanity.com/paper/2303.12534/) (2023)
- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [Approximation of Functionals by Neural Network without Curse of Dimensionality](https://mlsanity.com/paper/2205.14421/) (2022)
- [Diffusion Methods for Generating Transition Paths](https://mlsanity.com/paper/2309.10276/) (2023)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Supervised Learning and the Finite-Temperature String Method for Computing Committor Functions and Reaction Rates](https://mlsanity.com/paper/2107.13522/) (2021)
- [Deep Learning Method for Computing Committor Functions with Adaptive Sampling](https://mlsanity.com/paper/2404.06206/) (2024)
- [The fast committor machine: Interpretable prediction with kernels](https://mlsanity.com/paper/2405.10410/) (2024)
- [A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks](https://mlsanity.com/paper/2105.03480/) (2021)
- [Computing the Invariant Distribution of Randomly Perturbed Dynamical Systems Using Deep Learning](https://mlsanity.com/paper/2110.11538/) (2021)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [Computing non-equilibrium trajectories by a deep learning approach](https://mlsanity.com/paper/2210.04042/) (2022)
- [The Mori-Zwanzig formulation of deep learning](https://mlsanity.com/paper/2209.05544/) (2022)
- [Accelerated Sampling of Rare Events using a Neural Network Bias Potential](https://mlsanity.com/paper/2401.06936/) (2024)
- [Probing reaction channels via reinforcement learning](https://mlsanity.com/paper/2305.17531/) (2023)
- [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)
- [Conditional Tropical Cyclogenesis Rates via Rare-Event Sampling in a Neural Weather Emulator](https://mlsanity.com/paper/2606.30920/) (2026)
- [Computing Transition Pathways for the Study of Rare Events Using Deep Reinforcement Learning](https://mlsanity.com/paper/2404.05905/) (2024)

## Abstract (arXiv)

In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theory, which satisfies a high-dimensional Fokker-Planck equation. By working with the variational formulation of such partial differential equation and parameterizing the committor function in terms of a neural network, approximations can be obtained via optimizing the neural network weights using stochastic algorithms. The numerical examples show that moderate accuracy can be achieved for high-dimensional problems.