Source-linked AI summary
Solving for high dimensional committor functions using artificial neural networks
Yuehaw Khoo, Jianfeng Lu, Lexing Ying
TL;DR
The paper addresses computation of high-dimensional committor functions for stochastic transitions, where solving the associated Fokker–Planck equation is difficult. It parameterizes the committor with a neural network and optimizes the variational formulation, obtaining moderate accuracy in high-dimensional numerical examples while requiring architecture choices that handle singularities and sampling limitations.
Problem
Computing the committor function is difficult because its governing Fokker–Planck equation is high-dimensional, and singularities can occur in regions A and B.
Method
The method parameterizes the committor with a neural network and finds its parameters by optimizing the variational formulation using stochastic, Monte Carlo-based integration.
Results
The numerical experiments show that the proposed architecture recovers the expected singular behavior and achieves few-percent accuracy in a tested example.
Takeaways & Limitations
Neural-network parameterization provides a useful alternative approach for high-dimensional partial differential equations when the architecture is designed for the committor’s singularities.
Takeaways & Limitations
The learned committor depends crucially on sampling, and naive sampling becomes sparse between regions A and B at low temperature.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper frames the committor function as a central transition-path quantity whose high-dimensional Fokker–Planck equation is difficult to solve. It motivates neural-network parameterization as an alternative to finite-element, dimension-reduction, and diffusion-map approaches.
- The committor function gives the probability of reaching region B before region A and describes statistical properties of reaction trajectories.
- The committor satisfies a high-dimensional Fokker–Planck equation, making finite-element-type methods intractable.
- Existing alternatives include finite-temperature string methods, explicit dimension reduction, diffusion maps, and point-cloud discretization of the generator.
- The paper uses an artificial neural network to provide a low-dimensional parameterization of the committor function.
- The variational formulation converts the PDE into optimization over neural-network parameters, with Monte Carlo integration using samples generated by the stochastic process.
- Related neural-network PDE methods avoid specifying a basis and have shown low-dimensional success, but their high-dimensional performance remains unexplored.
2 Proposed method
The proposed method parameterizes the committor with a neural network and minimizes its variational formulation using stochastic optimization. Its architecture explicitly addresses high-temperature singularities and low-temperature sharp transitions, while numerical examples demonstrate the resulting approximations.
- 2 Proposed method: The variational formulation enforces q = 0 on ∂A, q = 1 on ∂B, and a boundary condition on ∂Ω.For the unbounded domain, subexponential growth of ∇q together with exponential decay of the equilibrium density ensures the required condition.
- 2 Proposed method: The method replaces the high-dimensional Fokker–Planck problem with optimization of a neural-network parameterization of the committor's variational formulation.The variational objective is approximated by sampling from the overdamped Langevin process and boundary measures, then optimized over neural-network weights.
- 2 Proposed method: The stochastic objective combines interior and boundary sampling through a mixture measure whose parameter α controls their relative sample proportions.Boundary constraints are imposed as soft constraints, with hardness tuned by ρ.
- 2.1 Challenge in high T regime: At high temperature, singularities near A and B can make the committor steep and difficult for a standard neural network to approximate.In the Laplace limit, the committor has singular behavior associated with the solution constructed using the method of images.
- 2.2 Challenge in low T regime: At T = 0.05 and d = 10, a single-hidden-layer tanh network gives a good approximation despite the committor being steep around x1 = 0.The example is intended to show qualitatively that the neural network can handle this low-temperature singularity issue.
- 2.3 Neural-network architecture: The proposed architecture removes dominant singularities by multiplying problem-dependent singular functions by neural-network outputs and combining them with a baseline network.For low-temperature transitions, tanh nonlinearities are used to represent a sharp committor change near the midpoint between the boundary regions.
3 Numerical experiments
The proposed neural-network method is evaluated on high-dimensional double-well, concentric-sphere, and rugged-Muller committor problems. The experiments emphasize architecture choices for singularities and report moderate accuracy, with sampling affecting performance.
- Evaluation: Across the experiments, the variational reaction-rate quantity is evaluated from the weighted gradient energy of the committor.The same quantity is identified as the energy minimized in the variational formulation.
- Double well: The first experiment solves a d = 10 double-well committor problem at temperatures T = 0.2 and 0.05 using tailored sampling near the saddle point.Uniform sampling in x1 is used to obtain sufficient coverage where low temperature makes equilibrium samples sparse.
- Concentric spheres: The concentric-spheres experiment compares neural networks with and without an explicitly included 1/|x|d−2 singularity for T = 2 and d = 6.The singular behavior motivates setting Ns = 1 with S1 = 1/|x|d−2, while Table 2 compares Ns = 0 and Ns = 1.
- Concentric spheres: Explicit singularity treatment is important for the concentric-spheres example, where the figure compares ground truth with neural-network committors along multiple radial directions.Figure 5 contrasts parameterizations including and omitting the singularity.
- Rugged Muller: The rugged-Muller experiment uses temperatures T = 40 and 22, two localized regions, and a neural network with two logarithmic singularities.The reported results achieve few-percent accuracy, but lower temperature requires more samples because the equilibrium distribution is less smooth.
4 Conclusion
The note develops a neural-network representation of high-dimensional committor functions by optimizing the variational Fokker-Planck formulation. Numerical experiments show usefulness for high-dimensional problems, while accuracy depends crucially on sampling, especially at low temperature.
- Method: The method represents the high-dimensional committor with a neural network whose parameters are optimized through the variational form of the Fokker-Planck equation.The conclusion identifies this as the paper’s central approach.
- Result: Numerical experiments show the proposed approach is useful for solving high-dimensional partial differential equations.The conclusion presents this as the supported outcome of the experiments.
- Limitation: Learned-committor quality depends crucially on sampling, and naive low-temperature sampling can produce transitions that are too sharp.Enhanced sampling guided by the learned network is identified as a future direction.