Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths

TL;DR

This work addresses the challenge of estimating high-dimensional committor functions by introducing Deep Adaptive Sampling on Rare Transition Paths (DASTR), which adaptively concentrates training data in transition regions using a data distribution $p_{V,q}(\boldsymbol{x}) \propto |\nabla q(\boldsymbol{x})|^2 e^{-\beta V(\boldsymbol{x})}$ and optionally a bias $V_{\text{bias}}$. A KRnet-based flow model approximates this distribution to generate effective transition-state samples, which are used to train a neural committor solver under a variational loss. To handle very high-dimensional problems, DASTR is extended to latent spaces via autoencoders, enabling sampling in latent CVs either directly or with umbrella sampling for physical validity. Numerical experiments on rugged Mueller potentials, standard Brownian motion, and alanine dipeptide demonstrate substantial accuracy and efficiency gains, with latent-variable strategies offering a practical path toward scalable, physics-aware rare-event learning.

Abstract

The committor functions are central to investigating rare but important events in molecular simulations. It is known that computing the committor function suffers from the curse of dimensionality. Recently, using neural networks to estimate the committor function has gained attention due to its potential for high-dimensional problems. Training neural networks to approximate the committor function needs to sample transition data from straightforward simulations of rare events, which is very inefficient. The scarcity of transition data makes it challenging to approximate the committor function. To address this problem, we propose an efficient framework to generate data points in the transition state region that helps train neural networks to approximate the committor function. We design a Deep Adaptive Sampling method for TRansition paths (DASTR), where deep generative models are employed to generate samples to capture the information of transitions effectively. In particular, we treat a non-negative function in the integrand of the loss functional as an unnormalized probability density function and approximate it with the deep generative model. The new samples from the deep generative model are located in the transition state region and fewer samples are located in the other region. This distribution provides effective samples for approximating the committor function and significantly improves the accuracy. We demonstrate the effectiveness of the proposed method through both simulations and realistic examples.

Figures (16)

• Figure 1: The schematic of DASTR for computing the committor function. Training a deep neural network $q_{\boldsymbol{\theta}}$ to approximate the high-dimensional committor function must use a high-quality dataset (i.e. data points from the transition area). Typically, the data points from Langevin dynamics are not in the transition state region since the transition between two metastable states is rare and difficult to sample. The proposed DASTR method can adaptively generate effective data points on the transition area according to the information of the current approximate solution. The key point is to define a sampling distribution $p_{V,q}$ dependent on the current approximate solution and the potential. Effective data points in the transition area are generated by sampling from $p_{V,q}$, which is achieved through training a deep generative model.
• Figure 2: Molecular configurations generated by two different settings in DASTR: (a) the inputs of KRnet are the coordinates of heavy atoms (b) the inputs of KRnet are the latent CVs. The hydrogen atoms are completed by the software package PyMOL schrodinger2015pymol. This figure demonstrates that using the latent collective variables to conduct DASTR is more effective.
• Figure 3: The schematic of adaptive sampling in the latent space. We first train an autoencoder to obtain the latent variables as the collective variables (CVs), and then use KRnet to approximate the distribution of the CVs. After training KRnet, we use a random sample $\boldsymbol{z}_0$ from the standard Gaussian distribution to generate a new sample of latent CVs. We can feed this new sample of latent CVs into the decoder to obtain a new sample of molecules after the post-processing step. Such a new sample of molecules is located in the transition state region. The autoencoder not only provides an effective way to automatically choose the collective variables, but also enhances the sampling efficiency of molecules in the transition state region.
• Figure 4: DASTR, samples for the 10-dimensional rugged Mueller potential problem. The red line denotes the test points from the $1/2$-isosurface ($q \approx 1/2$) projected onto the $x_1$-$x_2$ plane.
• Figure 5: Solutions, $10$-dimensional rugged Mueller potential test problem.

Theorems & Definitions (1)

• Remark