Accelerated Markov Chain Monte Carlo Simulation via Neural Network-Driven Importance Sampling

TL;DR

An importance sampling method designed to accelerate the time scale of Markov chain Monte Carlo (MCMC) simulations by employing a bias potential, which enables the sampling of rare transition events while preserving the relative probabilities of distinct transition pathways.

Abstract

Atomistic simulations provide valuable insights into the physical processes governing material behavior. However, their applicability is fundamentally constrained by the limited time scales accessible to brute-force simulations. This bottleneck often stems from complex energy landscapes where the systems stay trapped in metastable states for long periods of time. Yet, the long-term evolution is controlled by the transitions between the metastable states, which are rare events and difficult to observe. We present an importance sampling method designed to accelerate the time scale of Markov chain Monte Carlo (MCMC) simulations. By employing a bias potential, our approach enhances the sampling of rare transition events while preserving the relative probabilities of distinct transition pathways. The bias potential is represented by a neural network which enables the flexibility needed for high-dimensional systems. We propose a rigorous formulation to obtain the original transition rates between metastable states using transition paths obtained from the biased simulation. We further use a branching random walk (BRW) technique to enhance efficiency and to reduce variance. The proposed methodology is validated on 2-dimensional and 14-dimensional systems, demonstrating its accuracy and scalability.

Figures (8)

• Figure 1: Domain $\Omega$ is a set of allowed microscopic states (e.g. labeled as $i$ and $j$) correspond to points on a regular grid in a two-dimensional region. The colored background corresponds to the energy landscape $E(i)$. Microscopic states A and B corresponding to the local minima of the energy landscape. Microscopic states S$_1$ and S$_2$ are the saddle points of the energy landscape. The white path through the microscopic states indicates a transition path from A to B.
• Figure 2: Transition from state F to S. Failure (i.e., coming back to F after starting from F) occurs $N_{\textrm{F}}$ times before the success (i.e., arriving at S before hitting F after starting from F) takes place. The first passage time can be considered as a sum of the duration of failure and success paths. In the case of rare events, the number of failures before success is very large.
• Figure 3: Neural network bias potential over a 2-dimensional domain after (a) training for 30 epochs and (d) 100 epochs (at each temperature). (b) Parity plots between the neural network bias potential and the exact optimal bias potential after training for (b) 30 epochs and (e) 100 epochs (at each temperature). Differences between the neural network bias potential and the exact solution after training for (c) 30 epochs and (f) 100 epocs (at each temperature).
• Figure 4: Weight distribution of successful paths for importance sampled success paths using neural network bias potentials trained for (a) 30 epochs and (c) 100 epochs (at each temperature). Distribution of success probability estimates for bias potentials trained for (b) 30 epochs and (d) 100 epochs (at each temperature).
• Figure 5: Weight distributions of the sampled paths (a) without and (c) with the BRW process. Distribution of success probability estimates (b) without and (d) with the BRW process.