Journal
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
Volume 129, Issue -, Pages -Publisher
ELSEVIER
DOI: 10.1016/j.cnsns.2023.107701
Keywords
Transition path theory; Optimal stochastic control; Transition path process; Committor; Transition rate; Langevin dynamics; Overdamped Langevin dynamics; Collective variables; Sampling; Neural network; Lennard-Jones-7; Duffing oscillator
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This study focuses on modeling various processes in nature using stochastic differential equations with small white noise. It investigates rare transitions in such systems through transition path theory and proposes a methodology to estimate transition rates and construct effective controllers through the computation of committor functions using neural networks.
Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, and many others are modeled using stochastic differential equations with small white noise. The study of rare transitions between metastable states in such systems is of great interest and importance. The direct simulation of rare transitions is difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its crucial component is the committor function, the solution to a boundary value problem for the backward Kolmogorov equation. The key fact exploited in this work is that the optimal controller constructed from the committor leads to the generation of transition trajectories exclusively. We prove this fact for a broad class of stochastic differential equations. Moreover, we demonstrate that the committor computed for a dimensionally reduced system and then lifted to the original phase space still allows us to construct an effective controller and estimate the transition rate with reasonable accuracy. Furthermore, we propose an all-the -way-through scheme for computing the committor via neural networks, sampling the transition trajectories, and estimating the transition rate without meshing the space. We apply the proposed methodology to four test problems: the overdamped Langevin dynamics with Mueller's potential and the rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and Lennard-Jones-7 in 2D.
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