From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach

TL;DR

This paper introduces a novel framework for learning the spectral properties of stochastic processes from biased simulation data, effectively overcoming sampling limitations in molecular dynamics.

Contribution

It presents an infinitesimal generator-based approach that can accurately learn eigenfunctions and eigenvalues from biased data, outperforming transfer operator methods.

Findings

Effective in estimating spectral properties from limited biased data

Outperforms transfer operator approaches in experiments

Recovers transition mechanisms even with sub-optimal biasing

Abstract

We investigate learning the eigenfunctions of evolution operators for time-reversal invariant stochastic processes, a prime example being the Langevin equation used in molecular dynamics. Many physical or chemical processes described by this equation involve transitions between metastable states separated by high potential barriers that can hardly be crossed during a simulation. To overcome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. We propose a framework for learning from biased simulations rooted in the infinitesimal generator of the process and the associated resolvent operator. We contrast our approach to more common ones based on the transfer operator, showing that it can provably learn the spectral properties of the unbiased system from biased data. In experiments, we highlight the advantages of our method over transfer operator approaches and recent developments based on generator learning, demonstrating its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we show that even with datasets containing only a few relevant transitions due to sub-optimal biasing, our approach recovers relevant information about the transition mechanism.