From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach

Timothée Devergne, Vladimir R Kostic, Michele Parrinello, Massimiliano Pontil

Keywords: Molecular dynamics, stochastic differential equations

TL;DR: we prove that learning the true dynamics rom biased simulations is possible via the Infinitesimal Generator and develop the method that does it efficently

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.

Primary Area: Machine learning for physical sciences (for example: climate, physics)

Submission Number: 18933