Deep learning Markov and Koopman models with physical constraints

TL;DR

This paper introduces deep learning Markov and Koopman models that incorporate physical constraints, providing a universal approximation for reversible processes and improved performance on biased data.

Contribution

It develops theory and methods for deep learning models with physical constraints, extending linear approaches to nonlinear latent spaces.

Findings

Model is a universal approximator for reversible Markov processes.

Optimizable via maximum likelihood or VAMP methods.

Performs better on biased data compared to existing approaches.

Abstract

The long-timescale behavior of complex dynamical systems can be described by linear Markov or Koopman models in a suitable latent space. Recent variational approaches allow the latent space representation and the linear dynamical model to be optimized via unsupervised machine learning methods. Incorporation of physical constraints such as time-reversibility or stochasticity into the dynamical model has been established for a linear, but not for arbitrarily nonlinear (deep learning) representations of the latent space. Here we develop theory and methods for deep learning Markov and Koopman models that can bear such physical constraints. We prove that the model is an universal approximator for reversible Markov processes and that it can be optimized with either maximum likelihood or the variational approach of Markov processes (VAMP). We demonstrate that the model performs equally well for equilibrium and systematically better for biased data compared to existing approaches, thus providing a tool to study the long-timescale processes of dynamical systems.