Deep learning Markov and Koopman models with physical constraints

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.

References in corpus (6)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Variational cross-validation of slow dynamical modes in molecular kinetics
• Nonlinear Discovery of Slow Molecular Modes using State-Free Reversible VAMPnets
• Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
• Graph Dynamical Networks for Unsupervised Learning of Atomic Scale Dynamics in Materials
• Deep Generative Markov State Models

Cited by in corpus (4)

• GraphVAMPNet, using graph neural networks and variational approach to markov processes for dynamical modeling of biomolecules
• Kernel Embedding based Variational Approach for Low-dimensional Approximation of Dynamical Systems
• Internal-Coordinate Density Modelling of Protein Structure: Covariance Matters
• Progress in deep Markov State Modeling: Coarse graining and experimental data restraints