Deep learning Markov and Koopman models with physical constraints

Andreas Mardt, Luca Pasquali, Frank Noé, Hao Wu
Proceedings of The First Mathematical and Scientific Machine Learning Conference, PMLR 107:451-475, 2020.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v107-mardt20a, title = {Deep learning {M}arkov and {K}oopman models with physical constraints}, author = {Mardt, Andreas and Pasquali, Luca and No\'{e}, Frank and Wu, Hao}, booktitle = {Proceedings of The First Mathematical and Scientific Machine Learning Conference}, pages = {451--475}, year = {2020}, editor = {Lu, Jianfeng and Ward, Rachel}, volume = {107}, series = {Proceedings of Machine Learning Research}, month = {20--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v107/mardt20a/mardt20a.pdf}, url = {https://proceedings.mlr.press/v107/mardt20a.html}, 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. } }
Endnote
%0 Conference Paper %T Deep learning Markov and Koopman models with physical constraints %A Andreas Mardt %A Luca Pasquali %A Frank Noé %A Hao Wu %B Proceedings of The First Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2020 %E Jianfeng Lu %E Rachel Ward %F pmlr-v107-mardt20a %I PMLR %P 451--475 %U https://proceedings.mlr.press/v107/mardt20a.html %V 107 %X 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.
APA
Mardt, A., Pasquali, L., Noé, F. & Wu, H.. (2020). Deep learning Markov and Koopman models with physical constraints. Proceedings of The First Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 107:451-475 Available from https://proceedings.mlr.press/v107/mardt20a.html.

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