Optimal Reaction Coordinates: Variational Characterization and Sparse Computation
Abstract.
Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.
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Acknowledgment
We thank the anonymous reviewers for their helpful criticism and constructive suggestions.
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Title of paper: Optimal Reaction Coordinates: Variational Characterization and Sparse Computation
Authors: Andreas Bittracher, Mattes Mollenhauer, Péter Koltai, and Christof Schütte
File: supplement.zip
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Contents: Matlab, Python and Mathematica scripts used to create the numerical results in the paper.
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Submitted: 24 September 2021
Accepted: 28 October 2022
Published online: 25 April 2023
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Deutsche Forschungsgemeinschaft (DFG): CRC 1114, 235221301, B03, A01, EXC 2046, AA1-2, EF4-8
Funding: This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems,” project 235221301, project B03 “Multilevel coarse graining of multiscale problems,” project A01 “Coupling a multiscale stochastic precipitation model to large scale atmospheric flow dynamics,” grant EXC 2046 ”MATH+,” project 390685689, project AA1-2 “Learning Transition Manifolds and Effective Dynamics of Biomolecules,” and project EF4-8 “Concentration effects and collective variables in agent-based systems.”
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