Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

Abstract

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

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• Kernel methods for detecting coherent structures in dynamical data
• Maximally predictive states: from partial observations to long timescales
• Data-driven Computation of Molecular Reaction Coordinates
• Data-driven construction of stochastic reduced dynamics encoded with non-Markovian features
• Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain
• Efficient Approximation of Molecular Kinetics using Random Fourier Features
• Fluctuating landscapes and heavy tails in animal behavior
• The numerical computation of unstable manifolds for infinite dimensional dynamical systems by embedding techniques
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• Flow Matching for Optimal Reaction Coordinates of Biomolecular System
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• Predicting trajectory behaviour via machine-learned invariant manifolds
• Learning Interpretable Collective Variables for Spreading Processes on Networks
• Collective variables between large-scale states in turbulent convection
• Random walk based snapshot clustering for detecting community dynamics in temporal networks
• The Augmented Jump Chain -- a sparse representation of time-dependent Markov jump processes
• Accelerated Simulations of Molecular Systems through Learning of their Effective Dynamics