Abstract
We propose a universal method for data-driven modeling of complex nonlinear dynamics from time-resolved snapshot data without prior knowledge. Complex nonlinear dynamics govern many fields of science and engineering. Data-driven dynamic modeling often assumes a low-dimensional subspace or manifold for the state. We liberate ourselves from this assumption by proposing cluster-based network modeling (CNM) bridging machine learning, network science, and statistical physics. CNM describes short- and long-term behavior and is fully automatable, as it does not rely on application-specific knowledge. CNM is demonstrated for the Lorenz attractor, ECG heartbeat signals, Kolmogorov flow, and a high-dimensional actuated turbulent boundary layer. Even the notoriously difficult modeling benchmark of rare events in the Kolmogorov flow is solved. This automatable universal data-driven representation of complex nonlinear dynamics complements and expands network connectivity science and promises new fast-track avenues to understand, estimate, predict, and control complex systems in all scientific fields.
Highlights
Climate, epidemiology, brain activity, financial markets, and turbulence constitute examples of complex systems
cluster-based network modeling (CNM) is applied to the Lorenz system, a widely used canonical chaotic dynamical system [26] defined by three coupled nonlinear differential equations
Too few centroids might oversimplify the dynamics, whereas too many might lead to a noisy solution
Summary
Epidemiology, brain activity, financial markets, and turbulence constitute examples of complex systems. They are characterized by a large range of time and spatial scales, intrinsic high dimensionality, and nonlinear dynamics. Dynamic modeling for the long-term features is a key enabler for understanding, state estimation from limited sensor signals, prediction, control, and optimization. Data- driven modeling has made tremendous progress in the past decades, driven by algorithmic advances, accessibility to large data, and hardware speedups. The modeling is based on a low-dimensional approximation of the state and system identification in that approximation. Autoencoders [5] represent a general nonlinear dimension reduction to a low-dimensional feature space. The dynamic system identification is substantially simplified in this feature space
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