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Physics-Informed Probabilistic Learning of Linear Embeddings of Nonlinear Dynamics with Guaranteed Stability
- Source :
- SIAM Journal on Applied Dynamical Systems. 19:480-509
- Publication Year :
- 2020
- Publisher :
- Society for Industrial & Applied Mathematics (SIAM), 2020.
-
Abstract
- The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models: differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.<br />Comment: 31 pages
- Subjects :
- FOS: Computer and information sciences
Mathematics::Dynamical Systems
Operator (physics)
Dynamics (mechanics)
Probabilistic logic
Stability (learning theory)
Machine Learning (stat.ML)
Dynamical Systems (math.DS)
01 natural sciences
010305 fluids & plasmas
Nonlinear dynamical systems
Nonlinear system
Statistics - Machine Learning
Modeling and Simulation
0103 physical sciences
FOS: Mathematics
Applied mathematics
Mathematics - Dynamical Systems
Linear embedding
Analysis
Subjects
Details
- ISSN :
- 15360040
- Volume :
- 19
- Database :
- OpenAIRE
- Journal :
- SIAM Journal on Applied Dynamical Systems
- Accession number :
- edsair.doi.dedup.....2a389f8aea593d2b0fe8d7d217a2400b