Data-driven prediction of multistable systems from sparse measurements
| Autor: | Mohammad Farazmand, Bryan Chu |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
FOS: Computer and information sciences
Optimization problem Computer science FOS: Physical sciences General Physics and Astronomy Machine Learning (stat.ML) Dynamical Systems (math.DS) Pattern Formation and Solitons (nlin.PS) 01 natural sciences 010305 fluids & plasmas Data-driven Statistics - Machine Learning Stability theory 0103 physical sciences FOS: Mathematics Applied mathematics Mathematics - Dynamical Systems 010306 general physics Mathematics - Optimization and Control Scaling Mathematical Physics Applied Mathematics Regular polygon Statistical and Nonlinear Physics State (functional analysis) Nonlinear Sciences - Pattern Formation and Solitons Optimization and Control (math.OC) Metric (mathematics) Equivariant map |
| Zdroj: | Chaos: An Interdisciplinary Journal of Nonlinear Science. 31:063118 |
| ISSN: | 1089-7682 1054-1500 |
| Popis: | We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) it is compatible with the precomputed library and (ii) it is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to the scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction–diffusion equation, arising in pattern formation, which has four asymptotically stable steady states, and a FitzHugh–Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction–diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95 % accuracy when a moderate number of labeled data are used. For the FitzHugh–Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90 % accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions. |
| Databáze: | OpenAIRE |
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