Exact Recovery of Chaotic Systems from Highly Corrupted Data

By Giang Tran, Rachel A. Ward. 2016

182 citations recorded by the source corpus.

Learning the governing equations in dynamical systems from time-varying measurements is of great interest across different scientific fields. This task becomes prohibitive when such data is moreover highly corrupted, for example, due to the recording mechanism failing over unknown intervals of time. When the underlying system exhibits chaotic behavior, such as sensitivity to initial conditions, it is crucial to recover the governing equations with high precision. In this work, we consider continuous time dynamical systems $\dot{x} = f(x)$ where each component of $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^d$ is a multivariate polynomial of maximal degree $p$; we aim to identify $f$ exactly from possibly highly corrupted measurements $x(t_1), x(t_2), \dots, x(t_m)$. As our main theoretical result, we show that if the system is sufficiently ergodic that this data satisfies a strong central limit theorem (as is known to hold for chaotic Lorenz systems), then the governing equations $f$ can be exactly recovered as the solution to an $\ell_1$ minimization problem -- even if a large percentage of the data is corrupted by outliers. Numerically, we apply the alternating minimization method to solve the corresponding constrained optimization problem. Through several examples of 3D chaotic systems and higher dimensional hyperchaotic systems, we illustrate the power, generality, and efficiency of the algorithm for recovering governing equations from noisy and highly corrupted measurement data.

Subfields: Applied Mathematics, Artificial Intelligence, Computer Vision and Pattern Recognition

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Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, methodology): “More recently, sparsity (Tibshirani, 1996) has been used to determine the governing dynamical system (Brunton et al., 2016; Mangan et al., 2016; Wang et al., 2011; Schaeffer et al., 2013; Ozoliņš et al., 2013; Mackey et al., 2014; Brunton et al., 2014; Proctor et al., 2014; Bai et al., 2014; Tran an”
WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS (2020, background): “The wide applicability, computational efficiency, and interpretability of the SINDy algorithm has spurred an explosion of interest in the problem of identifying nonlinear dynamical systems from data [7, 23, 9, 10, 12, 32, 19].”
Data-driven discovery of coordinates and governing equations (2019, background): “The algorithm can also be reformulated to include integral terms for noisy data [62] or handle incomplete or limited data [63, 64].”
Model selection for dynamical systems via sparse regression and information criteria (2017, methodology): “Since the original SINDy method, there have been numerous innovations and extensions to handle rational function nonlinearities [4], partial diﬀerential equations [2], highly corrupted data [27], and to build Galerkin regression models in ﬂuids [28].”
Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, methodology): “Many studies have been conducted to eﬀectively deal with noises in data ([7, 40]), corruptions in data ([48]), partial diﬀerential equations [38, 41], etc. Methods have also been developed in conjunction with model selection approach ([28]), Koopman theory ([6]), and Gaussian process regression…”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, background): “Robust variants of SINDY have been formulated to identify models despite large outliers and noise [56,57].”