Exact Recovery of Chaotic Systems from Highly Corrupted Data
Exact Recovery of Chaotic Systems from Highly Corrupted Data
复制标题
从高度损坏的数据中精确恢复混沌系统
DOI：
10.1137/16m1086637
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Rachel A. Ward
中科院分区：
文献类型：
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作者：
Giang Tran;Rachel A. Ward
从时变测量中学习动力系统的控制方程在不同科学领域都备受关注。当此类数据受到高度破坏时，例如由于记录机制在未知时间段内失效，这项任务就变得非常困难。当基础系统呈现出混沌行为，比如对初始条件敏感时，高精度地恢复控制方程就至关重要。在这项工作中，我们考虑连续时间动力系统\(\dot{x} = f(x)\)，其中\(f:\mathbb{R}^{d}\to\mathbb{R}^d\)的每个分量都是最高次数为\(p\)的多元多项式；我们的目标是从可能受到高度破坏的测量值\(x(t_1), x(t_2), \cdots, x(t_m)\)中准确地识别\(f\)。作为我们的主要理论结果，我们表明如果系统具有足够的遍历性，使得这些数据满足一个强中心极限定理（已知混沌洛伦兹系统满足该定理），那么控制方程\(f\)可以作为\(\ell_1\)最小化问题的解被准确恢复——即使很大比例的数据被异常值破坏。在数值上，我们应用交替最小化方法来解决相应的约束优化问题。通过几个三维混沌系统和更高维超混沌系统的例子，我们展示了该算法从有噪声且高度破坏的测量数据中恢复控制方程的能力、通用性和效率。
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.