Extracting governing equations from dynamic data is an essential task in
model selection and parameter estimation. The form of the governing equation is
rarely known a priori; however, based on the sparsity-of-effect principle one
may assume that the number of candidate functions needed to represent the
dynamics is very small. In this work, we leverage the sparse structure of the
governing equations along with recent results from random sampling theory to
develop methods for selecting dynamical systems from under-sampled data. In
particular, we detail three sampling strategies that lead to the exact recovery
of first-order dynamical systems when we are given fewer samples than unknowns.
The first method makes no assumptions on the behavior of the data, and requires
a certain number of random initial samples. The second method utilizes the
structure of the governing equation to limit the number of random
initializations needed. The third method leverages chaotic behavior in the data
to construct a nearly deterministic sampling strategy. Using results from
compressive sensing, we show that the strategies lead to exact recovery, which
is stable to the sparse structure of the governing equations and robust to
noise in the estimation of the velocity. Computational results validate each of
the sampling strategies and highlight potential applications.

利用稀疏效应原则和随机抽样理论开发了三种采样策略用于从稀疏数据中选择动力系统，能够稳定恢复第一阶动力系统且对速度估计的噪声具有健壮性。

从有限数据中提取稀疏高维动态

Extracting Sparse High-Dimensional Dynamics from Limited Data

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.

通过混有噪声和异常值得动力系统测量数据，我们提出了一种使用多项式恢复其控制方程的算法，该算法能够适用于混沌系统中通过强中心极限定理满足的数据，并进行完美的精度恢复。该算法使用交替最小化进行实现，并在多个 3D 混沌系统和更高维超混沌系统实例上表现出了强大的通用性和高效性。