Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Chaos. 2022 Jun;32(6):063101. doi: 10.1063/5.0066066.
Many natural systems exhibit chaotic behavior, including the weather, hydrology, neuroscience, and population dynamics. Although many chaotic systems can be described by relatively simple dynamical equations, characterizing these systems can be challenging due to sensitivity to initial conditions and difficulties in differentiating chaotic behavior from noise. Ideally, one wishes to find a parsimonious set of equations that describe a dynamical system. However, model selection is more challenging when only a subset of the variables are experimentally accessible. Manifold learning methods using time-delay embeddings can successfully reconstruct the underlying structure of the system from data with hidden variables, but not the equations. Recent work in sparse-optimization based model selection has enabled model discovery given a library of possible terms, but regression-based methods require measurements of all state variables. We present a method combining variational annealing-a technique previously used for parameter estimation in chaotic systems with hidden variables-with sparse-optimization methods to perform model identification for chaotic systems with unmeasured variables. We applied the method to ground-truth time-series simulated from the classic Lorenz system and experimental data from an electrical circuit with Lorenz-system like behavior. In both cases, we successfully recover the expected equations with two measured and one hidden variable. Application to simulated data from the Colpitts oscillator demonstrates successful model selection of terms within nonlinear functions. We discuss the robustness of our method to varying noise.
许多自然系统表现出混沌行为，包括天气、水文学、神经科学和人口动态。尽管许多混沌系统可以用相对简单的动力方程来描述，但由于对初始条件的敏感性以及区分混沌行为与噪声的困难，对这些系统进行特征描述可能具有挑战性。理想情况下，人们希望找到一组简洁的方程来描述一个动力系统。然而，当只有一部分变量可以通过实验获得时，模型选择就更加具有挑战性了。使用时滞嵌入的流形学习方法可以从具有隐藏变量的数据中成功地重建系统的底层结构，但无法重建方程。基于稀疏优化的模型选择的最新工作已经能够在给定可能项的库的情况下实现模型发现，但是基于回归的方法需要测量所有状态变量。我们提出了一种方法，将变分退火技术（一种以前用于具有隐藏变量的混沌系统参数估计的技术）与稀疏优化方法相结合，以对具有未测量变量的混沌系统进行模型识别。我们将该方法应用于经典 Lorenz 系统的真实时间序列和具有 Lorenz 系统行为的电路的实验数据。在这两种情况下，我们都成功地恢复了具有两个测量变量和一个隐藏变量的预期方程。将该方法应用于 Colpitts 振荡器的模拟数据表明，可以成功地在非线性函数中选择项。我们讨论了我们的方法对不同噪声的鲁棒性。
