同时识别非线性动力学和从数据中提取噪声概率分布的自动微分
Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data
摘要：
非线性动力学的稀疏识别（SINDy）是一个回归框架，用于从时间序列数据中发现简约的动力学模型和控制方程。与所有系统辨识方法一样，噪声测量损害模型发现过程的准确性和鲁棒性。在这项工作中，我们开发了一种变体的SINDy算法，该算法集成了自动微分和最近的时间步进约束，由Rudy等人激励。同时（i）对数据进行去噪，（ii）学习和参数化噪声概率分布，以及（iii）识别负责生成时间序列数据的底层简约动力系统。因此，在集成优化框架内，噪声可以从信号中分离出来，从而产生一种对噪声的鲁棒性大约是现有技术方法的两倍的架构，在给定的时间序列信号上处理多达40%的噪声，并显式地参数化噪声概率分布。我们证明了这种方法的几个数值例子，从Lotka-Volterra模型的时空洛伦兹96模型。此外，我们证明了该方法可以识别多种概率分布，包括高斯分布，均匀分布，伽玛分布和瑞利分布。
英文摘要：
The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work, we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al. for simultaneously (i) denoising the data, (ii) learning and parametrizing the noise probability distribution, and (iii) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can identify a diversity of probability distributions including Gaussian, uniform, Gamma, and Rayleigh.
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