Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov–Zhabotinsky (BZ) reaction.

中文翻译：

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SINDy-PI：一种用于非线性动力学并行隐式稀疏识别的鲁棒算法

根据测量数据对系统的非线性动力学进行准确建模是一个具有挑战性但又至关重要的课题。非线性动力学稀疏识别 (SINDy) 算法是一种从数据中发现动态系统模型的方法。尽管已经开发了扩展来识别隐式动态或由有理函数描述的动态，但这些扩展对噪声极为敏感。在这项工作中，我们开发了 SINDy-PI（并行、隐式），这是 SINDy 算法的一种鲁棒变体，用于识别隐式动态和有理非线性。SINDy-PI 框架包括多种优化算法和模型选择的原则方法。我们展示了该算法从有限和嘈杂的数据中学习隐式常微分方程和偏微分方程以及守恒定律的能力。特别是，