A significant challenge in many fields of science and engineering is making sense of time-dependent measurement data by recovering governing equations in the form of differential equations. We focus on finding parsimonious ordinary differential equation (ODE) models for nonlinear, noisy, and non-autonomous dynamical systems and propose a machine learning method for data-driven system identification. While many methods tackle noisy and limited data, non-stationarity - where differential equation parameters change over time - has received less attention. Our method, dynamic SINDy, combines variational inference with SINDy (sparse identification of nonlinear dynamics) to model time-varying coefficients of sparse ODEs. This framework allows for uncertainty quantification of ODE coefficients, expanding on previous methods for autonomous systems. These coefficients are then interpreted as latent variables and added to the system to obtain an autonomous dynamical model. We validate our approach using synthetic data, including nonlinear oscillators and the Lorenz system, and apply it to neuronal activity data from C. elegans. Dynamic SINDy uncovers a global nonlinear model, showing it can handle real, noisy, and chaotic datasets. We aim to apply our method to a variety of problems, specifically dynamic systems with complex time-dependent parameters.

本研究解决了在科学与工程领域中，时间依赖测量数据的理解及恢复相应微分方程模型的挑战。提出了一种名为动态SINDy的机器学习方法，针对非线性、噪声和非自主动力系统的稀疏常微分方程进行建模，并能够处理系统参数随时间变化的问题。验证结果表明，该方法在处理真实的、噪声和混沌数据集时表现出色，具有广泛的应用潜力。

用于识别噪声、非平稳动力系统的深度生成建模