Deep Generative Modeling for Identification of Noisy, Non-Stationary Dynamical Systems

Doris Voina, J. Nathan Kutz, Steven Brunton

Keywords: system identification, non-autonomous differential equations, dynamical systems, variational inference, variational autoencoders, SINDy, sparse regression, uncertainty quantification, latent variable discovery, biophysics applications, biology, neuroscience

TL;DR: We introduce a method for identifying non-autonomous differential equations and discovering latent variables in dynamic systems, validated on synthetic data (e.g., the Lorenz system) and applied to neuronal activity data.

Abstract:

An important 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 wide range of problems, specifically to dynamic systems where complex parametric time dependencies are expected.

Primary Area: learning on time series and dynamical systems

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Submission Number: 12372