Dr. Doris Voina (AI Institute in Dynamic Systems, Univ. of Washington)

9/5/2024  3:10pm

Abstract: 

An important challenge in many fields of science and engineering is making sense of a deluge of time-dependent measurement data by recovering the governing equations in the form of ordinary or partial differential equations. We are interested in finding interpretable,  parsimonious models in the form of closed-form differential equations for nonlinear, noisy, and non-stationary dynamical systems. To this end, we propose a machine learning method that performs system identification in such systems. An array of methods so far have addressed noisy and limited data; however, the problem of non-stationarity, when parameters describing the differential equations have explicit time dependence, has received considerably less attention. In this work, we use different variational autoencoder architectures whose outputs (coefficients) are constrained to be the time-varying parameters of a sparse system of ODEs that express the dynamical trajectories of interest. This variational framework enables uncertainty quantification of the ODE coefficients across time, expanding on previous methods for stationary systems. We interpret these coefficients as hidden latent variables whose time trajectories are then subsequently added to the system to obtain a stationary dynamical system. Our novel method – dynamic SINDy – combines variational inference with the SINDy (sparse identification of nonlinear dynamics) method to identify the parametric time dependencies and enable data-driven discovery of non-stationary stochastic differential equations. We test our method on synthetic data using simple canonical systems, including nonlinear oscillators with underlying time-dependent parameters, and the chaotic Lorenz system. Using dynamic SINDy, we further discover the dynamics of neuronal activity during the locomotion of the nematode C. elegans. Our method uncovers a global nonlinear model with control as found elsewhere in the literature, showing that extensions of SINDy can be used with real, noisy, and chaotic datasets.  Finally, we aim to extend dynamic SINDy to a wide range of problems, specifically on dynamic systems where complex parametric time dependencies are expected.