Multi-objective SINDy for parameterized model discovery from single transient trajectory data

TL;DR

This paper extends the SINDy method to incorporate attractor data as soft constraints, enabling more robust and data-efficient identification of parameterized dynamical systems from limited transient trajectory data.

Contribution

The authors develop a multi-objective sparse regression framework that integrates attractor data as soft constraints, reducing data requirements and improving robustness in system identification.

Findings

More robust to noisy data

Requires less transient trajectory data

Successfully identifies parameterized systems in numerical examples

Abstract

The sparse identification of nonlinear dynamics (SINDy) has been established as an effective technique to produce interpretable models of dynamical systems from time-resolved state data via sparse regression. However, to model parameterized systems, SINDy requires data from transient trajectories for various parameter values over the range of interest, which are typically difficult to acquire experimentally. In this work, we extend SINDy to be able to leverage data on fixed points and/or limit cycles to reduce the number of transient trajectories needed for successful system identification. To achieve this, we incorporate the data on these attractors at various parameter values as constraints in the optimization problem. First, we show that enforcing these as hard constraints leads to an ill-conditioned regression problem due to the large number of constraints. Instead, we implement soft constraints by modifying the cost function to be minimized. This leads to the formulation of a multi-objective sparse regression problem where we simultaneously seek to minimize the error of the fit to the transients trajectories and to the data on attractors, while penalizing the number of terms in the model. Our extension, demonstrated on several numerical examples, is more robust to noisy measurements and requires substantially less training data than the original SINDy method to correctly identify a parameterized dynamical system.