Data-driven Discovery of Delay Differential Equations with Discrete Delays

TL;DR

This paper extends the SINDy framework to identify delay differential equations with unknown delays using an augmented library and Bayesian optimization, significantly improving efficiency and expanding the class of discoverable models.

Contribution

The authors develop a novel method combining SINDy with Bayesian optimization to identify delay differential equations and unknown delays more efficiently.

Findings

Method reduces calls to SINDy compared to brute force

Successfully identifies multiple unknown delays and parameters

Supports various long-term behaviors in numerical tests

Abstract

The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian optimization. To identify a possibly unknown delay we minimize the reconstruction error over a set of candidates. The resulting methodology improves the overall performance by remarkably reducing the number of calls to SINDy with respect to a brute force approach. We also address a multivariate setting to identify multiple unknown delays and (non-multiplicative) parameters. Several numerical tests on delay differential equations with different long-term behavior, number of variables, delays, and parameters support the use of Bayesian optimization highlighting both the efficacy of the proposed methodology and its computational advantages. As a consequence, the class of discoverable models is significantly expanded.