Data-driven discovery of delay differential equations with discrete delays

• Published in: Journal of computational and applied mathematics 2025-06, Vol.461, p.116439, Article 116439
• Main Authors: Pecile, Alessandro, Demo, Nicola, Tezzele, Marco, Rozza, Gianluigi, Breda, Dimitri
• Format: Article
• Language: English
• Subjects: Bayesian optimization; Delay differential equations; Nonlinear dynamics; Sparse identification; Unknown delays
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2024.116439

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. •Data-driven system identification method to identify Delay Differential Equations.•Bayesian optimization to speedup the model discovery and the delay identification.•Multivariate setting to identify multiple unknown delays and non-multiplicative parameters.