Sparse Identification of Nonlinear Distributed-Delay Dynamics via the Linear Chain Trick

TL;DR

This paper extends the SINDy framework to identify nonlinear systems with distributed delays by integrating the Linear Chain Trick, enabling accurate, robust discovery of delay dynamics from time-series data.

Contribution

It introduces a novel method combining SINDy with the Linear Chain Trick to capture distributed delays, including delay mean and dispersion, from data.

Findings

Accurately reconstructs distributed delay dynamics in models.

Remains robust under noise and sparse data sampling.

Provides interpretable insights into delayed influences.

Abstract

The Sparse Identification of Nonlinear Dynamics (SINDy) framework has been frequently used to discover parsimonious differential equations governing natural and physical systems. This includes recent extensions to SINDy that enable the recovery of discrete delay differential equations, where delay terms are represented explicitly in the candidate library. However, such formulations cannot capture the distributed delays that naturally arise in biological, physical, and engineering systems. In the present work, we extend SINDy to identify distributed-delay differential equations by incorporating the Linear Chain Trick (LCT), which provides a finite-dimensional ordinary differential equation representing the distributed memory effects. Hence, SINDy can operate in an augmented state space using conventional sparse regression while preserving a clear interpretation of delayed influences via the chain trick. From time-series data, the proposed method jointly infers the governing equations, the mean delay, and the dispersion of the underlying delay distribution. We numerically verify the method on several models with distributed delay, including the logistic growth model and a Hes1--mRNA gene regulatory network model. We show that the proposed method accurately reconstructs distributed delay dynamics, remains robust under noise and sparse sampling, and provides a transparent, data-driven approach for discovering nonlinear systems with distributed-delay.