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

TL;DR

This work extends Sparse Identification of Nonlinear Dynamics (SINDy) to systems with distributed delays by integrating the Linear Chain Trick (LCT). The method augments the state with a chain-ODE representation of memory, enabling SINDy to identify distributed-delay differential equations while preserving interpretability through explicit delayed terms. By constructing a grid over mean delay $\tau$ and chain length $p$, driving LCT chains with data, and using STRidge with Bayesian model selection, the approach jointly infers governing equations, delay characteristics, and the distributed-memory structure. Numerical experiments on Hes1–mRNA and distributed-delay logistic models demonstrate accurate recovery of the dynamics, robustness to noise and sparse sampling, and the advantage of modeling memory as an Erlang/Gamma kernel over purely discrete delays. The framework provides a data-driven, mechanistic means to discover nonlinear systems with distributed-delay memory in biology and engineering, with public code available for replication.

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.

Figures (7)

• Figure 1: Discrete-delay SINDy identification results. (a) simulation of Hes1 protein. (b) simulation of mRNA. The parameters used to generate the synthetic data are given in Table \ref{['tab:parameters']}. We use a broadly distributed delay with $p = 2$.
• Figure 2: Discrete-delay SINDy identification results. (a) simulation of Hes1 protein. (b) simulation of mRNA. We use the same parameter values as in Figure \ref{['fig:discreteSINDyH']}, except with $p = 100$.
• Figure 3: $LCT-SIND^3y$ identification results for distributed-delay Hes1--mRNA dynamics. Panels (a) and (b) correspond to a broadly distributed delay with $p = 2$, while panels (c) and (d) correspond to a sharply distributed delay with $p = 100$. Synthetic data are generated using the parameters listed in Table \ref{['tab:parameters']}.
• Figure 4: Comparison of LCT robustness. (a) Reconstruction stability vs noise ($\Delta t = 5$ minutes). Relative error of the identified $z_p$ (blue) computed from data sampled every 5 minutes, plotted against noise level, along with derivative errors of $P$ (red) and $M$ (yellow). (b) Reconstruction stability vs sampling (noise = 0.00). Relative error of the identified $z_p$ obtained from noise-free data with varying sampling step sizes.
• Figure 5: LCT--SINDy identification results for noisy Hes1--mRNA data (noise intensity $0.5$) with sampling step $dt = 0.5\,\mathrm{min}$. Left column: noisy measurements (blue) together with the smoothed signals (red); the vertical dashed line indicates the split between training and validation data. Right column: comparison between the ground-truth trajectories (solid blue) and the trajectories generated by the identified LCT--SINDy model (dashed red). Top row: data generated with ground-truth parameters $\tau = 15$ and $n = 10$, correctly identified as $\tau^\ast = 15$ and $n^\ast = 10$. Bottom row: data generated with ground-truth parameters $\tau = 20$ and $n = 20$, correctly identified as $\tau^\ast = 20$ and $n^\ast = 20$. Despite substantial measurement noise, the identified models accurately recover both the delay and the chain length and reproduce the underlying dynamics.

Theorems & Definitions (5)

• Lemma 3.1: cf. smith2010linearchain
• proof
• Proposition 3.2: LCT: existence–uniqueness and representation; cf. smith2010linearchain, Prop. 7.3
• Proposition 3.3: Stability of the LCT delayed state to input perturbations
• proof