Sparse identification of time delay systems via pseudospectral collocation

TL;DR

This paper tackles data-driven discovery of time-delay dynamics from measurements, focusing on unknown discrete delays in delay differential equations (DDEs). It introduces P-SINDy, a pragmatic SINDy variant that reduces the DDE to a finite-dimensional ODE via pseudospectral collocation, enabling sparse identification while external optimization is limited to the maximum delay $\bar{\tau}$, with the collocation degree $M$ controlling accuracy. The approach uses an extended collocation library and a regression framework to fit the data while ensuring trajectory consistency, and it is benchmarked against E-SINDy across four DDE models using brute-force, Bayesian optimization, and particle swarm optimization. Results show that P-SINDy often matches or surpasses the accuracy of baselines with significantly fewer SINDy evaluations and computational cost, highlighting its practical utility for data-driven discovery of time-delay systems and its potential for extension to distributed delays via quadrature.

Abstract

We present a pragmatic approach to the sparse identification of nonlinear dynamics for systems with discrete delays. It relies on approximating the underlying delay model with a system of ordinary differential equations via pseudospectral collocation. To minimize the reconstruction error, the new strategy avoids optimizing all possible multiple unknown delays, identifying only the maximum one. The computational burden is thus greatly reduced, improving the performance of recent implementations that work directly on the delay system.

Figures (6)

• Figure 1: Trajectory reconstruction for \ref{['DLE']}, see text.
• Figure 2: Errors $\epsilon$ and $\epsilon_{10}$ for \ref{['DLE']} with BF.
• Figure 3: Errors $\epsilon$ and $\epsilon_{10}$ for \ref{['DLE']} with BO and PO.
• Figure 4: Trajectory reconstruction for \ref{['MG']}, see text.
• Figure 5: Trajectory reconstruction for \ref{['2tau']}, see text.