Sparse identification of delay equations with distributed memory

TL;DR

This paper extends the SINDy framework to identify delay differential equations with distributed delays and renewal equations by reconstructing kernel functions through sparse regression, enabling data-driven modeling of systems with memory.

Contribution

It introduces a novel method for directly reconstructing kernel functions in delay equations with distributed memory using sparse regression and quadrature, advancing data-driven discovery.

Findings

Effective in identifying accurate models

Capable of capturing systems with distributed memory

Produces interpretable results

Abstract

We present a novel extension of the SINDy framework to delay differential equations with {\it distributed delays} and {\it renewal equations}, where typically the dependence from the past manifests via integrals in which the history is weighted through specific functions that are in general nonautonomous. Using sparse regression following the application of suitable quadrature formulas, the proposed methodology aims at directly reconstructing these kernel functions, thereby capturing the dynamics of the underlying infinite-dimensional systems. Numerical experiments confirm the effectiveness of the presented approach in identifying accurate and interpretable models, thus advancing data-driven discovery towards systems with distributed memory.