Sparse identification of delay equations with distributed memory

We present a novel extension of the SINDy framework to delay differential equations with distributed delays and 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.