Barycentric rational approximation for learning the index of a dynamical system from limited data

TL;DR

This paper introduces a novel approach for data-driven identification of dynamical systems with non-standard high-frequency behavior, using rational surrogate models with prescribed relative degree to improve extrapolation from limited data.

Contribution

The authors develop a method to construct rational surrogate models with specified relative degree and a routine to estimate the system's degree from low-frequency data, enhancing modeling accuracy.

Findings

Improved high-frequency behavior modeling with prescribed relative degree.

Enhanced extrapolation capabilities from low-frequency data.

Robustness demonstrated through numerical tests.

Abstract

We consider the task of data-driven identification of dynamical systems, specifically for systems whose behavior at large frequencies is non-standard, as encoded by a non-trivial relative degree of the transfer function or, alternatively, a non-trivial index of a corresponding realization as a descriptor system. We develop novel surrogate modeling strategies that allow state-of-the-art rational approximation algorithms (e.g., AAA and vector fitting) to better handle data coming from such systems with non-trivial relative degree. Our contribution is twofold. On one hand, we describe a strategy to build rational surrogate models with prescribed relative degree, with the objective of mirroring the high-frequency behavior of the high-fidelity problem, when known. The surrogate model's desired degree is achieved through constraints on its barycentric coefficients, rather than through ad-hoc modifications of the rational form. On the other hand, we present a degree-identification routine that allows one to estimate the unknown relative degree of a system from low-frequency data. By identifying the degree of the system that generated the data, we can build a surrogate model that, in addition to matching the data well (at low frequencies), has enhanced extrapolation capabilities (at high frequencies). We showcase the effectiveness and robustness of the newly proposed method through a suite of numerical tests.