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

Davide Pradovera, Ion Victor Gosea, Jan Heiland

TL;DR

This work addresses learning the index of a dynamical system, encoded as the transfer-function relative degree $rdeg(r)$, from limited frequency-domain data, particularly for descriptor systems with non-proper high-frequency behavior. It develops a barycentric rational surrogate framework that can enforce a prescribed degree via linear constraints on barycentric weights and includes a data-driven degree-identification routine using AAA/least-squares fits, complemented by a stable two-region extrapolation strategy combining a barycentric form for low frequencies with an asymptotic expansion for large frequencies. The authors establish an algebraic link between degree and barycentric coefficients, introduce constrained AAA (and extensions to VF) for degree-preserving surrogate modeling, and provide complexity and consistency analyses alongside extensive numerical demonstrations on mass-spring, Oseen flow, and SLICOT MNA benchmarks. The proposed approach enables reliable high-frequency extrapolation and degree-aware MOR from limited data, with competitive runtimes and the ability to reveal the underlying system index without a priori degree information, albeit with noted sensitivity to noise and data bandwidth in practice.

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.

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

TL;DR

This work addresses learning the index of a dynamical system, encoded as the transfer-function relative degree , from limited frequency-domain data, particularly for descriptor systems with non-proper high-frequency behavior. It develops a barycentric rational surrogate framework that can enforce a prescribed degree via linear constraints on barycentric weights and includes a data-driven degree-identification routine using AAA/least-squares fits, complemented by a stable two-region extrapolation strategy combining a barycentric form for low frequencies with an asymptotic expansion for large frequencies. The authors establish an algebraic link between degree and barycentric coefficients, introduce constrained AAA (and extensions to VF) for degree-preserving surrogate modeling, and provide complexity and consistency analyses alongside extensive numerical demonstrations on mass-spring, Oseen flow, and SLICOT MNA benchmarks. The proposed approach enables reliable high-frequency extrapolation and degree-aware MOR from limited data, with competitive runtimes and the ability to reveal the underlying system index without a priori degree information, albeit with noted sensitivity to noise and data bandwidth in practice.

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.
Paper Structure (22 sections, 6 theorems, 33 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 22 sections, 6 theorems, 33 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The barycentric form
  3. Motivation: DAE index and relative degree
  4. Rational approximation using the barycentric form
  5. Relative degree of barycentric forms
  6. Rational approximation with given relative degree
  7. Stable extrapolation
  8. Data-driven identification of the relative degree
  9. Complexity
  10. Consistency
  11. Extensions to least-squares rational approximation
  12. Numerical examples
  13. Results with AAA
  14. Forward 3-mass system
  15. Inverted 3-mass system
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $\{\alpha_k\}_{k=0}^m\cup\{s_k\}_{k=0}^m\subset\mathbb C$. If $|s|>\max_{k=0,\ldots,m}|s_k|$, we have the series representation

Figures (5)

  • Figure 1: Numerical results for the forward transfer function of the $3$-mass chain. The gray band is the sampled frequency range. Black dots denote the point $R_{\text{cutoff}}$, where the barycentric rational form gives way to the asymptotic rational form. The orange dash-dotted curves pertain to the adaptive method's results using purely the barycentric form.
  • Figure 2: Numerical results for the inverted transfer function of the $3$-mass chain. The gray band is the sampled frequency range. Black dots denote the point $R_{\text{cutoff}}$, where the barycentric rational form gives way to the asymptotic rational form. The orange dash-dotted curves pertain to the adaptive method's results using purely the barycentric form.
  • Figure 3: Numerical results for two entries of the MNA_1 transfer-function matrix. The gray band is the sampled frequency range.
  • Figure 4: Numerical results for the entry $(2,3)$ of the MNA_1 transfer-function matrix. The gray band is the sampled frequency range.
  • Figure 5: Illustration of the sample mechanical system with $n=2$ masses.

Theorems & Definitions (16)

  • Remark 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.1
  • ...and 6 more