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Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference

Pawan Goyal, Igor Pontes Duff, Peter Benner

TL;DR

This work develops stability-guaranteed operator inference for quadratic dynamical systems by design. It introduces three stability classes—local asymptotic stability (LAS), global asymptotic stability (GAS), and attracting trapping regions (ATR)—via dedicated parameterizations that encode Lyapunov-based guarantees, including energy-preserving nonlinearities for GAS. An integral-form learning approach avoids derivative estimation, enabling robust learning from scarce or noisy data, and a gradient-based optimization framework under these parameterizations preserves stability at every iteration. The authors validate the methods on Burgers, Chafee-Infante, Lorenz, and MHD models, demonstrating enhanced stability and accuracy over traditional OpInf and sparse-regression baselines and enabling discovery of energy-preserving or bounded dynamics in a data-driven setting.

Abstract

Scientific machine learning for inferring dynamical systems combines data-driven modeling, physics-based modeling, and empirical knowledge. It plays an essential role in engineering design and digital twinning. In this work, we primarily focus on an operator inference methodology that builds dynamical models, preferably in low-dimension, with a prior hypothesis on the model structure, often determined by known physics or given by experts. Then, for inference, we aim to learn the operators of a model by setting up an appropriate optimization problem. One of the critical properties of dynamical systems is stability. However, this property is not guaranteed by the inferred models. In this work, we propose inference formulations to learn quadratic models, which are stable by design. Precisely, we discuss the parameterization of quadratic systems that are locally and globally stable. Moreover, for quadratic systems with no stable point yet bounded (e.g., chaotic Lorenz model), we discuss how to parameterize such bounded behaviors in the learning process. Using those parameterizations, we set up inference problems, which are then solved using a gradient-based optimization method. Furthermore, to avoid numerical derivatives and still learn continuous systems, we make use of an integral form of differential equations. We present several numerical examples, illustrating the preservation of stability and discussing its comparison with the existing state-of-the-art approach to infer operators. By means of numerical examples, we also demonstrate how the proposed methods are employed to discover governing equations and energy-preserving models.

Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference

TL;DR

This work develops stability-guaranteed operator inference for quadratic dynamical systems by design. It introduces three stability classes—local asymptotic stability (LAS), global asymptotic stability (GAS), and attracting trapping regions (ATR)—via dedicated parameterizations that encode Lyapunov-based guarantees, including energy-preserving nonlinearities for GAS. An integral-form learning approach avoids derivative estimation, enabling robust learning from scarce or noisy data, and a gradient-based optimization framework under these parameterizations preserves stability at every iteration. The authors validate the methods on Burgers, Chafee-Infante, Lorenz, and MHD models, demonstrating enhanced stability and accuracy over traditional OpInf and sparse-regression baselines and enabling discovery of energy-preserving or bounded dynamics in a data-driven setting.

Abstract

Scientific machine learning for inferring dynamical systems combines data-driven modeling, physics-based modeling, and empirical knowledge. It plays an essential role in engineering design and digital twinning. In this work, we primarily focus on an operator inference methodology that builds dynamical models, preferably in low-dimension, with a prior hypothesis on the model structure, often determined by known physics or given by experts. Then, for inference, we aim to learn the operators of a model by setting up an appropriate optimization problem. One of the critical properties of dynamical systems is stability. However, this property is not guaranteed by the inferred models. In this work, we propose inference formulations to learn quadratic models, which are stable by design. Precisely, we discuss the parameterization of quadratic systems that are locally and globally stable. Moreover, for quadratic systems with no stable point yet bounded (e.g., chaotic Lorenz model), we discuss how to parameterize such bounded behaviors in the learning process. Using those parameterizations, we set up inference problems, which are then solved using a gradient-based optimization method. Furthermore, to avoid numerical derivatives and still learn continuous systems, we make use of an integral form of differential equations. We present several numerical examples, illustrating the preservation of stability and discussing its comparison with the existing state-of-the-art approach to infer operators. By means of numerical examples, we also demonstrate how the proposed methods are employed to discover governing equations and energy-preserving models.
Paper Structure (24 sections, 9 theorems, 87 equations, 12 figures)

This paper contains 24 sections, 9 theorems, 87 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model Inference with Quadratic Non-linearity
  3. Inference of Locally Asymptotically Stable Quadratic Models
  4. Local asymptotic stabilty characterization
  5. Locally asymptotically stable model inference via suitable parameterization
  6. Inference of Globally Asymptotically Stable Quadratic Models
  7. Global asymptotic stability characterization
  8. Global-stability informed learning
  9. Quadratic Models with attracting trapping regions
  10. Boundedness and attracting trapping region characterization
  11. Attracting trapping regions informed learning
  12. Optimizing Through a Numerical Integration Scheme to Infer Operators
  13. Numerical Experiments
  14. Training set-up:
  15. Setting $\mathbf{Q}=\mathbf{I}$ for gasMI and atrMI:
  16. ...and 9 more sections

Key Result

Lemma 1

Consider a quadratic system as in eq:quad_model, where $\mathbf{A}$ takes the form given in eq:par_stabil, i.e., $\mathbf{A} = (\mathbf{J} -\mathbf{R})\mathbf{Q}$, where $\mathbf{J} = -\mathbf{J}^\top$, and $\mathbf{R}, \mathbf{Q}$ are SPD matrices. Then, the quadratic function is a Lyapunov function of the system when $\|\mathbf{x}\|_2 <r$, where with $\mathbf{L}\mathbf{L}^\top = \mathbf{Q}^\t

Figures (12)

  • Figure 1: Burgers' equation with Dirichlet boundary conditions: Decay of the singular values obtained using the training data. The orange graph indicates how much energy is captured by how many dominant modes.
  • Figure 2: Burgers' equation with Dirichlet boundary conditions: A performance over all the test data of the inferred models. The missing dots indicate instability of the model.
  • Figure 3: Burgers' equation with Dirichlet boundary conditions: A comparison of the time-domain simulations of the inferred models on an initial test condition.
  • Figure 4: Burgers' equation with Dirichlet boundary conditions: Comparisons of the qualities of the inferred models.
  • Figure 5: Chafee Infante equation: Decay of the singular values for the $v$ and $w := \tfrac{1}{2}v^2$ using the training data.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Corollary 2
  • ...and 9 more