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

2023/08/26 by Pawan Goyal, Goyal, Pawan, Igor Pontes Duff +3 · 20 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Artificial intelligence #Bounded function #Computer science #Control Systems and Identification #Dynamical Systems (math.DS) #Dynamical system (definition) #Dynamical systems theory #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Inference #Machine Learning (cs.LG) #Machine learning #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Operator (biology) #Quadratic equation #Stability (learning theory) #Theoretical computer science

paper · pdf · doi:10.48550/arxiv.2308.13819

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2023/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees
• On Boundedness of Quadratic Dynamics with Energy-Preserving Nonlinearity
• Physics-Informed Machine Learning for Characterizing System Stability
• Data-driven nonlinear aerodynamics models with certifiably optimal boundedness properties
• Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models
• Exact operator inference with minimal data
• Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms

Related

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• Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference
• On the representation of energy-preserving quadratic operators with application to Operator Inference
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