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

arXiv: https://arxiv.org/abs/2308.13819 · HTML: https://mlsanity.com/paper/2308.13819/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 58 references that are in the map.
- First posted on arXiv 26 Aug 2023.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Canonical and Noncanonical Hamiltonian Operator Inference](https://mlsanity.com/paper/2304.06262/) (2023)
- [Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition](https://mlsanity.com/paper/1808.02086/) (2018)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Operator Inference and Physics-Informed Learning of Low-Dimensional Models for Incompressible Flows](https://mlsanity.com/paper/2010.06701/) (2020)
- [Stability Domains for Quadratic-Bilinear Reduced-Order Models](https://mlsanity.com/paper/2009.02769/) (2020)
- [On computing the distance to stability for matrices using linear dissipative Hamiltonian systems](https://mlsanity.com/paper/1611.00595/) (2016)

## Built on by (most cited first, 11 of 11)

- [Energy-Preserving Reduced Operator Inference for Efficient Design and Control](https://mlsanity.com/paper/2401.02889/) (2024)
- [Data-driven identification of latent port-Hamiltonian systems](https://mlsanity.com/paper/2408.08185/) (2024)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2512.01984/) (2025)
- [Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems](https://mlsanity.com/paper/2601.12161/) (2026)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Learning reduced-order Quadratic-Linear models in Process Engineering using Operator Inference](https://mlsanity.com/paper/2402.17698/) (2024)
- [Stability-Certified Learning of Control Systems with Quadratic Nonlinearities](https://mlsanity.com/paper/2403.00646/) (2024)
- [Physics-Informed Machine Learning for Characterizing System Stability](https://mlsanity.com/paper/2511.08831/) (2025)
- [Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference](https://mlsanity.com/paper/2601.00038/) (2025)
- [SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs](https://mlsanity.com/paper/2609.33205/) (2026)

## Abstract (arXiv)

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.