# Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

arXiv: https://arxiv.org/abs/2107.02597 · HTML: https://mlsanity.com/paper/2107.02597/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 75 references that are in the map.
- First posted on arXiv 6 Jul 2021.

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

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Learning physics-based reduced-order models for a single-injector combustion process](https://mlsanity.com/paper/1908.03620/) (2019)
- [Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference](https://mlsanity.com/paper/1908.11233/) (2019)
- [Data-driven Structured Realization](https://mlsanity.com/paper/1611.02072/) (2016)
- [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)
- [Learning low-dimensional dynamical-system models from noisy frequency-response data with Loewner rational interpolation](https://mlsanity.com/paper/1910.00110/) (2019)
- [Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations](https://mlsanity.com/paper/2005.05890/) (2020)
- [Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories](https://mlsanity.com/paper/2103.01362/) (2021)
- [Representing model inadequacy: A stochastic operator approach](https://mlsanity.com/paper/1604.01651/) (2016)
- [Reduced-order modeling of LPV systems in the Loewner framework](https://mlsanity.com/paper/2104.10767/) (2021)

## Built on by (most cited first, 16 of 16)

- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method](https://mlsanity.com/paper/2409.01433/) (2024)
- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes](https://mlsanity.com/paper/2310.07485/) (2023)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [Data-Driven Discovery of PDEs via the Adjoint Method](https://mlsanity.com/paper/2401.17177/) (2024)
- [System stabilization with policy optimization on unstable latent manifolds](https://mlsanity.com/paper/2407.06418/) (2024)
- [Physically consistent predictive reduced-order modeling by enhancing Operator Inference with state constraints](https://mlsanity.com/paper/2502.03672/) (2025)
- [Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction](https://mlsanity.com/paper/2507.20463/) (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)
- [Inference of Continuous Linear Systems from Data with Guaranteed Stability](https://mlsanity.com/paper/2301.10060/) (2023)
- [Learning reduced-order Quadratic-Linear models in Process Engineering using Operator Inference](https://mlsanity.com/paper/2402.17698/) (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)

## Abstract (arXiv)

Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). This work focuses on the large class of physical systems that can be well described by models with quadratic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto quadratic models. The proposed regularizer is physics informed in the sense that it penalizes quadratic terms with large norms and so explicitly leverages the quadratic model form that is given by the underlying physics. This means that the proposed approach judiciously learns from data and physical insights combined, rather than from either data or physics alone. Additionally, a formulation of operator inference is proposed that enforces model constraints for preserving structure such as symmetry and definiteness in the linear terms. Numerical results demonstrate that models learned with operator inference and the proposed regularizer and structure preservation are accurate and stable even in cases where using no regularization or Tikhonov regularization leads to models that are unstable.