# Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction

arXiv: https://arxiv.org/abs/2507.20463 · HTML: https://mlsanity.com/paper/2507.20463/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 68 references that are in the map.
- First posted on arXiv 28 Jul 2025.

## Builds on (most cited first, 27 of 27)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (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)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference](https://mlsanity.com/paper/1908.11233/) (2019)
- [Bayesian operator inference for data-driven reduced-order modeling](https://mlsanity.com/paper/2204.10829/) (2022)
- [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)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [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)
- [Domain decomposition for data-driven reduced modeling of large-scale systems](https://mlsanity.com/paper/2311.00883/) (2023)
- [Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulations](https://mlsanity.com/paper/2401.05972/) (2024)
- [A Quadratic Manifold for Model Order Reduction of Nonlinear Structural Dynamics](https://mlsanity.com/paper/1610.09902/) (2016)
- [Data-driven balancing of linear dynamical systems](https://mlsanity.com/paper/2104.01006/) (2021)
- [Active operator inference for learning low-dimensional dynamical-system models from noisy data](https://mlsanity.com/paper/2107.09256/) (2021)
- [SOBMOR: Structured Optimization-Based Model Order Reduction](https://mlsanity.com/paper/2011.07567/) (2020)
- [Generalization of Quadratic Manifolds for Reduced Order Modeling of Nonlinear Structural Dynamics](https://mlsanity.com/paper/1610.09906/) (2016)
- [A quadratic decoder approach to nonintrusive reduced-order modeling of nonlinear dynamical systems](https://mlsanity.com/paper/2209.15412/) (2022)
- [Structured Optimization-Based Model Order Reduction for Parametric Systems](https://mlsanity.com/paper/2209.05101/) (2022)
- [Distributed computing for physics-based data-driven reduced modeling at scale: Application to a rotating detonation rocket engine](https://mlsanity.com/paper/2407.09994/) (2024)
- [Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2506.10224/) (2025)
- [Generalizations of data-driven balancing: What to sample for different balancing-based reduced models](https://mlsanity.com/paper/2312.12561/) (2023)

## Built on by (most cited first, 2 of 2)

- [Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems](https://mlsanity.com/paper/2601.12161/) (2026)
- [Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference](https://mlsanity.com/paper/2601.00038/) (2025)

## Abstract (arXiv)

Quadratic manifolds for nonintrusive reduced modeling are typically trained to minimize the reconstruction error on snapshot data, which means that the error of models fitted to the embedded data in downstream learning steps is ignored. In contrast, we propose a greedy training procedure that takes into account both the reconstruction error on the snapshot data and the prediction error of reduced models fitted to the data. Because our procedure learns quadratic manifolds with the objective of achieving accurate reduced models, it avoids oscillatory and other non-smooth embeddings that can hinder learning accurate reduced models. Numerical experiments on transport and turbulent flow problems show that quadratic manifolds trained with the proposed greedy approach lead to reduced models with up to two orders of magnitude higher accuracy than quadratic manifolds trained with respect to the reconstruction error alone.