# Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction

arXiv: https://arxiv.org/abs/2204.02462 · HTML: https://mlsanity.com/paper/2204.02462/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2022.

## Built on by (most cited first, 30 of 30)

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- [Interpretable learning of effective dynamics for multiscale systems](https://mlsanity.com/paper/2309.05812/) (2023)
- [Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning](https://mlsanity.com/paper/2211.00357/) (2022)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)
- [Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data](https://mlsanity.com/paper/2303.11379/) (2023)
- [The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models](https://mlsanity.com/paper/2410.04668/) (2024)
- [Hybrid coupling with operator inference and the overlapping Schwarz alternating method](https://mlsanity.com/paper/2511.20687/) (2025)
- [System stabilization with policy optimization on unstable latent manifolds](https://mlsanity.com/paper/2407.06418/) (2024)
- [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)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)
- [Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction](https://mlsanity.com/paper/2507.20463/) (2025)
- [Leveraging time and parameters for nonlinear model reduction methods](https://mlsanity.com/paper/2501.03853/) (2025)
- [Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs](https://mlsanity.com/paper/2505.00473/) (2025)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [KROM: Kernelized Reduced Order Modeling](https://mlsanity.com/paper/2603.00360/) (2026)
- [History-aware adaptive reduced-order models via incremental singular value decomposition](https://mlsanity.com/paper/2605.28684/) (2026)
- [Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design](https://mlsanity.com/paper/2403.18044/) (2024)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systems](https://mlsanity.com/paper/2506.19609/) (2025)
- [Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference](https://mlsanity.com/paper/2601.00038/) (2025)
- [Data-driven stochastic reduced-order modeling of parametrized dynamical systems](https://mlsanity.com/paper/2601.10690/) (2026)
- [Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks](https://mlsanity.com/paper/2605.27756/) (2026)
- [In-context learning from self-generated trajectories for adaptive model reduction](https://mlsanity.com/paper/2607.02937/) (2026)
- [Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem](https://mlsanity.com/paper/2607.27433/) (2026)
- [Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization](https://mlsanity.com/paper/2610.12801/) (2026)

## Abstract (arXiv)

A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation that is aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however a much smaller dimension -- typically, n2 ~ √ n1, where n2 and n1 denote the dimensions of the quadratic and traditional PROMs, respectively. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude.