# Nonlinear proper orthogonal decomposition for convection-dominated flows

arXiv: https://arxiv.org/abs/2110.08295 · HTML: https://mlsanity.com/paper/2110.08295/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 56 references that are in the map.
- First posted on arXiv 15 Oct 2021.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [An autoencoder-based reduced-order model for eigenvalue problems with application to neutron diffusion](https://mlsanity.com/paper/2008.10532/) (2020)
- [Efficient nonlinear manifold reduced order model](https://mlsanity.com/paper/2011.07727/) (2020)
- [Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation](https://mlsanity.com/paper/2007.12167/) (2020)
- [Neural Dynamic Mode Decomposition for End-to-End Modeling of Nonlinear Dynamics](https://mlsanity.com/paper/2012.06191/) (2020)
- [Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators](https://mlsanity.com/paper/2106.09658/) (2021)

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

- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning](https://mlsanity.com/paper/2202.05460/) (2022)
- [Digital Twins in Wind Energy: Emerging Technologies and Industry-Informed Future Directions](https://mlsanity.com/paper/2304.11405/) (2023)
- [textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications](https://mlsanity.com/paper/2207.11842/) (2022)
- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes](https://mlsanity.com/paper/2202.06170/) (2022)
- [Intrinsic Dimension Estimating Autoencoder (IDEA) Using CancelOut Layer and a Projected Loss](https://mlsanity.com/paper/2509.10011/) (2025)
- [Prospects of federated machine learning in fluid dynamics](https://mlsanity.com/paper/2208.07017/) (2022)
- [Slim multi-scale convolutional autoencoder-based reduced-order models for interpretable features of a complex dynamical system](https://mlsanity.com/paper/2501.03070/) (2025)
- [End-to-end data-driven prediction of urban airflow and pollutant dispersion](https://mlsanity.com/paper/2603.17606/) (2026)

## Abstract (arXiv)

Autoencoder techniques find increasingly common use in reduced order modeling as a means to create a latent space. This reduced order representation offers a modular data-driven modeling approach for nonlinear dynamical systems when integrated with a time series predictive model. In this letter, we put forth a nonlinear proper orthogonal decomposition (POD) framework, which is an end-to-end Galerkin-free model combining autoencoders with long short-term memory networks for dynamics. By eliminating the projection error due to the truncation of Galerkin models, a key enabler of the proposed nonintrusive approach is the kinematic construction of a nonlinear mapping between the full-rank expansion of the POD coefficients and the latent space where the dynamics evolve. We test our framework for model reduction of a convection-dominated system, which is generally challenging for reduced order models. Our approach not only improves the accuracy, but also significantly reduces the computational cost of training and testing.