# Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations

arXiv: https://arxiv.org/abs/2302.01278 · HTML: https://mlsanity.com/paper/2302.01278/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [RepVGG: Making VGG-style ConvNets Great Again](https://mlsanity.com/paper/2101.03697/) (2021)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Deep k-Means: Jointly clustering with k-Means and learning representations](https://mlsanity.com/paper/1806.10069/) (2018)
- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [Scheduling Dimension Reduction of LPV Models -- A Deep Neural Network Approach](https://mlsanity.com/paper/2005.06786/) (2020)
- [Robust output-feedback stabilization for incompressible flows using low-dimensional H∞-controllers](https://mlsanity.com/paper/2103.01608/) (2021)
- [Learning finite difference methods for reaction-diffusion type equations with FCNN](https://mlsanity.com/paper/2201.01854/) (2022)
- [Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis](https://mlsanity.com/paper/1812.02083/) (2018)

## Built on by (most cited first, 3 of 3)

- [Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design](https://mlsanity.com/paper/2403.18044/) (2024)
- [Slim multi-scale convolutional autoencoder-based reduced-order models for interpretable features of a complex dynamical system](https://mlsanity.com/paper/2501.03070/) (2025)
- [Generative prediction of laser-induced rocket ignition with dynamic latent space representations](https://mlsanity.com/paper/2603.07525/) (2026)

## Abstract (arXiv)

Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a certain compromise between accuracy and model complexity. However, for really low-dimensional parametrizations (for example for controller design) linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in two cylinder-wake scenarios modeled by the incompressible Navier-Stokes equations.