# Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flows

arXiv: https://arxiv.org/abs/2401.10620 · HTML: https://mlsanity.com/paper/2401.10620/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 51 references that are in the map.
- First posted on arXiv 19 Jan 2024.

## Builds on (most cited first, 13 of 13)

- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [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)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [DKM: Differentiable K-Means Clustering Layer for Neural Network Compression](https://mlsanity.com/paper/2108.12659/) (2021)
- [Differentiable Deep Clustering with Cluster Size Constraints](https://mlsanity.com/paper/1910.09036/) (2019)
- [An autoencoder-based reduced-order model for eigenvalue problems with application to neutron diffusion](https://mlsanity.com/paper/2008.10532/) (2020)
- [Scheduling Dimension Reduction of LPV Models -- A Deep Neural Network Approach](https://mlsanity.com/paper/2005.06786/) (2020)
- [Conditional variational autoencoder with Gaussian process regression recognition for parametric models](https://mlsanity.com/paper/2305.09625/) (2023)
- [Low-order Linear Parameter Varying Approximations for Nonlinear Controller Design for Flows](https://mlsanity.com/paper/2311.05305/) (2023)

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

- [Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspective](https://mlsanity.com/paper/2507.18220/) (2025)
- [Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design](https://mlsanity.com/paper/2403.18044/) (2024)

## Abstract (arXiv)

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network. Supported by several proofs, the model architecture ensures that all reconstructed states lie within a polytope, accompanied by a metric indicating the quality of the constructed polytopes, referred to as polytope error. Additionally, it offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition (POD). To validate our proposed model, we conduct simulations involving two flow scenarios with the incompressible Navier-Stokes equation. Numerical results demonstrate the guaranteed properties of the model, low reconstruction errors compared to POD, and the improvement in error using a clustering network.