# Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data

arXiv: https://arxiv.org/abs/2404.19660 · HTML: https://mlsanity.com/paper/2404.19660/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 44 references that are in the map.
- First posted on arXiv 25 Apr 2024.

## Builds on (most cited first, 12 of 12)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [An Intriguing Failing of Convolutional Neural Networks and the CoordConv Solution](https://mlsanity.com/paper/1807.03247/) (2018)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Learning in Sinusoidal Spaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2109.09338/) (2021)
- [All You Need is Beyond a Good Init: Exploring Better Solution for Training Extremely Deep Convolutional Neural Networks with Orthonormality and Modulation](https://mlsanity.com/paper/1703.01827/) (2017)
- [Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows](https://mlsanity.com/paper/2109.01514/) (2021)
- [Grasping Extreme Aerodynamics on a Low-Dimensional Manifold](https://mlsanity.com/paper/2305.08024/) (2023)
- [Interpretable deep-learning models to help achieve the Sustainable Development Goals](https://mlsanity.com/paper/2108.10744/) (2021)
- [Symmetry-Aware Autoencoders: s-PCA and s-nlPCA](https://mlsanity.com/paper/2111.02893/) (2021)
- [Auto-Encoded Reservoir Computing for Turbulence Learning](https://mlsanity.com/paper/2012.10968/) (2020)
- [On interpretability and proper latent decomposition of autoencoders](https://mlsanity.com/paper/2211.08345/) (2022)
- [Convolutional autoencoder for the spatiotemporal latent representation of turbulence](https://mlsanity.com/paper/2301.13728/) (2023)

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

- [Stability analysis of chaotic systems in latent spaces](https://mlsanity.com/paper/2410.00480/) (2024)
- [Online model learning with data-assimilated reservoir computers](https://mlsanity.com/paper/2504.16767/) (2025)
- [Latent attention on masked patches for flow reconstruction](https://mlsanity.com/paper/2603.02028/) (2026)

## Abstract (arXiv)

Turbulent flows are chaotic and multi-scale dynamical systems, which have large numbers of degrees of freedom. Turbulent flows, however, can be modelled with a smaller number of degrees of freedom when using the appropriate coordinate system, which is the goal of dimensionality reduction via nonlinear autoencoders. Autoencoders are expressive tools, but they are difficult to interpret. The goal of this paper is to propose a method to aid the interpretability of autoencoders. This is the decoder decomposition. First, we propose the decoder decomposition, which is a post-processing method to connect the latent variables to the coherent structures of flows. Second, we apply the decoder decomposition to analyse the latent space of synthetic data of a two-dimensional unsteady wake past a cylinder. We find that the dimension of latent space has a significant impact on the interpretability of autoencoders. We identify the physical and spurious latent variables. Third, we apply the decoder decomposition to the latent space of wind-tunnel experimental data of a three-dimensional turbulent wake past a bluff body. We show that the reconstruction error is a function of both the latent space dimension and the decoder size, which are correlated. Finally, we apply the decoder decomposition to rank and select latent variables based on the coherent structures that they represent. This is useful to filter unwanted or spurious latent variables, or to pinpoint specific coherent structures of interest. The ability to rank and select latent variables will help users design and interpret nonlinear autoencoders.