# Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

arXiv: https://arxiv.org/abs/2111.02893 · HTML: https://mlsanity.com/paper/2111.02893/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 4 Nov 2021.

## Builds on (most cited first, 5 of 5)

- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [From Principal Subspaces to Principal Components with Linear Autoencoders](https://mlsanity.com/paper/1804.10253/) (2018)

## Built on by (most cited first, 8 of 8)

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Data-driven low-dimensional dynamic model of Kolmogorov flow](https://mlsanity.com/paper/2210.16708/) (2022)
- [Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data](https://mlsanity.com/paper/2404.19660/) (2024)
- [Decoding complexity: how machine learning is redefining scientific discovery](https://mlsanity.com/paper/2405.04161/) (2024)
- [Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow](https://mlsanity.com/paper/2312.10235/) (2023)
- [Convolutional autoencoders for the reconstruction of three-dimensional interfacial multiphase flows](https://mlsanity.com/paper/2508.04084/) (2025)
- [An autoencoder compression approach for accelerating large-scale inverse problems](https://mlsanity.com/paper/2304.04781/) (2023)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)

## Abstract (arXiv)

Nonlinear principal component analysis (NLPCA) via autoencoders has attracted attention in the dynamical systems community due to its larger compression rate when compared to linear principal component analysis (PCA). These model reduction methods experience an increase in the dimensionality of the latent space when applied to datasets that exhibit invariant samples due to the presence of symmetries. In this study, we introduce a novel machine learning embedding for autoencoders, which uses Siamese networks and spatial transformer networks to account for discrete and continuous symmetries, respectively. The Siamese branches autonomously find a fundamental domain to which all samples are transformed, without introducing human bias. The spatial transformer network discovers the optimal slicing template for continuous translations so that invariant samples are aligned in the homogeneous direction. Thus, the proposed symmetry-aware autoencoder is invariant to predetermined input transformations. This embedding can be employed with both linear and nonlinear reduction methods, which we term symmetry-aware PCA (s-PCA) and symmetry-aware NLPCA (s-NLPCA). We apply the proposed framework to the Kolmogorov flow to showcase the capabilities for a system exhibiting both a continuous symmetry as well as discrete symmetries.