AutoEncoders latent space interpretability in the light of proper orthogonal decomposition: Machine learning of periodically forced fluid flows

Abstract

This work explores the learning and interpretability challenges of Autoencoders (AEs) and Variational Autoencoders (VAEs) when applied to the reconstruction of dynamic velocity fields governed by the Navier-Stokes equations. Throughout model training, the emphasis is on understanding how flow features are encoded into the latent space and how this impacts the interpretability and usability of the models. Based on a parametric study of forced flows, i.e. flows around an oscillating cylinder, as well as a von Kármán swirling flow, we first investigate the trade-offs between reconstruction accuracy and regularization in VAEs. We confirm that increasing the regularization parameter degrades reconstruction quality, which underscores a significant limitation of the Gaussian prior from this point of vue. A comparative analysis reveals that standard AEs exhibit quite robust training behaviour, while VAEs show a sharper transition between non-learning and learning regimes, depending on the amount of regularization. By leveraging Proper Orthogonal Decomposition (POD) to identify characteristic flow structures and frequencies, we establish connections between latent space organisations and POD modes. To address the interpretability challenge, we then perform a symmetry analysis of latent spaces, stating equivariance relations between latent and physical variables. Despite reduced reconstruction precision, VAEs show greater fidelity in preserving these relationships. Building on this, we propose a clustering-inspired method to interpret latent representations, identifying characteristic states from temporal POD time coefficients to provide deeper insights into latent space structure and untangling. This work highlights pathways for autoencoder's analysis methodological advancements, emphasizing the critical need to align latent space representations with physical interpretation for broader applicability in fluid dynamics.