β-Variational autoencoders for nonlinear and ortogonal reduced-order models in turbulence*

In this work we propose a deep-learning framework for learning a minimal and near-orthogonal set of non-linear modes in the context of turbulent flows. In particular, we focus on a high-fidelity numerical database of a simplified urban environment (Lazpita et al., Phys. Fluids 34, 051702, 2022). The proposed technique relies on β-variational autoencoders (β-VAEs) and convolutional neural networks (CNNs), which enable extracting non-linear modes while encouraging the learning of statistically-independent latent variables and penalizing the size of the latent vector. Moreover, we introduce an algorithm for ordering the resulting modes with respect to their contribution to the reconstruction. We demonstrate that by constraining the shape of the latent space, it is possible to motivate orthogonality and extract a set of parsimonious modes which enable high-quality reconstruction. Our results show the excellent performance of the method in the reconstruction against linear-theory-based decompositions, where the energy percentage captured by the proposed method from five modes is equal to 87.36% against 32.41% from POD. Furthermore, we show the ability of our approach to extract near-orthogonal modes with the determinant of the correlation matrix, which is equal to 0.99, thus enhancing the interpretability of the obtained reduced-order models (ROMs).