  We propose a deep probabilistic-neural-network architecture for learning a
minimal and near-orthogonal set of non-linear modes from high-fidelity
turbulent-flow-field data useful for flow analysis, reduced-order modeling, and
flow control. Our approach is based on $\beta$-variational autoencoders
($\beta$-VAEs) and convolutional neural networks (CNNs), which allow us to
extract non-linear modes from multi-scale turbulent flows while encouraging the
learning of independent latent variables and penalizing the size of the latent
vector. Moreover, we introduce an algorithm for ordering VAE-based modes with
respect to their contribution to the reconstruction. We apply this method for
non-linear mode decomposition of the turbulent flow through a simplified urban
environment, where the flow-field data is obtained based on well-resolved
large-eddy simulations (LESs). We demonstrate that by constraining the shape of
the latent space, it is possible to motivate the orthogonality and extract a
set of parsimonious modes sufficient for high-quality reconstruction. Our
results show the excellent performance of the method in the reconstruction
against linear-theory-based decompositions. Moreover, we compare our method
with available AE-based models. We show the ability of our approach in the
extraction of near-orthogonal modes that may lead to interpretability.
