Proper latent decomposition (PLD)*

The dynamics of fluids can be modelled on latent spaces, which are nonlinear manifolds that can be inferred by autoencoders. The latent space is, however, difficult to interpret, which calls for decomposition methods. Linear decomposition methods, such as POD, are not suitable for nonlinear manifolds. To generalise POD to nonlinear manifolds, we introduce the Proper Latent Decomposition (PLD) both mathematically and algorithmically. This approach allows us to explore the underlying flow structures in flows, without imposing a linear ansatz like in POD, and find a coordinate chart (Principal Geodesic Modes) on the latent space. We showcase the PLD on three systems: a laminar wake past a bluff body and the chaotic Kolmogorov flow. We find that the proposed methodology can extract physical modes and can provide a physically motivated reduced-order representation of turbulent flows, which is expressive and compact. This work provides a framework for decomposing turbulent flows with nonlinear methods, which opens opportunities for interpretability and reduced-order modelling.