Proper Latent Decomposition

TL;DR

This paper introduces Proper Latent Decomposition (PLD), a nonlinear reduced-order modeling technique using autoencoders and differential geometry to analyze high-dimensional flow data more effectively than traditional linear methods.

Contribution

The paper develops a novel nonlinear decomposition method on manifolds, combining autoencoders and geometric tools for improved reduced-order modeling of complex flows.

Findings

Successfully identified a semi-analytical solution for laminar flow.

Extracted dominant modes with physical structures in turbulent Kolmogorov flow.

Demonstrated the effectiveness of PLD compared to POD.

Abstract

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.