Proper Latent Decomposition

TL;DR

This work addresses the limitation of linear reduced-order models for nonlinear turbulent dynamics by introducing proper latent decomposition (PLD), a nonlinear ROM on latent manifolds inferred through autoencoders. It combines Fréchet-mean based tangent-space PCA with geodesic mappings on a learned manifold and a metric-aware Eikonal framework to compute distances and robust log_p maps, enabling principal geodesics that reflect underlying physics. The authors demonstrate PLD on a laminar wake past a bluff body and on the Kolmogorov turbulent flow, obtaining a dominant geodesic mode with physical structure and, in the laminar case, a pathway toward a semi-analytical Navier–Stokes description. This manifold-aware methodology offers a principled way to perform nonlinear reduced-order modeling and gain physical insight from high-dimensional data, with implications for interpretability and robustness in ROMs.

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.

Figures (6)

• Figure 1: Overview of proper latent decomposition. Given a manifold $M$ (one-dimensional here for pictorial purposes) and data $\mathcal{Z} \subset M$ (red dots), we wish to find modes, or principal geodesics, that best describe this data. This decomposition comprises three stages: (i) a mean $\mu \in M$ is computed on the manifold; (ii) data is mapped to the tangent space centered at the mean, $T_\mu M$, and singular value decomposition is performed to obtain an orthonormal basis in the tangent space; (iii) these basis vectors are mapped back down to the manifold, yielding principal geodesics.
• Figure 2: Proper latent decomposition of the laminar flow. Left and right panels show the manifold and the tangent space to the mean respectively. Encoded snapshots of vorticity are depicted as red points, and the resulting geodesic modes are shown in black. An ellipse has been fit to the data in the tangent space, and the resulting trajectory is shown in purple in both panels. The left panel displays contours of geodesic distance around the Fréchet mean, highlighting the nonlinearity of the manifold.
• Figure 3: Leading principal geodesic mode for the laminar wake. Snapshots are visualized along the trajectory of the principal geodesic, showing the transition from the Fréchet mean, to a physical snapshot of the laminar wake.
• Figure 4: Proper latent decomposition of the Kolmogorov flow. Left and right panels show the manifold and the tangent space to the mean respectively. Encoded snapshots of vorticity are depicted as white points on the manifold, with the colormap showing the distance from the Fréchet mean.
• Figure 5: Leading principal geodesic mode for the Kolmogorov flow. Snapshots are visualized along the trajectory of the principal geodesic, showing the transition from the Fréchet mean to a physical snapshot of turbulence.