Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data

TL;DR

This paper tackles the interpretability challenge of nonlinear autoencoders when applied to high-dimensional turbulent flows. It introduces the decoder decomposition, a post-processing method that links latent variables to coherent flow structures by leveraging POD modes and decoder sensitivities, and provides formulations for both standard AEs and mode-decomposing autoencoders (MD-AEs). Through analyses of a laminar cylinder wake and wind-tunnel turbulence data, it demonstrates that latent-space dimension strongly affects interpretability, while decoder size can compensate for limited latent variables in complex flows. The work also shows how to rank and filter latent variables to isolate specific structures, such as vortex shedding, enabling targeted, physically meaningful nonlinear reduced-order models. Overall, the decoder decomposition offers a practical, physics-informed approach to designing and interpreting nonlinear autoencoders for fluid-mechanical applications.

Abstract

Turbulent flows are chaotic and multi-scale dynamical systems, which have large numbers of degrees of freedom. Turbulent flows, however, can be modelled with a smaller number of degrees of freedom when using the appropriate coordinate system, which is the goal of dimensionality reduction via nonlinear autoencoders. Autoencoders are expressive tools, but they are difficult to interpret. The goal of this paper is to propose a method to aid the interpretability of autoencoders. This is the decoder decomposition. First, we propose the decoder decomposition, which is a post-processing method to connect the latent variables to the coherent structures of flows. Second, we apply the decoder decomposition to analyse the latent space of synthetic data of a two-dimensional unsteady wake past a cylinder. We find that the dimension of latent space has a significant impact on the interpretability of autoencoders. We identify the physical and spurious latent variables. Third, we apply the decoder decomposition to the latent space of wind-tunnel experimental data of a three-dimensional turbulent wake past a bluff body. We show that the reconstruction error is a function of both the latent space dimension and the decoder size, which are correlated. Finally, we apply the decoder decomposition to rank and select latent variables based on the coherent structures that they represent. This is useful to filter unwanted or spurious latent variables, or to pinpoint specific coherent structures of interest. The ability to rank and select latent variables will help users design and interpret nonlinear autoencoders.

Figures (28)

• Figure 1: Snapshots of the streamwise velocity of the laminar wake dataset at different times within the same period. A vortex shedding period is denoted with $T^{\hbox{lam}}$. The area bounded by the grey box is used for training.
• Figure 2: POD of the laminar wake $\pmb{U}$. Left: the percentage energy contained in the first six POD modes of the unsteady wake $\pmb{\Phi}^{\hbox{lam}}$ (data mode). Data modes 1&2, 3&4 and 5&6 contain similar flow energy and oscillate at the same frequency but out of phase. Centre: phase plot of the first two data time coefficients. Right: the frequency spectrum of the data and the data time coefficients 1, 3, 5 and 7, normalized by the standard deviation. The data contains the vortex shedding frequency and its harmonics. (Since each pair has the same frequency spectrum, only the odd data modes are shown here).
• Figure 3: Experimental set-up, reproduced from rigas2021experiment. The dimensions $x_1$ and $x_2$ are the measured location nondimensionalized by the diameter $D$. The black dots mark the location of the pressure sensors.
• Figure 4: The wind-tunnel pressure dataset $\pmb{P}$. Left: Mean pressure. Centre: RMS pressure. Right: The premultiplied PSD ($\hbox{St} \cdot$ PSD) of the wind-tunnel dataset, with peaks at $\hbox{St} \approx 0.002, 0.06$ and $\hbox{St} \approx 0.2$ and its harmonics. The peaks correspond to the three-dimensional rotation of the wake, the pulsation of the vortex core, and the vortex shedding and its harmonics, respectively.
• Figure 5: POD of the wind-tunnel dataset $\pmb{P}$. Left: percentage energy of the first 10 data modes. Right: cumulative percentage energy of POD modes. The reconstruction of the pressure dataset to 95% energy needs 21 data modes.