Slim multi-scale convolutional autoencoder-based reduced-order models for interpretable features of a complex dynamical system

TL;DR

This paper addresses the interpretability gap in data-driven reduced-order models for high-dimensional turbulent flows by introducing Slim multi-scale convolutional autoencoder (SMS-CAE). SMS-CAE uses a constrained dropout mechanism to prioritize large-scale structure in early latent features, enabling interpretable, energy-ranked representations with minimal architectural changes and strong reconstruction performance. Across three Rayleigh-Bénard convection datasets, SMS-CAE delivers up to +6.4% NMSS improvement over POD for 64 modes and up to +229.8% for very small latent spaces, while using about 2% of the parameters of competing interpretable CAEs. The approach yields lightweight, scalable, and interpretable ROMs suitable for very low-dimensional reconstructions with practical applicability to turbulent flow analysis.

Abstract

In recent years, data-driven deep learning models have gained significant interest in the analysis of turbulent dynamical systems. Within the context of reduced-order models (ROMs), convolutional autoencoders (CAEs) pose a universally applicable alternative to conventional approaches. They can learn nonlinear transformations directly from data, without prior knowledge of the system. However, the features generated by such models lack interpretability. Thus, the resulting model is a black-box which effectively reduces the complexity of the system, but does not provide insights into the meaning of the latent features. To address this critical issue, we introduce a novel interpretable CAE approach for high-dimensional fluid flow data that maintains the reconstruction quality of conventional CAEs and allows for feature interpretation. Our method can be easily integrated into any existing CAE architecture with minor modifications of the training process. We compare our approach to Proper Orthogonal Decomposition (POD) and two existing methods for interpretable CAEs. We apply all methods to three different experimental turbulent Rayleigh-Bénard convection datasets with varying complexity. Our results show that the proposed method is lightweight, easy to train, and achieves relative reconstruction performance improvements of up to 6.4% over POD for 64 modes. The relative improvement increases to up to 229.8% as the number of modes decreases. Additionally, our method delivers interpretable features similar to those of POD and is significantly less resource-intensive than existing CAE approaches, using less than 2% of the parameters. These approaches either trade interpretability for reconstruction performance or only provide interpretability to a limited extend.

Figures (17)

• Figure 1: A schematic sketch of the experimental Rayleigh-Bénard cell. The transparent heating plate provides the optical access to conduct stereoscopic particle image velocimetry in the horizontal mid-plane of the cell. The working fluid, compressed sulfur hexafluoride, is in turbulent motion between the top (in blue) and bottom plates (in red). The side faces, which close the cell, are thermally insulated such that all the supplied heat has to pass through the fluid layer, either by diffusion or by convection.
• Figure 2: Instantaneous turbulent velocity field in the horizontal mid-plane for a Rayleigh number Ra $= 5.5\times 10^6$. The color map indicates the out-of-plane velocity component $w$, while the in-plane components $u$ and $v$ are represented by the vectors. Note that only one out of four vectors in each direction is plotted.
• Figure 3: Sketch of the shift and crop operation applied to an example snapshot for data augmentation.
• Figure 4: Basic convolutional autoencoder (CAE) architecture. Illustrated are the data dimensions at the different stages of the encoding and decoding process. The CAE takes a snapshot $\mathbf{x}$ and encodes it into a latent representation $\mathbf{z}$. Subsequently it is expanded into the reconstructed snapshot $\hat{\mathbf{x}}$. The squared brackets contain the shape of input, output and features on the forward pass. Here, $\rho$ is a hyper-parameter that controls the number of filters in the convolutional layers. Thus it determines the number of channels of the corresponding output feature maps.
• Figure 5: Minimal schematic architectures that we study in this paper. Each model is visualized with only three latent features for illustration. Per architecture, the left hand side represents the features as they leave the encoder and while the right hand side shows how they enter the decoder. (a) is the basic CAE that is also shown in Figure \ref{['fig:cae_architecture']} in detail. It simply forwards the latent features as they are. (b) and (c) show the existing interpretable CAE methods that have sub-encoders or -decoders that are responsible for a specific subset of features. (c) represents our method that randomly sets values to zero at the lower end of the latent feature vector during training.