Physically Interpretable Representation Learning with Gaussian Mixture Variational AutoEncoder (GM-VAE)

TL;DR

The paper tackles the challenge of extracting physically meaningful, low-dimensional representations from high-dimensional turbulent and combustion data. It introduces a Gaussian Mixture Variational Autoencoder trained with an EM-inspired block-coordinate descent, paired with a graph-Laplacian based interpretability metric to ensure latent coordinates vary smoothly with key physical quantities. The approach yields latent spaces where regimes align with distinct physical states, demonstrated across surface-reaction bifurcations, wake flows, and Schlieren ignition images, and outperforms standard nonlinear DR methods in physical interpretability. This framework enables both regime discovery and generative sampling within physically coherent states, offering a robust tool for data-driven discovery in turbulent and reactive systems. Limitations include computational complexity, sensitivity to initialization, and the Gaussian-regime assumption, with future work targeting multi-modal integration and product-of-experts fusion to broaden applicability.

Abstract

Extracting compact, physically interpretable representations from high-dimensional scientific data is a persistent challenge due to the complex, nonlinear structures inherent in physical systems. We propose a Gaussian Mixture Variational Autoencoder (GM-VAE) framework designed to address this by integrating an Expectation-Maximization (EM)-inspired training scheme with a novel spectral interpretability metric. Unlike conventional VAEs that jointly optimize reconstruction and clustering (often leading to training instability), our method utilizes a block-coordinate descent strategy, alternating between expectation and maximization steps. This approach stabilizes training and naturally aligns latent clusters with distinct physical regimes. To objectively evaluate the learned representations, we introduce a quantitative metric based on graph-Laplacian smoothness, which measures the coherence of physical quantities across the latent manifold. We demonstrate the efficacy of this framework on datasets of increasing complexity: surface reaction ODEs, Navier-Stokes wake flows, and experimental laser-induced combustion Schlieren images. The results show that our GM-VAE yields smooth, physically consistent manifolds and accurate regime clustering, offering a robust data-driven tool for interpreting turbulent and reactive flow systems.

Figures (11)

• Figure 1: The GM-VAE framework enables both dimension reduction and generative sampling. In addition to the standard VAE pipeline, this framework introduces a Gaussian Mixture prior for the latent variables to impose a clustering structure in the latent space, which reflects distinct physical regimes. Conceptually, the GM-VAE assumes that our data naturally form clusters corresponding to distinct physical regimes or states. During training, the model jointly minimizes reconstruction error and encourages the latent representations to align with a Gaussian-mixture distribution. The model is thus encouraged to learn a latent manifold where nearby points share similar physical characteristics, and transitions between clusters reflect real physical transitions.
• Figure 2: Training samples (left), generated samples (middle), and latent representations (right) for surface reaction simulations. The GM-VAE compresses the data into 2 latent dimensions and achieves perfect clustering of the bifurcation outcomes.
• Figure 3: The evolution of the neural network predicted marginal cluster probability $\pi$, aggregated across trainings with different random initializations.
• Figure 4: Physical parameters of the simulations and 2D latent embeddings learned by tSNE, UMAP, and GM-VAE (up to an affine transformation computed from solving a linear regression problem, see details in Appendix \ref{['sec:affine-mapping']}). GM-VAE recovers the physical parameter space from data-driven learning by learning a latent manifold where the axes closely correlate with key physical quantities.
• Figure 5: Gap size and Reynolds number visualized in the 2D latent manifold learned by t-SNE, UMAP, and GM-VAE.