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

TL;DR

This paper introduces GM-VAE, a novel variational autoencoder framework with an EM-inspired training scheme and a spectral interpretability metric, enabling stable, physically meaningful representations of complex scientific data.

Contribution

The paper presents GM-VAE with a block-coordinate descent training method and a new interpretability metric, improving physical interpretability and stability over traditional VAEs.

Findings

GM-VAE produces smooth, physically consistent latent manifolds.

The method accurately clusters physical regimes in complex datasets.

Training stability is enhanced compared to conventional VAEs.

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.