Viscosity-Dependent Latent Space Structure in a PINN for Burger's Equation: Analysis via PCA and Fractal Dimension with a Renormalization Group Analogy

Abstract: Physics-Informed Neural Networks (PINNs) learn compressed representations of physical systems in their latent spaces, but how these representations encode physical parameters like viscosity is not fully understood. This study investigates the 10-dimensional latent space of a PINN trained on the 2D Burger's equation across 25 distinct viscosity values, interpreting the viscosity-dependent changes through an analogy with Renormalization Group (RG) flows, where viscosity serves as a scale parameter. Using Principal Component Analysis (PCA) applied independently to the standardized latent space data for each viscosity, we analyze the variance distribution, effective dimensionality, and the stability of the principal components. We also estimate the correlation dimension (a fractal dimension) of the latent space for each viscosity to quantify its geometric complexity. Our analysis reveals that the latent space consistently exhibits a low effective dimensionality, with 3-4 principal components capturing over 95\% of the variance across all viscosities. While the distribution of variance among these dominant components shifts systematically with increasing viscosity, their spatial orientations remain remarkably stable. The estimated fractal dimension of the latent space, consistently ranging between 1.5 and 1.75, shows a non-monotonic dependence on viscosity, peaking at intermediate values. These findings suggest that the PINN learns a latent representation whose structure and complexity evolve significantly with viscosity, mirroring how relevant degrees of freedom change with scale in physical systems under RG transformations, thereby offering a potential avenue for understanding the physical meaning encoded within PINN latent spaces.

Subjects:	physics.comp-ph; cs.LG
Cite as:	PX:2508.00070