Renormalization Group Analysis of PINN Latent Space Structure for the 2D Burger's Equation

Abstract: Understanding how Physics-Informed Neural Networks encode information about physical systems in their latent spaces, particularly across different scales and physical regimes determined by parameters like viscosity, is a key challenge. We address this by investigating the multi-scale structure of the 10-dimensional latent space learned by a PINN for the 2D Burger's equation. Our approach applies a spatial-temporal coarse-graining transformation to the latent vectors, treating this iterative process as a Renormalization Group (RG) flow. Using a dataset covering 25 viscosity values, we iteratively average latent vectors on the spatial-temporal grid and analyze the evolution of statistical properties derived from Principal Component Analysis (PCA)—including eigenvalues, effective dimensionality (ED\_99), and normalized Shannon entropy of the eigenvalue spectrum—as functions of the coarse-graining scale. Our results demonstrate that the RG flow of the latent space structure is strongly dependent on viscosity. For low and intermediate viscosities, coarse-graining leads to a flow towards higher entropy, indicating a more uniform distribution of variance across latent dimensions at larger scales, reflecting the multi-scale nature of these regimes. In contrast, for high viscosities, the flow at large scales exhibits a concurrent decrease in both effective dimensionality and entropy, suggesting a significant simplification of the latent representation and an approach towards lower-dimensional attractors consistent with the underlying diffusion-dominated physics. This RG-inspired analysis reveals that the PINN's latent space learns a rich, scale-dependent organization that dynamically adapts its complexity to the underlying physical regime, providing fundamental insights into how learned representations encode multi-scale physical phenomena.

Subjects:	physics.comp-ph; cs.LG
Cite as:	PX:2508.00073