Analyzing the Local Intrinsic Dimension of Physics-Informed Neural Network Latent Spaces for Burger's Equation

Abstract: Understanding how Physics-Informed Neural Networks (PINNs) encode complex physical phenomena, particularly challenging features like shocks, within their learned latent representations is crucial for interpreting and improving these models. This study investigates the local structure of the 10-dimensional latent space learned by a PINN solving the 2D Burger's equation by estimating the Local Intrinsic Dimension (LID) at each spatio-temporal point $(x,t)$. Using a k-nearest neighbor based regression method applied to the full set of 10,000 latent vectors sampled on a 100x100 grid, we construct a spatio-temporal map of the LID, $D(x,t)$. Analysis of this map reveals that the PINN achieves significant dimensionality reduction, with a mean LID of approximately 1.88, far below the embedding dimension of 10. Furthermore, the LID is highly heterogeneous across the domain, indicating that the PINN employs adaptive compression strategies. Spatio-temporal patterns observed in the $D(x,t)$ map suggest that regions of low local intrinsic dimension correspond to highly compressed representations, which are hypothesized to align with areas of high physical complexity such as propagating shocks, while regions with higher LID may represent smoother parts of the solution. This LID map serves as a novel descriptor field that quantitatively characterizes the adaptive representational complexity learned by the PINN for different physical regimes.

Subjects:	physics.comp-ph; cs.LG
Cite as:	PX:2508.00068