Mathematical Interpretation of PINN Latent Space for Burger's Equation: Learned Dynamics and Geometric Structure

Abstract: Interpreting the internal representations learned by Physics-Informed Neural Networks (PINNs) remains a significant challenge. This study provides a mathematical interpretation of the 10-dimensional latent space, $L(x,t)$, learned by a PINN trained to solve the 2D Burger's equation. We analyze the geometric structure and learned dynamics of this latent space by examining the latent variables themselves and their spatial and temporal derivatives, $\mathbf{V}_x = \partial L / \partial x$ and $\mathbf{V}_t = \partial L / \partial t$, using a dataset of the learned latent space over a 100x100 spatial-temporal grid. Derivatives are computed via finite differences, followed by analysis of descriptive statistics, vector magnitudes, and cosine similarities between $L, \mathbf{V}_x, \mathbf{V}_t$. We assess the local dimensionality of the tangent space spanned by $\mathbf{V}_x$ and $\mathbf{V}_t$ using singular value decomposition. Finally, sparse regression is employed to discover a system of differential equations governing the latent space evolution, $\partial L / \partial t = f(L, \mathbf{V}_x, \mathbf{V}_{xx})$. Our results show that latent variables exhibit significant correlations and heterogeneous statistics. Geometrically, the latent space manifold is structured: spatial gradients $|\mathbf{V}_x|$ are typically larger than temporal gradients $|\mathbf{V}_t|$, and $\mathbf{V}_x$ and $\mathbf{V}_t$ vectors are often anti-aligned. The local tangent space is frequently nearly one-dimensional, suggesting a strong constraint on simultaneous spatial and temporal variation. Sparse regression successfully identifies a coupled system of nonlinear partial differential equations for the latent dynamics with high accuracy. Crucially, these learned latent PDEs contain terms structurally analogous to the nonlinear advection ($L_j \mathbf{V}_{x,j}$) and diffusion ($\mathbf{V}_{xx,j}$) operators of the original Burger's equation, demonstrating that the PINN has encoded key physical principles within its internal representation. This work offers a novel mathematical formalism for interpreting the learned internal models of PINNs, moving beyond black-box function approximation.

Subjects:	physics.comp-ph; cs.LG
Cite as:	PX:2508.00066