Characterizing the Multi-Scale and Geometric Structure of PINN Latent Space via Wavelets and Ricci Scalar

Abstract: Understanding how Physics-Informed Neural Networks (PINNs) encode physical information within their internal representations, particularly the latent space, is key to their interpretability. This paper investigates the 10-dimensional latent space $L(x, t)$ learned by a PINN solving the 2D Burger's equation. We analyze each latent dimension $L_i(x, t)$ as a 2D function on a $100 \times 100$ spatio-temporal grid using two complementary mathematical tools. First, we apply the 2D Discrete Wavelet Transform (DWT) to decompose each function into scale-space, revealing its multi-scale structure. Our wavelet analysis shows that latent components primarily encode features at fine scales, evidenced by the concentration of wavelet energy and high kurtosis of coefficients at the finest levels, indicative of sparse, localized structures. Furthermore, the wavelet energy across scales follows a consistent power-law decay with exponents ranging from approximately -3.13 to -2.56, demonstrating self-affine, fractal-like properties. Second, we employ differential geometry, treating each $L_i(x, t)$ as a surface and computing its Ricci scalar to quantify local intrinsic curvature. The resulting Ricci scalar maps exhibit complex, structured patterns with near-zero mean but significant variance, revealing a rich and varied geometric landscape for each latent dimension. Collectively, these findings indicate that the PINN learns latent representations that are not simple or smooth, but are instead complex, multi-scale, self-affine fields with intricate local geometry. Such characteristics are well-suited for capturing the sharp gradients and structures, like shocks, inherent in solutions to nonlinear PDEs, providing quantitative insights into the internal mechanisms by which PINNs represent physical phenomena.

Subjects:	physics.comp-ph; cs.LG
Cite as:	PX:2508.00067