Discovering Scaling Exponents with Physics-Informed Müntz-Szász Networks

Physical systems near singularities, interfaces, and critical points exhibit power-law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics-informed Müntz-Szász Networks (MSN-PINN), a power-law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as $O(|\mu - \alpha|^2)$. Across experiments, MSN-PINN achieves single-exponent recovery with 1–5\% error under noise and sparse sampling. It recovers corner singularity exponents for the two-dimensional Laplace equation with 0.009\% error, matches the classical result of Kondrat'ev (1967), and recovers forcing-induced exponents in singular Poisson problems with 0.03\% and 0.05\% errors. On a 40-configuration wedge benchmark, it reaches a 100\% success rate with 0.022\% mean error. Constraint-aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training.

Many physical systems show power-law behavior near cracks, corners, interfaces, or other special points. The key scaling number in these power laws is often important because it reveals how the system behaves, but standard neural networks usually hide this number inside many learned parameters. This paper introduces a physics-informed neural network that learns these scaling numbers directly. The method uses physical equations to guide learning and outputs both the solution and its scaling structure. Across several test problems, including corner singularities and singular forcing, it accurately recovers known physical scaling laws. This makes neural network solutions more interpretable and useful for scientific discovery.