Discovering Scaling Exponents with Physics-Informed M\"untz-Sz\'asz Networks

TL;DR

This paper introduces MSN-PINN, a physics-informed neural network that learns power-law scaling exponents in physical systems, achieving high accuracy and interpretability even with noisy and sparse data.

Contribution

The paper presents a novel neural network architecture that explicitly models and learns scaling exponents as trainable parameters, with proven identifiability and superior performance in physical problems.

Findings

Achieves 1-5% error in single-exponent recovery under noise.

Recovers corner singularity exponents with 0.009% error.

Successfully applies to complex wedge benchmark with 0.022% mean error.

Abstract

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 M"untz-Sz'asz 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. By combining the expressiveness of neural networks with the interpretability of asymptotic analysis, MSN-PINN produces learned parameters with direct physical meaning.