Discovering Scaling Exponents with Physics-Informed Müntz-Szász Networks

TL;DR

This paper introduces MSN-PINN, a physics-informed framework that identifies scaling exponents directly from PDEs by using a Müntz expansion with learnable exponents. By making the exponents explicit parameters, the method provides interpretable physical quantities and enables simultaneous recovery of the solution and its scaling structure, supported by identifiability and stability theory. A constraint-aware training scheme encodes BCs and problem-specific physics to resolve degeneracies in exponent learning, yielding dramatic accuracy improvements on wedge corner singularities and other singular problems. The approach achieves state-of-the-art exponent recovery across singular ODEs, corner singularities (matching Kondrat'ev 1967), and singular forcing, with a 100% success rate on a wedge benchmark and sub-0.1% errors in several settings. Overall, MSN-PINN offers a principled path to integrate asymptotic structure into neural models, enabling reliable extrapolation and direct physical interpretation of learned exponents.

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 $O(|μ- α|^2)$. 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.

Figures (6)

• Figure 1: MSN-PINN discovers scaling exponents directly from physics. Unlike standard neural networks that hide exponents within millions of weights, the MSN ansatz $u(x) = \sum_k c_k x^{\mu_k}$ outputs the exponents $\{\mu_k\}$ as explicit parameters. Physics-informed losses enforce PDE residuals and BCs to guide exponent learning. For wedge singularities, constraint-aware training adds $\mathcal{L}_{\mathrm{constraint}} = \sum_k |c_k| \sin^2(\mu_k \omega)$, which drives the exponents toward physically valid values $\mu = n\pi/\omega$---improving accuracy from 14.6% to 0.009% error.
• Figure 2: Corner singularity: MSN-PINN recovers the result of kondrat1967boundary with a 0.009% error. Top: Learned solution $u(r,\theta) = r^{2/3}\sin(2\theta/3)$ on a 270° wedge domain, where color indicates solution magnitude, with blue at 0 and yellow at 1. Middle: Exponent trajectory during training---without constraints, not shown, $\mu$ drifts to 0.57 with a 14.6% error; with constraints, $\mu$ converges to the theoretical value of 0.6667. Bottom: Training loss showing rapid convergence once the constraint term activates around epoch 2000.
• Figure 3: Singular forcing: MSN-PINN accurately discovers both solution exponents with $<$0.1% error. The Poisson equation, $-u" = x^{-1/2}$, has the solution $u(x) = \frac{4}{3}x - \frac{4}{3}x^{3/2}$ containing exponents 1.0 for the homogeneous term and 1.5 for the particular term. Panel (a) shows the training loss convergence over 10,000 epochs. Panel (b) shows four MSN exponents initialized randomly; two converge to 1.0 with 0.03% error, and two to 1.5 with 0.05% error. The redundant convergence indicates robust discovery. Panel (c) shows the learned solution, dashed, matching the analytical solution, solid, across the domain.
• Figure 5: Single Exponent Recovery. Top: Function fit comparing the true $x^{0.5}$ with the MSN approximation. Bottom: Exponent trajectory during training, converging to 0.4927 from a random initialization.
• Figure 6: Noise Robustness: Exponent recovery remains stable across noise levels from $\sigma=0$ to $\sigma=0.05$. The power-law ansatz provides natural regularization.

Theorems & Definitions (17)

• Definition 3.1: MSN Ansatz
• Proposition 3.2: Exact Representation
• Theorem 4.1: Universal Approximation, nguessan2025msn
• Theorem 4.2: Approximation Rate, nguessan2025msn
• Definition 4.3: Exponent Identifiability
• Theorem 4.4: Identifiability
• Theorem 4.5: Stability and Separation
• Remark 4.6: On the Sampling Assumption
• Proposition 4.7: PDE-Induced Exponent Constraints
• Theorem 4.8: Constraint-Aware Critical Points