Radial Müntz-Szász Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities

TL;DR

Radial Müntz-Szász Networks (RMN)Address radial singularities by using learnable radial power bases $r^{\mu}$ with negative exponents and a limit-stable log-primitive to realize $\log r$. A separability obstruction shows coordinate-wise power bases cannot efficiently approximate non-quadratic radial functions, motivating RMN's architecture that directly parameterizes radial structure. RMN variants include RMN-Angular for angular dependence via spherical harmonics and RMN-MC for learnable singularity centers, with closed-form derivatives enabling physics-informed training. Across 10 benchmarks in 2D and 3D, RMN achieves up to $50\times$ lower RMSE with far fewer parameters than MLPs and SIREN, while preserving interpretability through learned exponent spectra that reflect physical singularities. The approach offers a compact, structure-aware alternative for singular problems, with clear limitations for strongly non-radial targets and opportunities for future physics-informed and time-dependent extensions.

Abstract

Radial singular fields, such as $1/r$, $\log r$, and crack-tip profiles, are difficult to model for coordinate-separable neural architectures. We show that any $C^2$ function that is both radial and additively separable must be quadratic, establishing a fundamental obstruction for coordinate-wise power-law models. Motivated by this result, we introduce Radial Müntz-Szász Networks (RMN), which represent fields as linear combinations of learnable radial powers $r^μ$, including negative exponents, together with a limit-stable log-primitive for exact $\log r$ behavior. RMN admits closed-form spatial gradients and Laplacians, enabling physics-informed learning on punctured domains. Across ten 2D and 3D benchmarks, RMN achieves 1.5$\times$--51$\times$ lower RMSE than MLPs and 10$\times$--100$\times$ lower RMSE than SIREN while using 27 parameters, compared with 33,537 for MLPs and 8,577 for SIREN. We extend RMN to angular dependence (RMN-Angular) and to multiple sources with learnable centers (RMN-MC); when optimization converges, source-center recovery errors fall below $10^{-4}$. We also report controlled failures on smooth, strongly non-radial targets to delineate RMN's operating regime.

Figures (13)

• Figure 1: The separability obstruction visualized. (a) The target $\log r$ has circular level sets. (b) The coordinate-separable MSN produces diamond-shaped artifacts due to its axis-aligned structure. (c) RMN precisely matches the radial structure, achieving a 72$\times$ lower error.
• Figure 2: Three RMN architectural variants. RMN-Direct: purely radial singularities. RMN-Angular: adds spherical harmonics for angular-dependent fields. RMN-MC: learns multiple singularity locations.
• Figure 3: RMN achieves orders-of-magnitude improvement on singular functions. Top: RMSE comparison across four benchmarks. Middle left: Improvement factors. Middle right: Pareto frontier. Bottom left: Convergence. Bottom right: Architecture summary.
• Figure 4: RMSE heatmap across all experiments and methods. Green = low error (good), red = high error (poor). RMN variants dominate on radial singularities; MLP excels on smooth functions.
• Figure 5: Pareto frontier: accuracy--model size trade-off. RMN occupies the optimal efficient region with low error and low parameter count.

Theorems & Definitions (42)

• Definition 3.1: Radial function
• Corollary 3.2
• Proposition 3.3: $L^p$ integrability of radial powers
• proof
• Remark 3.4: Punctured Domains and Evaluation Protocol
• Theorem 3.5: Müntz-Szász muntz1914szasz1916
• Example 3.6: Satisfying and violating the Müntz condition
• Definition 3.7: Radial Müntz space
• Theorem 3.8: Radial Müntz-Szász Density
• proof