# M"untz-Sz'asz Networks: Neural Architectures with Learnable Power-Law Bases

arXiv: https://arxiv.org/abs/2512.22222 · HTML: https://mlsanity.com/paper/2512.22222/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)

## Built on by (most cited first, 1 of 1)

- [Radial M"untz-Sz'asz Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities](https://mlsanity.com/paper/2602.08419/) (2026)

## Abstract (arXiv)

Standard neural network architectures employ fixed activation functions (ReLU, tanh, sigmoid) that are poorly suited for approximating functions with singular or fractional power behavior, a structure that arises ubiquitously in physics, including boundary layers, fracture mechanics, and corner singularities. We introduce M"untz-Sz'asz Networks (MSN), a novel architecture that replaces fixed smooth activations with learnable fractional power bases grounded in classical approximation theory. Each MSN edge computes φ(x) = sumk ak |x|μk + sumk bk sign(x)|x|λk, where the exponents μk, λk are learned alongside the coefficients. We prove that MSN inherits universal approximation from the M"untz-Sz'asz theorem and establish novel approximation rates: for functions of the form |x|α, MSN achieves error O(|μ - α|2) with a single learned exponent, whereas standard MLPs require O(ε-1/α) neurons for comparable accuracy. On supervised regression with singular target functions, MSN achieves 5-8x lower error than MLPs with 10x fewer parameters. Physics-informed neural networks (PINNs) represent a particularly demanding application for singular function approximation; on PINN benchmarks including a singular ODE and stiff boundary-layer problems, MSN achieves 3-6x improvement while learning interpretable exponents that match the known solution structure. Our results demonstrate that theory-guided architectural design can yield dramatic improvements for scientifically-motivated function classes.