{"ID":2826442,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.22222","arxiv_id":"2512.22222","title":"Müntz-Szász Networks: Neural Architectures with Learnable Power-Law Bases","abstract":"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 Müntz-Szász Networks (MSN), a novel architecture that replaces fixed smooth activations with learnable fractional power bases grounded in classical approximation theory. Each MSN edge computes $φ(x) = \\sum_k a_k |x|^{μ_k} + \\sum_k b_k \\mathrm{sign}(x)|x|^{λ_k}$, where the exponents $\\{μ_k, λ_k\\}$ are learned alongside the coefficients. We prove that MSN inherits universal approximation from the Müntz-Szász theorem and establish novel approximation rates: for functions of the form $|x|^α$, MSN achieves error $\\mathcal{O}(|μ- α|^2)$ with a single learned exponent, whereas standard MLPs require $\\mathcal{O}(ε^{-1/α})$ 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.","short_abstract":"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 introd...","url_abs":"https://arxiv.org/abs/2512.22222","url_pdf":"https://arxiv.org/pdf/2512.22222v3","authors":"[\"Gnankan Landry Regis N'guessan\"]","published":"2025-12-22T23:04:18Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\"]","methods":"[]","has_code":false}