用可学习的幂律基函数替代传统激活函数，显著提升奇异函数逼近能力。
Müntz-Szász Networks: Neural Architectures with Learnable Power-Law Bases
- 以可学习的分数阶幂律作为网络基函数，理论支撑强。
- 在奇异函数逼近上误差降低5-8倍，参数减少10倍。
- 特别适合物理信息神经网络，能学出符合物理规律的指数。
标准神经网络使用固定激活函数（如ReLU、tanh），难以有效逼近具有奇点或分数幂结构的函数，这类结构在物理学中普遍存在，如边界层、断裂力学和角奇点。我们提出穆尼茨-萨扎斯网络（Müntz-Szász Networks, MSN），将固定光滑激活替换为基于经典逼近论的可学习分数幂基函数。每个MSN边计算ϕ(x) = ∑a_k |x|^{μ_k} + ∑b_k sign(x)|x|^{λ_k}，其中指数{μ_k, λ_k}与系数一同学习。证明了MSN继承了穆尼茨-萨扎斯定理的通用逼近性，并建立了新逼近率：对|x|^α形式函数，单个学习指数下误差为O(|μ−α|²)，而标准MLP需O(ε^{-1/α})个神经元才能达到类似精度。在带奇异目标函数的监督回归任务中，MSN误差比MLP低5-8倍，且参数量少10倍。物理信息神经网络（PINNs）是奇异函数逼近的严苛挑战；在包含奇异常微分方程和刚性边界层问题的基准测试中，MSN实现3-6倍改进，同时学习到与已知解结构一致的可解释指数。结果表明，理论指导的架构设计能显著提升科学相关函数类的性能。
原文摘要 · Abstract (English)
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.
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