径向穆恩茨-萨兹网络：用于多维奇点的可学习幂基神经架构
Radial Müntz-Szász Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities
AI总结：
RMN通过可学习的径向幂基和对数原函数，有效建模多维奇点，其参数效率高且在多个基准测试中表现优异。
摘要：
径向奇异场，如1/r、log r和裂纹尖端轮廓，难以用当前的坐标分离神经架构建模。我们正式证明这一结果：任何既是径向又是加法可分离的C²函数必须是二次函数，这为坐标-wise幂律模型设定了根本性的障碍。受此结果启发，我们引入了径向穆恩茨-萨兹网络（RMN），它将场表示为可学习的径向幂r^μ的线性组合，包括负指数，以及一个极限稳定的对数原函数以实现精确的log r行为。RMN允许闭合形式的空间梯度和拉普拉斯算子，使在穿孔域上进行物理指导的学习成为可能。在十个2D和3D基准测试中，RMN的RMSE比MLP低1.5到51倍，比SIREN低10到100倍，而仅使用27个参数，相比MLP的33,537个参数和SIREN的8,577个参数。我们扩展RMN以包含角度依赖性（RMN-Angular）和处理多个源的可学习中心（RMN-MC），其源-中心恢复误差低于10^-4。我们还报告了在平滑且强烈非径向目标上的受控失败，以界定RMN的运行范围。
英文摘要：
Radial singular fields, such as $1/r$, $\log r$, and crack-tip profiles, are difficult to model with current coordinate-separable neural architectures. We formally establish this result: 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 between 1.5 and 51 times lower RMSE than MLPs and between 10 and 100 times lower RMSE than SIREN, while using only 27 parameters, compared with 33,537 for MLPs and 8,577 for SIREN. We extend RMN to incorporate angular dependence (RMN-Angular) and to handle multiple sources with learnable centers (RMN-MC), whose 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
