提出可学习幂次的径向神经网络，高效建模奇异场如1/r和log r。
Radial Müntz-Szász Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities
- 用可学习的径向幂次r^μ构建网络，支持负指数和对数项精确表达
- 在2D/3D基准上误差比MLP低1.5~51倍，参数量仅27个
- 适合处理带奇点的物理问题，如裂纹尖端、点源场等场景
径向奇异场（如 $1/r$、$/log r$ 及裂纹尖端分布）难以用现有坐标可分离神经架构建模。我们正式证明：任何同时为径向且加法可分离的 $C^2$ 函数必为二次函数，揭示了坐标分量幂律模型的根本局限。为此，我们提出径向 Müntz-Szász 网络（RMN），将场表示为可学习径向幂次 $r^μ$ 的线性组合（含负指数），并引入极限稳定的对数原函数以精确捕捉 $/log r$ 行为。RMN 具备闭式空间梯度与拉普拉斯算子，支持在穿孔域上的物理信息学习。在十个2D和3D基准测试中，RMN 的均方根误差（RMSE）比 MLP 低1.5至51倍，比 SIREN 低10至100倍，仅使用27个参数，而 MLP 和 SIREN 分别为33,537和8,577。我们进一步扩展为含角向依赖的 RMN-Angular 和多源可学习中心的 RMN-MC，其源中心恢复误差低于 $10^{-4}$。同时通过平滑非径向目标的可控失败实验，明确划分了 RMN 的适用范围。
原文摘要 · Abstract (English)
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.
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