几何感知的R结构Kolmogorov-Arnold网络
Geometry-Aware R-Structured Kolmogorov-Arnold Networks
AI总结：
提出GRS-KAN混合架构，将R函数集成到KAN中，通过可微逻辑运算编码几何约束，在回归任务中相比标准KAN测试RMSE降低高达67%，并提升可解释性。
摘要：
我们提出了一种新颖的混合神经架构——几何感知的R结构Kolmogorov-Arnold网络（GRS-KAN），它将http this URL的R函数集成到Kolmogorov-Arnold网络（KAN）框架中。该方法结合了两种互补的建模机制：平滑的非线性结构由KAN分支学习，而已知的几何或逻辑约束则使用可微的R函数解析地编码。这使得在可训练神经架构中能够显式表示不连续性、可行区域和隐式几何边界。该框架通过R-合取和R-析取实现可微的逻辑运算，允许复杂几何支撑被解析地表示并直接融入回归模型。引入了多种GRS-KAN变体，包括加法、乘法和不可知分支加权架构。该方法在涉及圆形和矩形支撑不连续性的回归问题上进行了演示。数值实验表明，与标准KAN相比，显式几何编码显著提高了预测精度和边界定位。在所考虑的基准测试中，几何感知的GRS-KAN模型将测试RMSE降低了高达67%，同时通过所学几何结构的显式解析表示提高了可解释性。不可知变体进一步展示了自动确定几何先验是否对给定学习任务有益的能力。
英文摘要：
We propose a novel hybrid neural architecture, the Geometry-aware R-Structured Kolmogorov-Arnold Network (GRS-KAN), which integrates V.L.Rvachev's R-functions into the Kolmogorov-Arnold Network (KAN) framework. The proposed approach combines two complementary modeling mechanisms: smooth nonlinear structure is learned by KAN branches, while known geometric or logical constraints are encoded analytically using differentiable R-functions. This enables explicit representation of discontinuities, feasible regions, and implicit geometric boundaries within a trainable neural architecture. The framework implements differentiable logical operations through R-conjunctions and R-disjunctions, allowing complex geometric supports to be represented analytically and incorporated directly into regression models. Several GRS-KAN variants are introduced, including additive, multiplicative, and agnostic branch-weighted architectures. The method is demonstrated on regression problems involving discontinuities with circular and rectangular supports. Numerical experiments show that explicit geometric encoding substantially improves predictive accuracy and boundary localization compared with standard KANs. In the considered benchmarks, geometry-aware GRS-KAN models reduce test RMSE by up to 67% while simultaneously improving interpretability through explicit analytical representation of the learned geometric structure. The agnostic variant further demonstrates the ability to automatically determine whether geometric priors are beneficial for a given learning task.
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。
