用威兰德径向基函数设计新激活函数，提升模型训练稳定性和泛化能力。
A parametric activation function based on Wendland RBF
- 基于威兰德RBF构造可调参数激活函数，结合线性与指数项增强灵活性。
- 在正弦波拟合和MNIST等任务中表现优于传统激活函数，尤其在回归任务中精度更高。
- 适合追求高稳定性与泛化能力的深度学习研究者，尤其是非线性建模场景。
本文提出一种基于威兰德径向基函数（Wendland RBF）的新型参数化激活函数，用于深度神经网络。威兰德RBF具备紧支集、光滑性和正定性等数学优势，被用来克服ReLU、Sigmoid和Tanh等传统激活函数的局限。所提出的增强型威兰德激活函数融合标准威兰德成分与线性、指数项，实现可调局部性、改善梯度传播并提升训练稳定性。理论分析揭示其光滑性与自适应特性；实验在合成任务（如正弦波逼近）及基准数据集（MNIST、Fashion-MNIST）上验证了其竞争力。结果表明，该激活函数在特定场景下，特别是在回归任务中，达到更高精度，同时保持计算效率。研究将经典RBF理论与现代深度学习结合，表明威兰德激活可通过局部平滑变换缓解过拟合、提升泛化性能。未来工作包括混合架构与领域适配。
原文摘要 · Abstract (English)
This paper introduces a novel parametric activation function based on Wendland radial basis functions (RBFs) for deep neural networks. Wendland RBFs, known for their compact support, smoothness, and positive definiteness in approximation theory, are adapted to address limitations of traditional activation functions like ReLU, sigmoid, and tanh. The proposed enhanced Wendland activation combines a standard Wendland component with linear and exponential terms, offering tunable locality, improved gradient propagation, and enhanced stability during training. Theoretical analysis highlights its mathematical properties, including smoothness and adaptability, while empirical experiments on synthetic tasks (e.g., sine wave approximation) and benchmark datasets (MNIST, Fashion-MNIST) demonstrate competitive performance. Results show that the Wendland-based activation achieves superior accuracy in certain scenarios, particularly in regression tasks, while maintaining computational efficiency. The study bridges classical RBF theory with modern deep learning, suggesting that Wendland activations can mitigate overfitting and improve generalization through localized, smooth transformations. Future directions include hybrid architectures and domain-specific adaptations.
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