发表机构
SBA Research gGmbH; University of Vienna（SBA研究有限责任公司; 维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在统一实验框架下实证探究分数阶优化器与分形激活函数的配对效果，发现正则化式分数缩放与选定分形激活函数适配性较好，自适应记忆优于普通记忆替换，可控分数阶记忆具研究潜力。
AI 中文摘要
分数阶优化方法与分形激活函数是提升神经网络训练的两个独立方向。分数阶优化器通过分数阶导数和记忆效应扩展一阶优化，而分形激活函数基于自相似的魏尔斯特拉斯型和布朗芒型函数引入多尺度非线性表示。本文在统一实验框架中研究二者的相互作用：在阿克利和希梅尔布劳基准曲面（含标准形式与加性魏尔斯特拉斯型扰动）上评估分数阶优化器族，随后在含传统及分形激活函数的前馈神经网络上，针对十个分类数据集开展实验。对比方法包括标准方法、正则化式优化器、显式与自适应基于记忆的分数阶优化器及其他代表性文献方法。总体而言，分数阶优化与分形激活函数存在有用但具选择性的配对：正则化式分数缩放在网络训练中与选定分形激活函数表现良好，而格伦瓦尔德-列特尼科夫记忆在扰动曲面中最具相关性；自适应记忆在若干案例中优于普通记忆替换，表明可控分数阶记忆是有前景的方向而非通用替代方案。
英文摘要
Fractional optimization methods and fractal activation functions are two independent directions for improving neural network training. Fractional optimizers extend first-order optimization through fractional derivatives and memory effects, whereas fractal activations introduce multi-scale nonlinear representations based on self-similar Weierstrass- and Blancmange-type functions. Here, we investigate their interaction within a unified experimental framework. We evaluate fractional optimizer families on Ackley and Himmelblau benchmark surfaces, in standard form and with additive Weierstrass-type perturbations, and then in feed-forward neural networks with conventional and fractal activations on ten classification datasets. The comparison includes standard methods, regularization-style optimizers, explicit and adaptive memory-based fractional optimizers, and other representative literature methods. Overall, fractional optimization and fractal activations show useful but selective pairings. Regularization-style fractional scaling performs well with selected fractal activations in network training, while Grünwald--Letnikov memory is most relevant on perturbed surfaces. Adaptive memory improves plain memory substitution in several cases, supporting controlled fractional memory as a promising direction rather than a universal replacement.
CommentsQuite extensive paper, more than 100 pages
