Standard Gradient Descent and its modern variants assume local, Markovian weight updates, making them highly susceptible to noise and overfitting. This limitation becomes critically severe in extremely imbalanced datasets such as financial fraud detection where dominant class gradients systematically overwrite the subtle signals of the minority class. In this paper, we introduce a novel optimization algorithm grounded in Fractional Calculus. By isolating the core memory engine of the generalized fractional derivative, the Weighted Fractional Weyl Integral, we replace the instantaneous gradient with a dynamically weighted historical sequence. This fractional memory operator acts as a natural regularizer. Empirical evaluations demonstrate that our method prevents overfitting in medical diagnostics and achieves an approximately 40 percent improvement in PR-AUC over classical optimizers in financial fraud detection, establishing a robust bridge between pure fractional topology and applied Machine Learning.

翻译：标准的梯度下降及其现代变体假设权重更新是局部且马尔可夫的，这使得它们极易受到噪声影响并导致过拟合。这一局限在极度不平衡的数据集（如金融欺诈检测）中变得尤为严重，其中主导类别的梯度会系统地覆盖少数类别的细微信号。本文提出了一种基于分数阶微积分的新型优化算法。通过分离广义分数阶导数的核心记忆引擎——加权分数阶Weyl积分，我们用动态加权的历史序列替代瞬时梯度。该分数阶记忆算子充当了天然的规范化器。实证评估表明，我们的方法在医学诊断中防止了过拟合，并在金融欺诈检测中相较于经典优化器实现了约40%的PR-AUC提升，从而在纯分数阶拓扑与应用机器学习之间建立了坚实的桥梁。