CFDNN

Implements CFDNN (Conformable Fractional Deep Neural Network), a deep network trained by conformable fractional gradient descent in place of standard backpropagation.

The conformable fractional derivative of order \(\alpha\) replaces the limit definition of the ordinary derivative with \(D^\alpha f(t) = t^{1-\alpha} f'(t)\), which contains no Gamma function and no memory integral — it is purely local. Applied to a weight, the conformable fractional gradient is the ordinary gradient rescaled by \(\theta^{1-\alpha}\). CFDNN trains in the super-integer regime \(\alpha \in [1.2, 1.8]\), where this rescaling smooths the loss landscape and accelerates convergence. An optional weight-decay term enters with the same conformable scaling.

\[ \begin{aligned} g_t &= \nabla_\theta E(\theta_t), \\ \theta_{t+1} &= \theta_t - \eta \left( \theta_t^{\,1-\alpha}\, g_t - \lambda\, \theta_t^{\,2-\alpha} \right). \end{aligned} \]

where \(\theta\) are the weights, \(\eta\) the learning rate, \(g_t\) the gradient of the loss \(E\), \(\alpha \in [1.2, 1.8]\) the conformable fractional order, \(\lambda\) the weight-decay coefficient, and \(\theta^{1-\alpha}\) the conformable rescaling factor (setting \(\lambda = 0\) and \(\alpha = 1\) recovers ordinary gradient descent).

Reference: B. Ajarmah and H. Iwidat, "Conformable Fractional Deep Neural Networks (CFDNN) for high-speed cyber-attack detection", Scientific Reports 2026. https://doi.org/10.1038/s41598-026-45213-w

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