Fractional Derivative in LSTM Networks: Adaptive Neuron Shape Modeling with the Grünwald–Letnikov Method

The incorporation of fractional-order derivatives into neural networks presents a novel approach to improving gradient flow and adaptive learning dynamics. This paper introduces a fractional-order LSTM model, leveraging the Grünwald–Letnikov (GL) method to modify both activation functions and backpropagation mechanics. By redefining the transition functions of LSTM gates with fractional derivatives, the model achieves a smoother gradient adaptation while maintaining consistency across forward and backward passes. This is the first study integrating the Grünwald–Letnikov operator directly into both forward and backward LSTM computations, ensuring a consistent fractional framework throughout the entire learning process. We apply this approach to anomaly detection in fiber optic cable manufacturing, where small deviations in production parameters can significantly impact quality. A dataset containing time-series sensor measurements was used to train the fractional LSTM, demonstrating improved generalization and stability compared to classical LSTM models. Numerical stability analysis confirms that the fractional derivative framework allows convergent learning, preventing both vanishing and exploding gradients. Experimental results show that the fractional-order LSTM outperforms standard architectures in detecting manufacturing anomalies, with the optimal fractional order ν=0.95 providing a balance between accuracy and computational complexity. The findings suggest that fractional calculus can enhance deep learning architectures by introducing a continuous and flexible transition between neuron activations, paving the way for adaptive neural networks with tunable memory effects.