Neural fractional differential networks for modeling complex dynamical systems

Neural fractional differential equations offer an innovative approach to modeling complex dynamical systems by integrating Caputo fractional derivatives into deep neural networks. This integration effectively captures memory effects and long-range dependencies, which are essential for analyzing irregular time series and fractal-like signals. In this study, we employ a predictor-corrector method for solution approximation, along with the Adam optimization algorithm for parameter tuning, significantly enhancing the neural networks' ability to model intricate non-local dynamics and anomalous behaviors. Our findings demonstrate that neural fractional differential equations outperform traditional models in representing the complexities of data, leading to improved predictive accuracy. By extending neural ordinary differential equations to fractional orders, this research broadens the applicability of neural networks, addresses the limitations of existing methods, and establishes a robust framework for real-world applications across diverse dynamical systems.