Physics-informed neural fractional differential equations

This study introduces physics-informed neural fractional differential equations, a novel approach that integrates neural ODE, fractional calculus, and physics-informed machine learning to advance the modeling of dynamical systems. Traditional methods often struggle to capture intricate systems' long-range dependencies and memory effects. Physics-informed neural fractional differential equations address these challenges by incorporating the Caputo fractional derivative into neural networks, combining the flexibility of neural ODEs with the power of fractional calculus and physical laws. This integration enhances the accuracy and efficiency of modeling systems with complicated behaviors and nonlocal effects. We use the predictor-corrector method to solve fractional differential equations and the Adam optimization method to update neural network parameters. Numerical examples, including the Van der Pol equation, a spring-mass system, and CO₂ emission modeling, show that physics-informed neural fractional differential equations outperform traditional methods in accuracy and computational efficiency. This research bridges the gap between fractional calculus and deep learning, providing a powerful tool for modeling and predicting intricate dynamical systems in various scientific and engineering fields.