Fractional Derivative
Fractional derivatives extend the concept of derivatives to non-integer orders, enabling the modeling of systems with memory effects and long-range dependencies not captured by traditional integer-order models. Current research focuses on applying fractional derivatives within various machine learning frameworks, including physics-informed neural networks (PINNs), Kolmogorov-Arnold networks (KANs), and graph neural networks, often employing techniques like fractional gradient descent and operational matrices for efficient computation. These advancements are significantly impacting diverse fields, improving the accuracy and efficiency of solving fractional differential equations in areas such as signal processing, control systems, and image analysis, as well as enabling more robust modeling of complex physical phenomena.
Papers
Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks
Zheyuan Hu, Kenji Kawaguchi, Zhongqiang Zhang, George Em Karniadakis
Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations
Zheyuan Hu, Zhongqiang Zhang, George Em Karniadakis, Kenji Kawaguchi