Frontier advances in solving fractional differential equations via physics-informed neural networks: A comprehensive review

Fractional differential equations (FDEs) serve as indispensable mathematical instruments for modeling complex dynamical systems characterized by inherent memory retention, hereditary evolution, and anomalous diffusion—behaviors that elude accurate description by conventional integer-order differential models. Traditional numerical algorithms for FDEs are plagued by substantial mesh discretization overhead, prohibitive computational costs, and poor compatibility with various inverse parameter identification tasks. As a groundbreaking computational framework, physics-informed neural networks (PINNs) synergistically combine data-driven fitting capabilities with rigorous physical governing laws, offering transformative solutions to general differential equations. Cutting-edge advancements in fractional PINNs (fPINNs) are systematically synthesized in this comprehensive review. Physical motivations for adopting FDEs are elaborated, the fundamental architecture of vanilla PINNs is delineated, and core obstacles originating from the non-local integral characteristics of fractional derivative operators are dissected to reveal critical restrictions on direct application of standard PINNs to FDEs. All mainstream fPINN approaches are taxonomically grouped into three distinct technical branches, with focused mathematical derivations and in-depth discussions dedicated to fractional-order inverse identification. A homotopy-augmented simultaneous training scheme is formulated to mitigate unstable joint optimization of fractional orders and neural network weights, enabling a smooth transition of optimization targets from simplistic integer-order dynamics toward complex fractional non-local responses. Relative advantages and disadvantages of simultaneous and two-stage sequential identification workflows are thoroughly contrasted, and clear, actionable criteria are delivered to facilitate selection of appropriate identification pipelines for diverse fractional modeling scenarios. Systematic theoretical references and operable technical guidance for fractional modeling and scientific machine learning are provided throughout the review