Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation
To address the high memory consumption and computational complexity arising from forward-mode differentiation in training Neural Fractional Differential Equations (Neural FDEs), this work introduces, for the first time, adjoint-based backpropagation into the Neural FDE training framework. By formulating and solving an augmented fractional-order adjoint equation, our method enables efficient time-reversed gradient computation, overcoming the scalability limitations of conventional forward-mode differentiation in large-scale settings. The approach integrates fractional calculus, the adjoint state method, and neural differential equation theory, and is compatible with mainstream numerical FDE solvers. Experiments on tasks such as graph representation learning demonstrate performance on par with baseline models, while reducing memory usage by over 60% and accelerating training by 2–3×. This advancement significantly enhances the feasibility of Neural FDEs for large-scale dynamical system modeling.