Fractional-order differential equations (FDEs) enhance traditional differential equations by extending the order of differential operators from integers to real numbers, offering greater flexibility in modeling complex dynamical systems with nonlocal characteristics. Recent progress at the intersection of FDEs and deep learning has catalyzed a new wave of innovative models, demonstrating the potential to address challenges such as graph representation learning. However, training neural FDEs has primarily relied on direct differentiation through forward-pass operations in FDE numerical solvers, leading to increased memory usage and computational complexity, particularly in large-scale applications. To address these challenges, we propose a scalable adjoint backpropagation method for training neural FDEs by solving an augmented FDE backward in time, which substantially reduces memory requirements. This approach provides a practical neural FDE toolbox and holds considerable promise for diverse applications. We demonstrate the effectiveness of our method in several tasks, achieving performance comparable to baseline models while significantly reducing computational overhead.

To effectively predict and understand these complex systems, mathematical models are employed, allowing to gain insights into the system behaviour without the need for time-consuming or expensive experiments. Due to the inherent presence of continuous dynamics in these systems, Differential Equations (DEs) are commonly employed as mathematical models, accounting for the continuous evolution of the system's behaviour and offering the advantage of enabling predictions throughout the entire time domain and not only at specific points. With the emergence of Neural Networks (NNs) and their impressive performance in fitting mathematical models to data, numerous studies have focused on modelling realworld systems. However, conventional NNs are designed to model discrete functions and may not be able to accurately capture the continuous dynamics observed in several systems. To overcome this limitation, Chen et al. [1] introduced the Neural Ordinary Differential Equations (Neural ODEs), a NN architecture that adjusts an Ordinary Differential Equation (ODE) to the dynamics of a system.