Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation

TL;DR

This work addresses the memory-intensive training of neural fractional-order differential equations by introducing an adjoint backpropagation method that solves an augmented FDE backward in time. The approach derives a backward-in-time adjoint system using a right-sided Caputo derivative to compute gradients efficiently, and provides a practical neural FDE toolbox. Empirical results on fractional Lotka–Volterra parameter recovery, MNIST-like image classification, and large-scale graph node classification show comparable performance to baselines with substantially reduced training memory. This work enables scalable neural FDE training and broadens their applicability in memory-constrained settings.

Abstract

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.

Figures (1)

• Figure 1: The visualization of the adjoint backpropagation method for training neural FDEs by solving an augmented FDE backward in time.