Neural Variable-Order Fractional Differential Equation Networks

Abstract: Neural differential equation models have garnered significant attention in recent years for their effectiveness in machine learning applications.Among these, fractional differential equations (FDEs) have emerged as a promising tool due to their ability to capture memory-dependent dynamics, which are often challenging to model with traditional integer-order approaches.While existing models have primarily focused on constant-order fractional derivatives, variable-order fractional operators offer a more flexible and expressive framework for modeling complex memory patterns. In this work, we introduce the Neural Variable-Order Fractional Differential Equation network (NvoFDE), a novel neural network framework that integrates variable-order fractional derivatives with learnable neural networks.Our framework allows for the modeling of adaptive derivative orders dependent on hidden features, capturing more complex feature-updating dynamics and providing enhanced flexibility. We conduct extensive experiments across multiple graph datasets to validate the effectiveness of our approach.Our results demonstrate that NvoFDE outperforms traditional constant-order fractional and integer models across a range of tasks, showcasing its superior adaptability and performance.