Neural Variable-Order Fractional Differential Equation Networks

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.