让微分方程的阶数自动学习，更灵活地捕捉动态变化
Neural Variable-Order Fractional Differential Equation Networks
- 用可学习的变阶分数阶微分构建神经网络，动态调整导数阶数
- 在多个图数据集上超越固定阶数模型，提升建模灵活性与性能
- 适合需要捕捉复杂记忆依赖关系的研究者或工业应用
神经微分方程模型近年来在机器学习中备受关注。其中，分数阶微分方程（FDE）因其能捕捉记忆依赖动态而成为有前景的工具，尤其在传统整数阶方法难以建模的场景下表现优异。现有工作多聚焦于固定阶分数导数，而变阶分数算子能提供更灵活、更丰富的建模能力。本文提出神经变阶分数微分方程网络（NvoFDE），将可学习的变阶分数导数与神经网络结合，使导数阶数能随隐藏特征自适应变化，从而更好地捕捉复杂的特征更新动态。我们在多个图数据集上进行了广泛实验，结果表明，NvoFDE在各类任务中均优于传统的固定阶分数及整数阶模型，展现出更强的适应性与性能优势。
原文摘要 · Abstract (English)
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.
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