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

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.

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.

Identifying the fragment structure of the organic compounds by deeply learning the original NMR data

We preprocess the raw NMR spectrum and extract key characteristic features by using two different methodologies, called equidistant sampling and peak sampling for subsequent substructure pattern recognition; meanwhile may provide the alternative strategy to address the imbalance issue of the NMR dataset frequently encountered in dataset collection of statistical modeling and establish two conventional SVM and KNN models to assess the capability of two feature selection, respectively. Our results in this study show that the models using the selected features of peak sampling outperform the ones using the other. Then we build the Recurrent Neural Network (RNN) model trained by Data B collected from peak sampling. Furthermore, we illustrate the easier optimization of hyper parameters and the better generalization ability of the RNN deep learning model by comparison with traditional machine learning SVM and KNN models in detail.