The Time-Dependent Traveling Salesman Problem (TDTSP) extends the classical TSP by allowing dynamic edge weights that vary with departure time, reflecting real-world scenarios such as transportation networks, where travel times fluctuate due to congestion patterns. TDTSP violates symmetry, triangle inequality, and cyclic invariance properties of classical TSP, creating unique computational challenges. In this paper, we propose a neural model that extends MatNet from static asymmetric TSP to time-dependent settings by using an adjacency tensor to capture temporal variations, followed by a time-aware decoder. Our architecture addresses the unique challenge of asymmetry and triangle inequality violations that change dynamically over time. Beyond architectural innovations, our research reveals a critical evaluation insight: many practical TDTSP instances maintain the same optimal solution regardless of time-dependent edge weights.

The integration of differential equations with neural networks has created powerful tools for modeling complex dynamics effectively across diverse machine learning applications. While standard integer-order neural ordinary differential equations (ODEs) have shown considerable success, they are limited in their capacity to model systems with memory effects and historical dependencies. Fractional calculus offers a mathematical framework capable of addressing this limitation, yet most current fractional neural networks use static memory weightings that cannot adapt to input-specific contextual requirements. This paper proposes a generalized neural Fractional Attention Differential Equation (FADE), which combines the memory-retention capabilities of fractional calculus with contextual learnable attention mechanisms. Our approach replaces fixed kernel functions in fractional operators with neural attention kernels that adaptively weight historical states based on their contextual relevance to current predictions. This allows our framework to selectively emphasize important temporal dependencies while filtering less relevant historical information. Our theoretical analysis establishes solution boundedness, problem well-posedness, and numerical equation solver convergence properties of the proposed model. Furthermore, through extensive evaluation on tasks such as fluid flow, graph learning problems and spatio-temporal traffic flow forecasting, we demonstrate that our adaptive attention-based fractional framework outperforms both integer-order neural ODE models and existing fractional approaches. The results confirm that our framework provides superior modeling capacity for complex dynamics with varying temporal dependencies.

In the supplementary material, we provide additional information and details in A.1. This section covers the introduction of data, key parameter settings, comparisons with baselines, optimization methods, and the algorithm process of our method. Furthermore, A.2 presents supplementary experiments for our model, including visualization experiments and replication studies. Additionally, we discuss the reasons behind utilizing hypergraphs as the temporal encoder in A.3. Finally, the limitations and broader impacts of our work are discussed in A.4. A.1 Data and Implementation Details Data. The statistical information of the aforementioned four real-world datasets is presented in Table 4.

Neural Information Processing Systems http://nips.cc/

The topic of Multivariate Time Series Anomaly Detection (MTSAD) has grown rapidly over the past years, with a steady rise in publications and Deep Learning (DL) models becoming the dominant paradigm. To address the lack of systematization in the field, this study introduces a novel and unified taxonomy with eleven dimensions over three parts (Input, Output and Model) for the categorization of DL-based MTSAD methods. The dimensions were established in a two-fold approach. First, they derived from a comprehensive analysis of methodological studies. Second, insights from review papers were incorporated. Furthermore, the proposed taxonomy was validated using an additional set of recent publications, providing a clear overview of methodological trends in MTSAD. Results reveal a convergence toward Transformer-based and reconstruction and prediction models, setting the foundation for emerging adaptive and generative trends. Building on and complementing existing surveys, this unified taxonomy is designed to accommodate future developments, allowing for new categories or dimensions to be added as the field progresses. This work thus consolidates fragmented knowledge in the field and provides a reference point for future research in MTSAD.

Neural Information Processing Systems http://nips.cc/

Here the concept of Granger causality is employed to propose a new criterion for unsupervised learning that is appropriate in the case of temporally-dependent source signals. The basic idea is to identify two projections of a multivariate time series such that the Granger causality among the resulting pair of components is maximized.