GMDNet: A Graph-Based Mixture Density Network for Estimating Packages’ Multimodal Travel Time Distribution

Abstract In the logistics network, accurately estimating packages' Travel Time Distribution (TTD) given the routes greatly benefits both consumers and platforms. Although recent works perform well in predicting an expected time or a time distribution in a road network, they could not be well applied to estimate TTD in logistics networks. Because TTD prediction in the logistics network requires modeling packages' multimodal TTD (MTTD, i.e., there can be more than one likely output with a given input) while leveraging the complex correlations in the logistics network. To this end, this work opens appealing research opportunities in studying MTTD learning conditioned on graph-structure data by investigating packages' travel time distribution in the logistics network. We propose a Graph-based Mixture Density Network, named GMDNet, which takes the benefits of both graph neural network and mixture density network for estimating MTTD conditioned on graph-structure data (i.e., the logistics network). Furthermore, we adopt the Expectation-Maximization (EM) framework in the training process to guarantee local convergence and thus obtain more stable results than gradient descent. Extensive experiments on two real-world datasets demonstrate the superiority of our proposed model. Corrigendum Notice In the initial publication of this article, the authors (Mao et al. 2023) acknowledged that although it referred to an earlier paper already presented and published in ICML-21 (Errica et al. 2021), it insufficiently acknowledged the extent to which it incorporated and made extensive use of techniques therein. We are providing a Corrigendum Note, "PDF (2024-09-25)," alongside the original published version. The Corrigendum Note summarizes the main novel contributions of this paper. Errica, F.; Bacciu, D.; and Micheli, A. 2021. Graph Mixture Density Networks. In Proceedings of the 38th International Conference on Machine Learning (PMLR-28), 3025–3035. PMLR.Mao, X.; Wan, H.; Wen, H.; Wu,