{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T05:26:55Z","timestamp":1761629215650,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,26]],"date-time":"2025-10-26T00:00:00Z","timestamp":1761436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Open Research Fund of Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment","award":["VECS2024K09"],"award-info":[{"award-number":["VECS2024K09"]}]},{"name":"Fundamental Research Funds for the Central Public-interest Scientific Institution","award":["2024YSKY-03"],"award-info":[{"award-number":["2024YSKY-03"]}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2022YFC3701802"],"award-info":[{"award-number":["2022YFC3701802"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Accurate heavy-duty vehicle load estimation is crucial for transportation and environmental regulation, yet current methods lack precision in data accuracy and practicality for field implementation. We propose a Self-Supervised Reconstruction Heterogeneous Graph Convolutional Network (SSR-HGCN) for load estimation using On-Board Diagnostics (OBD) data. The method integrates physics-constrained heterogeneous graph construction based on vehicle speed, acceleration, and engine parameters, leveraging graph neural networks\u2019 information propagation mechanisms and self-supervised learning\u2019s adaptability to low-quality data. The method comprises three modules: (1) a physics-constrained heterogeneous graph structure that, guided by the symmetry (invariance) of physical laws, introduces a structural asymmetry by treating kinematic and dynamic features as distinct node types to enhance model interpretability; (2) a self-supervised reconstruction module that learns robust representations from noisy OBD streams without extensive labeling, improving adaptability to data quality variations; and (3) a multi-layer feature extraction architecture combining graph convolutional networks (GCNs) and graph attention networks (GATs) for hierarchical feature aggregation. On a test set of 800 heavy-duty vehicle trips, SSR-HGCN demonstrated superior performance over key baseline models. Compared with the classical time-series model LSTM, it achieved average improvements of 20.76% in RMSE and 41.23% in MAPE. It also outperformed the standard graph model GraphSAGE, reducing RMSE by 21.98% and MAPE by 7.15%, ultimately achieving &lt; 15% error for over 90% of test samples. This method provides an effective technical solution for heavy-duty vehicle load monitoring, with immediate applications in fleet supervision, overloading detection, and regulatory enforcement for environmental compliance.<\/jats:p>","DOI":"10.3390\/sym17111802","type":"journal-article","created":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T02:50:48Z","timestamp":1761533448000},"page":"1802","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Physics-Constrained Heterogeneous GNN Guided by Physical Symmetry for Heavy-Duty Vehicle Load Estimation"],"prefix":"10.3390","volume":"17","author":[{"given":"Lizhuo","family":"Luo","sequence":"first","affiliation":[{"name":"Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Vehicle Emission Control Center, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Institute of Advanced Technology, University of Science and Technology of China, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leqi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Vehicle Emission Control Center, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongli","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Vehicle Emission Control Center, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunjing","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Vehicle Emission Control Center, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hang","family":"Yin","sequence":"additional","affiliation":[{"name":"Key Laboratory for Vehicle Emission Control and Simulation of the Ministry of Ecology and Environment, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"},{"name":"Vehicle Emission Control Center, Chinese Research Academy of Environmental Sciences, Beijing 100012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,26]]},"reference":[{"key":"ref_1","unstructured":"Ministry of Transport of China (2025, June 04). Summary of National Logistics Operation Data for the First 10 Months of 2024. China Standard News, Available online: https:\/\/www.stdaily.com\/web\/gdxw\/2024-11\/29\/content_266208.html."},{"key":"ref_2","first-page":"208","article-title":"Strain Gauge Based Sensor for Real-Time Truck Freight Monitoring","volume":"44","author":"Tosoongnoen","year":"2017","journal-title":"Eng. Appl. Sci. Res."},{"key":"ref_3","first-page":"167","article-title":"Fluxgate Sensors for Onboard Weighing Systems of Heavy-Duty Dump Trucks","volume":"19","author":"Nikishechkin","year":"2021","journal-title":"World Transp. Transp."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Marszalek, Z., Duda, K., Piwowar, P., Stencel, M., Zeglen, T., and Izydorczyk, J. (2023). Load Estimation of Moving Passenger Cars Using Inductive-Loop Technology. Sensors, 23.","DOI":"10.3390\/s23042063"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Torabi, S., Wahde, M., and Hartono, P. (2019, January 20\u201322). Road Grade and Vehicle Mass Estimation for Heavy-Duty Vehicles Using Feedforward Neural Networks. Proceedings of the 2019 4th International Conference on Intelligent Transportation Engineering (ICITE), Singapore.","DOI":"10.1109\/ICITE.2019.8880261"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107982","DOI":"10.1016\/j.ymssp.2021.107982","article-title":"Mass Estimation of Ground Vehicles Based on Longitudinal Dynamics Using IMU and CAN-Bus Data","volume":"162","author":"Jensen","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12536","DOI":"10.1109\/TVT.2020.3023115","article-title":"Trailer Mass Estimation Using System Model-Based and Machine Learning Approaches","volume":"69","author":"Korayem","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_8","first-page":"2456","article-title":"Truck Load Estimation Model Integrating Multi-Head Attention Mechanism","volume":"53","author":"Gu","year":"2023","journal-title":"J. Jilin Univ. (Eng. Technol. Ed.)"},{"key":"ref_9","first-page":"295","article-title":"Research on Real-Time Estimation Method of Truck Load Based on Deep Learning","volume":"35","author":"Han","year":"2022","journal-title":"China J. Highw. Transp."},{"key":"ref_10","first-page":"712","article-title":"Load Estimation Method for Heavy-Duty Trucks Based on Internet of Vehicles Big Data","volume":"44","author":"Li","year":"2024","journal-title":"Trans. Beijing Inst. Technol. (Nat. Sci. Ed.)"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"11588","DOI":"10.1109\/JSEN.2020.3038186","article-title":"Application of Computer Vision for Estimation of Moving Vehicle Weight","volume":"21","author":"Feng","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1111\/mice.12677","article-title":"Probabilistic Vehicle Weight Estimation Using Physics-Constrained Generative Adversarial Network","volume":"36","author":"Yu","year":"2021","journal-title":"Comput.-Aided Civ. Infrastruct. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"\u0130\u015fbitirici, A., Giarr\u00e9, L., Xu, W., and Falcone, P. (2024). LSTM-Based Virtual Load Sensor for Heavy-Duty Vehicles. Sensors, 24.","DOI":"10.3390\/s24010226"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1458","DOI":"10.1038\/s42256-023-00761-y","article-title":"Physics-Enhanced Deep Surrogates for Partial Differential Equations","volume":"5","author":"Pestourie","year":"2023","journal-title":"Nat. Mach. Intell."},{"key":"ref_15","unstructured":"Tian, J., Liang, Y., Xu, R., Chen, P., Guo, C., Zhou, A., Pan, L., Rao, Z., and Yang, B. (2024). Air Quality Prediction with Physics-Informed Dual Neural ODEs in Open Systems. arXiv."},{"key":"ref_16","unstructured":"Zou, Z., Wang, Z., and Karniadakis, G.E. (2025). Learning and Discovering Multiple Solutions Using Physics-Informed Neural Networks with Random Initialization and Deep Ensemble. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"104695","DOI":"10.1016\/j.trc.2024.104695","article-title":"End-to-End Heterogeneous Graph Neural Networks for Traffic Assignment","volume":"165","author":"Liu","year":"2024","journal-title":"Transp. Res. Part C Emerg. Technol."},{"key":"ref_18","first-page":"575","article-title":"The Principled Design of Large-Scale Recursive Neural Network Architectures\u2014DAG-RNNs and the Protein Structure Prediction Problem","volume":"4","author":"Baldi","year":"2003","journal-title":"J. Mach. Learn. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1126\/science.1127647","article-title":"Reducing the Dimensionality of Data with Neural Networks","volume":"313","author":"Hinton","year":"2006","journal-title":"Science"},{"key":"ref_20","unstructured":"Li, Y., Yu, R., Shahabi, C., and Liu, Y. (2017). Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting. arXiv."},{"key":"ref_21","unstructured":"Chen, C., Chen, X., Yang, Y., and Hang, P. Sparse Attention Graph Convolution Network for Vehicle Trajectory Prediction, IEEE Trans. Veh. Technol., in press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1802\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T05:21:14Z","timestamp":1761628874000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,26]]},"references-count":21,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111802"],"URL":"https:\/\/doi.org\/10.3390\/sym17111802","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,10,26]]}}}