{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T18:43:34Z","timestamp":1787165014032,"version":"build-2736575974"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T00:00:00Z","timestamp":1783641600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Yunnan Provincial Colleges and Universities Science and Technology Project Serving Key Industries: Research on Intelligent Inspection Method of Logistics Equipment Based on UAV Swarm Image Recognition","award":["FWCY-ZNT2025015"],"award-info":[{"award-number":["FWCY-ZNT2025015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Long-tailed data are very common in industrial scenarios because equipment failures occur with a low probability, resulting in far fewer faulty samples than normal ones. However, when facing long-tailed data distributions, existing deep learning methods suffer from a significant degradation in performance and exhibit high bias. To overcome this limitation, this paper proposes a network that combines balanced adaptive logit-compensated cross-entropy loss with quadratic convolution (BALQNet) to improve diagnostic performance under long-tailed data conditions. The proposed method mainly consists of a balanced adaptive logit-compensated cross-entropy loss (BAL) and a quadratic convolution backbone. By jointly incorporating logit compensation, label smoothing, and class reweighting, BAL enhances the optimization of minority-class samples, thereby improving the classifier\u2019s ability to distinguish different categories without introducing additional architectural complexity. Meanwhile, quadratic convolution further improves the effectiveness of feature representation learning. Finally, experiments are conducted on self-built bearing, gear, and motor datasets. The results show that BALQNet maintains strong diagnostic performance when handling long-tailed data. In addition, the ablation results provide further evidence for the effectiveness of the proposed approach.<\/jats:p>","DOI":"10.3390\/e28070783","type":"journal-article","created":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T13:16:28Z","timestamp":1783689388000},"page":"783","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Balanced Adaptive Logit-Compensated Cross-Entropy and Quadratic Convolutional Network for Intelligent Fault Diagnosis Under Long-Tailed Data Distribution"],"prefix":"10.3390","volume":"28","author":[{"given":"Wenbin","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Kunming University, Kunming 650214, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-8104-0267","authenticated-orcid":false,"given":"Zikang","family":"Cao","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haijian","family":"Wu","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dewei","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Honghe University, Mengzi 661199, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yasong","family":"Pu","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Honghe University, Mengzi 661199, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"117323","DOI":"10.1016\/j.measurement.2025.117323","article-title":"Recognition of rolling bearing life status based on GI performance degradation indicator and AGNCN","volume":"252","author":"Guo","year":"2025","journal-title":"Measurement"},{"key":"ref_2","first-page":"95","article-title":"A method for extracting early fault features of rolling bearings","volume":"43","author":"Cui","year":"2023","journal-title":"Noise Vib. Control"},{"key":"ref_3","first-page":"1252","article-title":"Multi-source fusion perception intelligent fault diagnosis methods for critical components of industrial robots","volume":"38","author":"Han","year":"2025","journal-title":"J. Vib. Eng."},{"key":"ref_4","first-page":"289","article-title":"Fault diagnosis method for rolling bearing based on AVMD and MMDE-SMFF in noisy environment","volume":"44","author":"Zhao","year":"2025","journal-title":"J. Vib. Shock"},{"key":"ref_5","first-page":"1064","article-title":"Fault diagnosis of rolling bearings under variable speed conditions based on adaptive window rotation optimization short-time Fourier transform","volume":"37","author":"Zhao","year":"2024","journal-title":"J. Vib. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"24301","DOI":"10.1109\/ACCESS.2017.2766232","article-title":"Adaptive mode decomposition methods and their applications in signal analysis for machinery fault diagnosis: A review with examples","volume":"5","author":"Feng","year":"2017","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"110796","DOI":"10.1016\/j.ymssp.2023.110796","article-title":"Uncertainty quantification in machine learning for engineering design and health prognostics: A tutorial","volume":"205","author":"Nemani","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhang, W., Peng, G., Li, C., Chen, Y., and Zhang, Z. (2017). A new deep learning model for fault diagnosis with good anti-noise and domain adaptation ability on raw vibration signals. Sensors, 17.","DOI":"10.20944\/preprints201701.0132.v1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2023.3326161","article-title":"Attention-embedded quadratic network (Qttention) for effective and interpretable bearing fault diagnosis","volume":"72","author":"Liao","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"110532","DOI":"10.1016\/j.engappai.2025.110532","article-title":"A novel adaptive gating neurons model with physical features weighted for bearing fault diagnosis under strong noise","volume":"149","author":"Guo","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dai, C., He, D., Jin, Z., Zhang, X., Chen, G., Wu, J., Zhao, J., and Xu, Y. (2026). Digital twin-assisted graph contrastive domain adaptation for small-sample bearing fault diagnosis. Struct. Health Monit., Epub ahead of printing.","DOI":"10.1177\/14759217261440798"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xu, Y., He, D., Sun, H., Jin, Z., and Zhao, M. (2026). Self-supervised learning for train bearing fault diagnosis based on time\u2013frequency dual domain prediction. Struct. Health Monit., Epub ahead of printing.","DOI":"10.1177\/14759217251405584"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Cheng, Y., Liu, L., Liao, Z., Chen, B., Yan, J., and Chen, Z. (2026). A novel knowledge distillation framework for bearing fault diagnosis under imbalanced samples. Struct. Health Monit., Epub ahead of printing.","DOI":"10.1177\/14759217261418304"},{"key":"ref_14","first-page":"96","article-title":"Multi-condition fault diagnosis method of rolling bearing based on enhanced deep convolutional neural network","volume":"38","author":"Guo","year":"2025","journal-title":"J. Vib. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3935","DOI":"10.1080\/10589759.2024.2404489","article-title":"Small sample gearbox fault diagnosis based on improved deep forest in noisy environments","volume":"40","author":"Shao","year":"2025","journal-title":"Nondestruct. Test. Eval."},{"key":"ref_16","first-page":"1829","article-title":"Research on gearbox fault diagnosis method based on parameter optimized VMD and CNN","volume":"41","author":"Xi","year":"2022","journal-title":"Mech. Sci. Technol. Aerosp. Eng."},{"key":"ref_17","first-page":"149","article-title":"Motor fault diagnosis based on improved SDP feature fusion images","volume":"33","author":"Luo","year":"2025","journal-title":"Comput. Meas. Control"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Xu, Y., He, D., Luo, X., Jin, Z., and Sun, H. (2026). CSSCL: A cross-sensor self-contrastive learning for nondestructive fault detection of heavy locomotive motors in label-scarce scenarios. Nondestruct. Test. Eval.","DOI":"10.1080\/10589759.2026.2671370"},{"key":"ref_19","first-page":"93","article-title":"Wind turbine bearing fault diagnosis method based on LSTM-SVM","volume":"11","author":"Zhao","year":"2025","journal-title":"China Instrum."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Han, Y., Zhang, F., Li, Z., Wang, Q., Li, C., Lai, P., Li, T., Teng, F., and Jin, Z. (IEEE Trans. Instrum. Meas., 2024). Mt-ConvFormer: A multitask bearing fault diagnosis method using a combination of CNN and transformer, IEEE Trans. Instrum. Meas., in press.","DOI":"10.1109\/TIM.2024.3502821"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"110008","DOI":"10.1016\/j.knosys.2022.110008","article-title":"Imbalance fault diagnosis under long-tailed distribution: Challenges, solutions and prospects","volume":"258","author":"Chen","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"10795","DOI":"10.1109\/TPAMI.2023.3268118","article-title":"Deep long-tailed learning: A survey","volume":"45","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"112895","DOI":"10.1016\/j.ymssp.2025.112895","article-title":"A novel sparse-aware contrastive learning network with adaptive gating neurons for extreme class imbalance diagnosis scenarios","volume":"235","author":"Guo","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Khoshgoftaar, T.M., and Gao, K. (2009, January 13\u201315). Feature selection with imbalanced data for software defect prediction. Proceedings of the International Conference on Machine Learning and Applications, Miami, FL, USA.","DOI":"10.1109\/ICMLA.2009.18"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"104358","DOI":"10.1016\/j.conengprac.2020.104358","article-title":"An improved ensemble fusion autoencoder model for fault diagnosis from imbalanced and incomplete data","volume":"98","author":"Yang","year":"2020","journal-title":"Control Eng. Pract."},{"key":"ref_26","first-page":"1","article-title":"Research on probabilistic contrastive learning and quadratic convolutional neural network for intelligent diagnosis of long-tailed bearing data","volume":"74","author":"Cao","year":"2025","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_27","unstructured":"Wang, X., Lian, L., Miao, Z., Liu, Z., and Yu, S.X. (2020). Long-tailed recognition by routing diverse distribution-aware experts. arXiv."},{"key":"ref_28","first-page":"6396","article-title":"Decoupled contrastive learning for long-tailed recognition","volume":"38","author":"Xuan","year":"2024","journal-title":"Proc. AAAI Conf. Artif. Intell."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.neunet.2020.01.007","article-title":"Universal approximation with quadratic deep networks","volume":"124","author":"Fan","year":"2020","journal-title":"Neural Netw."},{"key":"ref_30","unstructured":"Fan, F.L., Li, M., Wang, F., Lai, R., and Wang, G. (IEEE Trans. Neural Netw. Learn. Syst., 2023). On expressivity and trainability of quadratic networks, IEEE Trans. Neural Netw. Learn. Syst., in press."},{"key":"ref_31","unstructured":"Yu, W.E., Sun, J., Zhang, S., Zhang, X., and Liao, J.X. (2023). A class-weighted supervised contrastive learning long-tailed bearing fault diagnosis approach using quadratic neural network. arXiv."},{"key":"ref_32","unstructured":"Menon, A.K., Jayasumana, S., Rawat, A.S., Jain, H., Veit, A., and Kumar, S. (2020). Long-tail learning via logit adjustment. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/7\/783\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T13:17:50Z","timestamp":1783689470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/7\/783"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,10]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2026,7]]}},"alternative-id":["e28070783"],"URL":"https:\/\/doi.org\/10.3390\/e28070783","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,10]]}}}