{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T09:58:50Z","timestamp":1790675930941,"version":"4.1.0"},"reference-count":30,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T00:00:00Z","timestamp":1790640000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Accelerometers built on micro-electromechanical systems (MEMS) play a critical role in structural monitoring and machinery diagnostics; however, their accuracy suffers from intrinsic nonlinearities\u2014dead zones, hysteresis, saturation, and colored noise. Conventional physics-based correction methods are interpretable yet cannot capture complicated hysteretic behavior, while purely neural-network approaches generalize poorly and lack a physical foundation. This paper proposes a hybrid architecture that combines a residual convolutional neural network with a physics-guided low-pass filter prior, fused through an attention-gated mechanism. The CNN learns only the residual nonlinearity; the filter supplies a steady, band-limited baseline. We validate the model on two simulated scenarios\u2014a noise-dominant track and a nonlinear-dominant track\u2014across three random seeds. The resulting Hybrid LPF-CNN outperforms a standalone CNN by 15.2% and an LSTM by 33.5% on the severely nonlinear track, reaching a mean R2 of 0.9970 and an RMSE of 0.0179 g. On the noise-dominant track, it reaches R2 = 0.9407 and RMSE = 0.0800 g, surpassing both CNN and LSTM baselines. The model is also stable across seeds (\u03c3=0.0001 in R2) and gives a legible breakdown of the correction it applies. Our systematic architectural search revealed that a dual-encoder design with attention-gated fusion\u2014where raw and filtered signals are processed separately and combined via a learnable spatial gate\u2014provides the optimal balance between stability, accuracy, and interpretability. Even basic physics priors substantially improve the performance, stability, and interpretability of deep learning models for sensor error correction.<\/jats:p>","DOI":"10.3390\/s26196165","type":"journal-article","created":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T09:10:54Z","timestamp":1790673054000},"page":"6165","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hybrid Physics-Guided Neural Network for Vibration Sensor Nonlinearity Correction"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0149-7587","authenticated-orcid":false,"given":"Alexander P.","family":"Lyapin","sequence":"first","affiliation":[{"name":"Department of Computational and Information Technology, Siberian Federal University, Krasnoyarsk 660041, Russia"},{"name":"Department of Economics, Shenzhen MSU-BIT University, Shenzhen 518712, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"project-administration","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-5798-3160","authenticated-orcid":false,"given":"Faizulddin","family":"Ebrahimi","sequence":"additional","affiliation":[{"name":"Department of Fuel Supply and Lubricants, Siberian Federal University, Krasnoyarsk 660041, Russia"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1822-1245","authenticated-orcid":false,"given":"Evgeny D.","family":"Agafonov","sequence":"additional","affiliation":[{"name":"Department of Fuel Supply and Lubricants, Siberian Federal University, Krasnoyarsk 660041, Russia"},{"name":"Department of System Analysis and Operations Research, Reshetnev Siberian State University of Science and Technology, Krasnoyarsk 660037, Russia"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"supervision","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0531-065X","authenticated-orcid":false,"given":"Viktor S.","family":"Ratushnyak","sequence":"additional","affiliation":[{"name":"Department of Train Traffic Management Systems, Institute of Railway Transport, Krasnoyarsk 660028, Russia"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"software","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-3144-6965","authenticated-orcid":false,"given":"Julia","family":"Schnitzer","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Media, Brandenburg University of Applied Sciences, 14770 Brandenburg an der Havel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"funding-acquisition","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mardanshahi, A., Sreekumar, A., Yang, X., Barman, S.K., and Chronopoulos, D. (2025). Sensing Techniques for Structural Health Monitoring: A State-of-the-Art Review on Performance Criteria and New-Generation Technologies. Sensors, 25.","DOI":"10.3390\/s25051424"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Qiao, H., Guan, H., and Zhu, Y. (2025). Footbridge structural health monitoring\u2014A review of current research and future directions. Struct. Infrastruct. Eng., 1\u201324.","DOI":"10.1080\/15732479.2025.2547349"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"202401745","DOI":"10.1002\/adem.202401745","article-title":"Advanced Sensors and Sensing Systems for Structural Health Monitoring in Aerospace Composites","volume":"26","author":"Ogunleye","year":"2024","journal-title":"Adv. Eng. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1797","DOI":"10.3390\/eng4030102","article-title":"A Review on Vibration Monitoring Techniques for Predictive Maintenance of Rotating Machinery","volume":"4","author":"Romanssini","year":"2023","journal-title":"Eng"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.3390\/vibration7040054","article-title":"Vibration Signal Analysis for Intelligent Rotating Machinery Diagnosis and Prognosis: A Comprehensive Systematic Literature Review","volume":"7","author":"Bagri","year":"2024","journal-title":"Vibration"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sintoni, M., Macrelli, E., Bellini, A., and Bianchini, C. (2023). Condition Monitoring of Induction Machines: Quantitative Analysis and Comparison. Sensors, 23.","DOI":"10.3390\/s23021046"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10845-023-02103-6","article-title":"Prognostics and health management for induction machines: A comprehensive review","volume":"35","author":"Huang","year":"2024","journal-title":"J. Intell. Manuf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Dong, W., Lu, C., Bao, L., Li, W., Shin, K., and Han, C. (2024). A Planar Multi-Inertial Navigation Strategy for Autonomous Systems for Signal-Variable Environments. Sensors, 24.","DOI":"10.3390\/s24041064"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chiominto, L., D\u2019Emilia, G., Gaspari, A., and Natale, E. (2025). Dynamic Multi-Axis Calibration of MEMS Accelerometers for Sensitivity and Linearity Assessment. Sensors, 25.","DOI":"10.20944\/preprints202501.1810.v1"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kim, H.J., and Jung, H.K. (2025). Temperature Hysteresis Calibration Method of MEMS Accelerometer. Sensors, 25.","DOI":"10.3390\/s25196131"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ali, G., and Mohd-Yasin, F. (2024). Comprehensive Noise Modeling of Piezoelectric Charge Accelerometer with Signal Conditioning Circuit. Micromachines, 15.","DOI":"10.3390\/mi15020283"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1038\/s41378-024-00849-4","article-title":"A 14 \u03bcHz\/Hz resolution and 32 \u03bcHz bias instability MEMS quartz resonant accelerometer with a low-noise oscillating readout circuit","volume":"10","author":"Bu","year":"2024","journal-title":"Microsyst. Nanoeng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ding, L., and Wen, C. (2024). High-Order Extended Kalman Filter for State Estimation of Nonlinear Systems. Symmetry, 16.","DOI":"10.3390\/sym16050617"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1177\/00202940241258481","article-title":"Improving the performance of a MEMS-IMU system based on a false state-space model by using a fading factor adaptive Kalman filter","volume":"57","year":"2024","journal-title":"Meas. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"12472","DOI":"10.1038\/s41598-024-61596-0","article-title":"Advancing state estimation for lithium-ion batteries with hysteresis through systematic extended Kalman filter tuning","volume":"14","author":"Knox","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Wang, S., Pitts, J., Purohit, R., and Shah, H. (2025). The Influence of Motion Data Low-Pass Filtering Methods in Machine-Learning Models. Appl. Sci., 15.","DOI":"10.3390\/app15042177"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Liu, Y., Ping, M., Han, J., Cheng, X., Qin, H., and Wang, W. (2024). Neural Network Methods in the Development of MEMS Sensors. Micromachines, 15.","DOI":"10.3390\/mi15111368"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1007\/s41315-025-00438-2","article-title":"Artificial neural network-based MEMS accelerometer array calibration","volume":"9","author":"Pesti","year":"2025","journal-title":"Int. J. Intell. Robot. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2386","DOI":"10.1021\/acssensors.5c01075","article-title":"Informing Deep Learning of Sensing Data with Physics and Chemistry","volume":"10","author":"Mao","year":"2025","journal-title":"ACS Sens."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"115128","DOI":"10.1016\/j.sna.2024.115128","article-title":"Micromechanical gyroscope temperature compensation based on combined LSTM-SVM-DBN algorithm","volume":"369","author":"Ouyang","year":"2024","journal-title":"Sens. Actuators A Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Parziale, M., Lomazzi, L., Giglio, M., and Cadini, F. (2023). Physics-Informed Neural Networks for the Condition Monitoring of Rotating Shafts. Sensors, 24.","DOI":"10.20944\/preprints202311.1450.v1"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20220400","DOI":"10.1098\/rsta.2022.0400","article-title":"Physics-informed deep learning for structural vibration identification and its application on a benchmark structure","volume":"382","author":"Zhang","year":"2024","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s10462-025-11322-7","article-title":"Physics-informed neural networks for PDE problems: A comprehensive review","volume":"58","author":"Luo","year":"2025","journal-title":"Artif. Intell. Rev."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"124678","DOI":"10.1016\/j.eswa.2024.124678","article-title":"Physics-informed machine learning: A comprehensive review on applications in anomaly detection and condition monitoring","volume":"255","author":"Wu","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.camwa.2025.02.022","article-title":"Optimizing Variational Physics-Informed Neural Networks Using Least Squares","volume":"185","author":"Uriarte","year":"2025","journal-title":"Comput. Math. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, Y., Sun, H., Li, J., Ma, C., Zhang, Y., Wang, X., and Feng, Q. (2026). A Nonlinear Error Compensation Method for Heterodyne Interferometry Based on Self-Supervised Physics-Informed Neural Networks with Frequency-Domain Priors. Sensors, 26.","DOI":"10.3390\/s26103000"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"111195","DOI":"10.1016\/j.ymssp.2024.111195","article-title":"Development of multi-sensor data fusion and in-process expert system for monitoring precision in thin wall lens barrel turning","volume":"210","author":"Tang","year":"2024","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"113226","DOI":"10.1016\/j.cma.2020.113226","article-title":"Physics-informed multi-LSTM networks for metamodeling of nonlinear structures","volume":"369","author":"Zhang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"104948","DOI":"10.1016\/j.aei.2026.104948","article-title":"Physics-informed graph neural networks for sim-to-real structural response prediction","volume":"76","author":"Xin","year":"2026","journal-title":"Adv. Eng. Inform."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"523367","DOI":"10.1007\/s10409-023-23367-x","article-title":"Phase deviation of semi-active suspension control and its compensation with inertial suspension","volume":"40","author":"Yang","year":"2024","journal-title":"Acta Mech. Sin."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6165\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T09:14:48Z","timestamp":1790673288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6165"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,29]]},"references-count":30,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2026,10]]}},"alternative-id":["s26196165"],"URL":"https:\/\/doi.org\/10.3390\/s26196165","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,29]]}}}