{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:43Z","timestamp":1760146963549,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, to study the influence of neural networks\u2019 morphology symmetry, a mathematical model is developed that considers dynamic symmetry for diagnosing complex dynamic objects. The developed mathematical model includes the symmetric architecture concept with adaptive parameters, according to which the neural network is represented by a function that relates the input data to the diagnostic outputs. A dynamic symmetry function is introduced to the neural networks\u2019 weight change depending on the systems\u2019 state. To achieve symmetric training, the loss function is minimised with regularisation considering deviations from the symmetric state. The theorem \u201cOn the symmetric neural network optimisation stability\u201d is formulated and proven, demonstrating the symmetric neural network optimisation stability, which is confirmed by the weights\u2019 stability and the loss functions\u2019 global optimisation, which includes symmetric regularisation, which stabilises the weights and reduces their sensitivity to minor disturbances. It is shown that in the training process, gradient descent with symmetric regularisation contributes to stable convergence and a decrease in weight asymmetry. In this case, an energy function that tends to zero with the optimal weights\u2019 achievement is introduced. The analysis showed that symmetric regularisation minimises the weights\u2019 deviation and prevents their overtraining. It was experimentally established that the optimal regularisation coefficient \u03bb = 1.0 ensures a balance between the models\u2019 symmetry and flexibility, minimising the diagnostic error. The results show that symmetric regularisation contributes to practical training and increases the diagnostic models\u2019 accuracy.<\/jats:p>","DOI":"10.3390\/sym17010035","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T13:26:25Z","timestamp":1735651585000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Influence of the Neural Network Morphology Symmetry on the Complex Dynamic Objects\u2019 Diagnostics"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8009-5254","authenticated-orcid":false,"given":"Serhii","family":"Vladov","sequence":"first","affiliation":[{"name":"Kremenchuk Flight College, Kharkiv National University of Internal Affairs, 27, L. Landau Avenue, 61080 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6417-3689","authenticated-orcid":false,"given":"Victoria","family":"Vysotska","sequence":"additional","affiliation":[{"name":"Information Systems and Networks Department, Lviv Polytechnic National University, 12, Bandera Street, 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Viktor","family":"Vasylenko","sequence":"additional","affiliation":[{"name":"Kremenchuk Flight College, Kharkiv National University of Internal Affairs, 27, L. Landau Avenue, 61080 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9676-0180","authenticated-orcid":false,"given":"Vasyl","family":"Lytvyn","sequence":"additional","affiliation":[{"name":"Information Systems and Networks Department, Lviv Polytechnic National University, 12, Bandera Street, 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6528-9867","authenticated-orcid":false,"given":"Mariia","family":"Nazarkevych","sequence":"additional","affiliation":[{"name":"Information Systems and Networks Department, Lviv Polytechnic National University, 12, Bandera Street, 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8170-3001","authenticated-orcid":false,"given":"Olha","family":"Fedevych","sequence":"additional","affiliation":[{"name":"Information Systems and Networks Department, Lviv Polytechnic National University, 12, Bandera Street, 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","first-page":"160","article-title":"Modified Neural Network Fault-Tolerant Closed Onboard Helicopters Turboshaft Engines Automatic Control System","volume":"3387","author":"Vladov","year":"2023","journal-title":"CEUR Workshop Proc."},{"key":"ref_2","first-page":"40","article-title":"A Neuro-Fuzzy Expert System for the Control and Diagnostics of Helicopters Aircraft Engines Technical State","volume":"3013","author":"Vladov","year":"2021","journal-title":"CEUR Workshop Proc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"186","DOI":"10.12913\/22998624\/156205","article-title":"New Design of the Hatch Cover to Increase the Carrying Capacity of the Gondola Car","volume":"16","author":"Baranovskyi","year":"2022","journal-title":"Adv. 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