Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics
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Published
Oct 26, 2025
Ibai Ramirez
Jokin Alcibar
Joel Pino
Mikel Sanz
David Pardo
Jose Aizpurua
Abstract
Scientific Machine Learning (SciML) integrates physics and data into the learning process, offering improved generalization compared with purely data-driven models. Despite its potential, applications of SciML in prognostics remain limited, partly due to the complexity of incorporating partial differential equations (PDEs) for ageing physics and the scarcity of robust uncertainty quantification methods. This work introduces a Bayesian Physics-Informed Neural Network (B-PINN) framework for probabilistic prognostics estimation. By embedding Bayesian Neural Networks into the PINN architecture, the proposed approach produces principled, uncertainty-aware predictions. The method is applied to a transformer ageing case study, where insulation degradation is primarily driven by thermal stress. The heat diffusion PDE is used as the physical residual, and different prior distributions are investigated to examine their impact on predictive posterior distributions and their ability to encode a priori physical knowledge. The framework is validated against a finite element model developed and tested with real measurements from a solar power plant. Results, benchmarked against a dropout-PINN baseline, show that the proposed B-PINN delivers more reliable prognostic predictions by accurately quantifying predictive uncertainty. This capability is crucial for supporting robust and informed maintenance decision-making in critical power assets.
How to Cite
Ramirez, I., Alcibar, J., Pino, J., Sanz, M., Pardo, D., & Aizpurua, J. (2025). Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics. Annual Conference of the PHM Society, 17(1). https://doi.org/10.36001/phmconf.2025.v17i1.4344
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Keywords
Bayesian Neural Networks, Electrical Transformers, Prognostics, Physics Informed Neural Networks, Scientific Machine Learning
References
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learning prognostics. Reliability Engineering & System Safety, 253:110513, 2025.
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neural networks using error bounds and solution bundles. In The Conference on Uncertainty in Artificial
Intelligence, 2025. doi:10.48550/arXiv.2505.06459.
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quantification. Engineering Applications of Artificial Intelligence, 118:105685, 2023. ISSN 0952-1976.
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solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational
Physics, 378:686–707, 2019b. ISSN 0021-9991. doi:10.1016/j.jcp.2018.10.045.
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Learn. Comput. Sci. Eng., 1(15), 2025. doi:10.1007/s44379-025-00015-1.
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Conference of the PHM Society, volume 7, 2015. doi:10.36001/phmconf.2015.v7i1.2646.
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Reliability Engineering & System Safety, 218:108119, 2022. ISSN 0951-8320. doi:10.1016/j.ress.2021.108119.
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reliability and systems safety applications: State of the art and challenges. Reliability Engineering & System Safety,
230:108900, 2023. ISSN 0951-8320. doi:10.1016/j.ress.2022.108900.
Manuel Arias Chao, Chetan Kulkarni, Kai Goebel, and Olga Fink. Fusing physics-based and deep learning
models for prognostics. Reliability Engineering & System Safety, 217:107961, 2022. ISSN 0951-8320.
doi:10.1016/j.ress.2021.107961.
Matthew Daigle, Indranil Roychoudhury, and Anibal Bregon. Model-based prognostics of hybrid systems. In Annual
Conference of the PHM Society, volume 7, 2015. doi:10.36001/phmconf.2015.v7i1.2586.
Jokin Alcibar, Jose I. Aizpurua, Ekhi Zugasti, and Oier Peñagarikano. A hybrid probabilistic battery health management
approach for robust inspection drone operations. Engineering Applications of Artificial Intelligence, 146:110246,
2025. ISSN 0952-1976. doi:10.1016/j.engappai.2025.110246.
Fujin Wang, Zhi Zhai, Zhibin Zhao, Yi Di, and Xuefeng Chen. Physics-informed neural network for
lithium-ion battery degradation stable modeling and prognosis. Nature Communications, 15(1):4332, 2024.
doi:https://doi.org/10.1038/s41467-024-48779-z.
Ibai Ramirez, Joel Pino, David Pardo, Mikel Sanz, Luis del Rio, Alvaro Ortiz, Kateryna Morozovska, and Jose I.
Aizpurua. Residual-based attention physics-informed neural networks for spatio-temporal ageing assessment of
transformers operated in renewable power plants. Engineering Applications of Artificial Intelligence, 139:109556,
2025. ISSN 0952-1976. doi:10.1016/j.engappai.2024.109556.
Renato G Nascimento, Felipe AC Viana, Matteo Corbetta, and Chetan S Kulkarni. A framework for li-ion battery
prognosis based on hybrid bayesian physics-informed neural networks. Scientific Reports, 13(1):13856, 2023.
doi:10.1038/s41598-023-33018-0.
Juan Fernández, Juan Chiachío, Manuel Chiachío, José Barros, and Matteo Corbetta. Physics-guided bayesian neural
networks by abc-ss: Application to reinforced concrete columns. Engineering Applications of Artificial Intelligence,
119:105790, 2023. ISSN 0952-1976. doi:10.1016/j.engappai.2022.105790.
Shankar Sankararaman. Significance, interpretation, and quantification of uncertainty in prognostics and remaining
useful life prediction. Mechanical Systems and Signal Processing, 52:228–247, 2015.
Luca Biggio, Alexander Wieland, Manuel Arias Chao, Iason Kastanis, and Olga Fink. Uncertainty-aware prognosis via
deep gaussian process. IEEE Access, 9:123517–123527, 2021.
Alireza Javanmardi and Eyke Hüllermeier. Conformal prediction intervals for remaining useful lifetime estimation.
International Journal of Prognostics and Health Management, 14(2), 2023.
Jokin Alcibar, Jose Ignacio Aizpurua, and Ekhi Zugasti. Towards a probabilistic fusion approach for robust battery
prognostics. In PHM Society European Conference, volume 8, 2024. doi:10.36001/phme.2024.v8i1.3981.
Venkat Nemani, Luca Biggio, Xun Huan, Zhen Hu, Olga Fink, Anh Tran, Yan Wang, Xiaoge Zhang, and Chao Hu.
Uncertainty quantification in machine learning for engineering design and health prognostics: A tutorial. Mechanical
Systems and Signal Processing, 205:110796, 2023.
Luis Basora, Arthur Viens, Manuel Arias Chao, and Xavier Olive. A benchmark on uncertainty quantification for deep
learning prognostics. Reliability Engineering & System Safety, 253:110513, 2025.
Kevin Linka, Amelie Schäfer, Xuhui Meng, Zongren Zou, George Em Karniadakis, and Ellen Kuhl. Bayesian physics
informed neural networks for real-world nonlinear dynamical systems. Computer Methods in Applied Mechanics
and Engineering, 402:115346, 2022. ISSN 0045-7825. doi:10.1016/j.cma.2022.115346.
Pablo Flores, Olga Graf, Pavlos Protopapas, and Karim Pichara. Improved uncertainty quantification in physicsinformed
neural networks using error bounds and solution bundles. In The Conference on Uncertainty in Artificial
Intelligence, 2025. doi:10.48550/arXiv.2505.06459.
Shailesh Garg and Souvik Chakraborty. Vb-deeponet: A bayesian operator learning framework for uncertainty
quantification. Engineering Applications of Artificial Intelligence, 118:105685, 2023. ISSN 0952-1976.
doi:10.1016/j.engappai.2022.105685.
Zhiwei Gao and George Em Karniadakis. Scalable bayesian physics-informed kolmogorov-arnold networks. arXiv
preprint arXiv:2501.08501, 2025. doi:10.48550/arXiv.2501.08501.
Jinwu Li, Xiangyun Long, Xinyang Deng, Wen Jiang, Kai Zhou, Chao Jiang, and Xiaoge Zhang. A principled
distance-aware uncertainty quantification approach for enhancing the reliability of physics-informed neural network.
Reliability Engineering & System Safety, 245:109963, 2024b. ISSN 0951-8320. doi:10.1016/j.ress.2024.109963.
Apostolos F Psaros, Xuhui Meng, Zongren Zou, Ling Guo, and George Em Karniadakis. Uncertainty quantification in
scientific machine learning: Methods, metrics, and comparisons. Journal of Computational Physics, 477:111902,
2023. doi:10.1016/j.jcp.2022.111902.
Ashish S. Nair, Bruno Jacob, Amanda A. Howard, Jan Drgona, and Panos Stinis. E-pinns: Epistemic physics-informed
neural networks, 2025.
Lena Podina, Mahdi Torabi Rad, and Mohammad Kohandel. Conformalized physics-informed neural networks. In
ICLR 2024 Workshop on AI4Differential Equations In Science, 2024. doi:10.48550/arXiv.2405.08111.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep
learning. In Maria Florina Balcan and Kilian Q. Weinberger, editors, Proceedings of The 33rd International
Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pages 1050–1059, New
York, New York, USA, 20–22 Jun 2016. PMLR. doi:10.48550/arXiv.1506.02142.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty
estimation using deep ensembles. In International Conference on Neural Information Processing Systems,
NIPS’17, page 6405–6416, Red Hook, NY, USA, 2017. Curran Associates Inc. ISBN 9781510860964.
doi:10.48550/arXiv.1612.01474.
Dongkun Zhang, Lu Lu, Ling Guo, and George Em Karniadakis. Quantifying total uncertainty in physics-informed
neural networks for solving forward and inverse stochastic problems. Journal of Computational Physics, 397:108850,
2019. ISSN 0021-9991. doi:10.1016/j.jcp.2019.07.048.
Xinchao Jiang, Xin Wang, Ziming Wen, Enying Li, and Hu Wang. Practical uncertainty quantification for spacedependent
inverse heat conduction problem via ensemble physics-informed neural networks. International Communications
in Heat and Mass Transfer, 147:106940, 2023. doi:10.1016/j.icheatmasstransfer.2023.106940.
Zongren Zou, Zhicheng Wang, and George Em Karniadakis. Learning and discovering multiple solutions using
physics-informed neural networks with random initialization and deep ensemble. arXiv preprint arXiv:2503.06320,
2025. doi:10.48550/arXiv.2503.06320.
Jose Ignacio Aizpurua, Ibai Ramirez, Iker Lasa, Luis del Rio, Alvaro Ortiz, and Brian G. Stewart. Hybrid transformer
prognostics framework for enhanced probabilistic predictions in renewable energy applications. IEEE Trans. Pow.
Del., 38(1):599–609, 2023. doi:10.1109/TPWRD.2022.3203873.
International Electrotechnical Commission. Loading guide for oil-immersed power transformers. IEC 60076-7, 2018.
URL https://webstore.iec.ch/.
M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for
solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational
Physics, 378:686–707, 2019b. ISSN 0021-9991. doi:10.1016/j.jcp.2018.10.045.
Logan G Wright, Tatsuhiro Onodera, Martin M Stein, Tianyu Wang, Darren T Schachter, Zoey Hu, and Peter L
McMahon. Deep physical neural networks trained with backpropagation. Nature, 601(7894):549–555, 2022.
doi:10.1038/s41586-021-04223-6.
Sokratis J. Anagnostopoulos, Juan Diego Toscano, Nikolaos Stergiopulos, and George Em Karniadakis. Residual-based
attention in physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 421:
116805, 2024. ISSN 0045-7825. doi:10.1016/j.cma.2024.116805.
Vincent Fortuin, Adrià Garriga-Alonso, Mark van der Wilk, and Laurence Aitchison. Bnnpriors: A library for bayesian
neural network inference with different prior distributions. Software Impacts, 9:100079, 2021. ISSN 2665-9638.
doi:10.1016/j.simpa.2021.100079.
Peter M. Williams. Bayesian regularization and pruning using a laplace prior. Neural Computation, 7(1):117–143,
1995. doi:10.1162/neco.1995.7.1.117.
Alex Kendall and Yarin Gal. What uncertainties do we need in bayesian deep learning for computer vision? In I. Guyon,
U. Von Luxburg, S. Bengio, H.Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information
Processing Systems, volume 30, pages 1–11. Curran Associates, Inc., 2017. doi:10.48550/arXiv.1703.04977.
Sifan Wang, Shyam Sankaran, Hanwen Wang, and Paris Perdikaris. An expert’s guide to training physics-informed
neural networks, 2023.
Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012. ISBN 978-0262018029.
Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, volume 4. Springer, 2006.
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