# Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics

arXiv: https://arxiv.org/abs/2509.15933 · HTML: https://mlsanity.com/paper/2509.15933/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 50 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 23 of 23)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Fusing Physics-based and Deep Learning Models for Prognostics](https://mlsanity.com/paper/2003.00732/) (2020)
- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [BNNpriors: A library for Bayesian neural network inference with different prior distributions](https://mlsanity.com/paper/2105.06964/) (2021)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Residual-based Attention Physics-informed Neural Networks for Spatio-Temporal Ageing Assessment of Transformers Operated in Renewable Power Plants](https://mlsanity.com/paper/2405.06443/) (2024)
- [A Benchmark on Uncertainty Quantification for Deep Learning Prognostics](https://mlsanity.com/paper/2302.04730/) (2023)
- [Conformalized Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.08111/) (2024)
- [Conformal Prediction Intervals for Remaining Useful Lifetime Estimation](https://mlsanity.com/paper/2212.14612/) (2022)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
- [A Hybrid Probabilistic Battery Health Management Approach for Robust Inspection Drone Operations](https://mlsanity.com/paper/2405.00055/) (2024)
- [Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.08501/) (2025)
- [Towards a Probabilistic Fusion Approach for Robust Battery Prognostics](https://mlsanity.com/paper/2405.15292/) (2024)

## Built on by (most cited first, 3 of 3)

- [PAD-Hand: Physics-Aware Diffusion for Hand Motion Recovery](https://mlsanity.com/paper/2603.26068/) (2026)
- [Physics-Informed Machine Learning for Transformer Condition Monitoring -- Part II: Physics-Informed Neural Networks and Uncertainty Quantification](https://mlsanity.com/paper/2512.22189/) (2025)
- [Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics](https://mlsanity.com/paper/2601.03673/) (2026)

## Abstract (arXiv)

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.