Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics

TL;DR

The paper addresses the need for uncertainty-aware prognostics in physics-informed learning by introducing a heteroscedastic Bayesian PINN (B-PINN) that explicitly disentangles epistemic and aleatoric uncertainty in a spatiotemporal insulation ageing model. By combining Bayesian neural inference with PDE-constrained learning, the method outputs full predictive posteriors $p(u|x,t)$ and separates sources of uncertainty through $\sigma^2(u|x) = \sigma^2_\theta[E_{u|x,\theta}(u|x,\theta)] + E_\theta[\sigma^2_{u|x,\theta}(u|x,\theta)]$, including input-dependent noise via a learned $\sigma^2_{\hat{u}}$. The approach improves predictive accuracy and calibration over deterministic PINNs and dropout-based PINNs, verified on transformer insulation ageing with FEM-grounded physics and real-field data, and is supported by a sensitivity analysis on sampling strategies. These results enable more reliable, uncertainty-aware decision-making in transformer asset management and PHM contexts. The work highlights the practical value of total UQ in SciML for critical infrastructure aging and points to broader extensions with alternative Bayesian inference techniques and priors.

Abstract

Physics-Informed Neural Networks (PINNs) provide a framework for integrating physical laws with data. However, their application to Prognostics and Health Management (PHM) remains constrained by the limited uncertainty quantification (UQ) capabilities. Most existing PINN-based prognostics approaches are deterministic or account only for epistemic uncertainty, limiting their suitability for risk-aware decision-making. This work introduces a heteroscedastic Bayesian Physics-Informed Neural Network (B-PINN) framework that jointly models epistemic and aleatoric uncertainty, yielding full predictive posteriors for spatiotemporal insulation material ageing estimation. The approach integrates Bayesian Neural Networks (BNNs) with physics-based residual enforcement and prior distributions, enabling probabilistic inference within a physics-informed learning architecture. The framework is evaluated on transformer insulation ageing application, validated with a finite-element thermal model and field measurements from a solar power plant, and benchmarked against deterministic PINNs, dropout-based PINNs (d-PINNs), and alternative B-PINN variants. Results show that the proposed B-PINN provides improved predictive accuracy and better-calibrated uncertainty estimates than competing approaches. A systematic sensitivity study further analyzes the impact of boundary-condition, initial-condition, and residual sampling strategies on accuracy, calibration, and generalization. Overall, the findings highlight the potential of Bayesian physics-informed learning to support uncertainty-aware prognostics and informed decision-making in transformer asset management.

Figures (13)

• Figure 1: Proposed B-PINN approach for the probabilistic spatiotemporal transformer thermal model, including spatiotemporal inputs, $(x, t)$, a BNN with probability distribution function over its parameters, two output variables over the solution ($u$) of the PDE under study ($\mu_{\hat{u}}$, $\sigma_{\hat{u}}^2$), the physics residual loss, and initial and boundary condition losses integrated in the total loss, which is used to update BNN parameter distributions.
• Figure 2: Overall framework for probabilistic transformer insulation ageing estimation. The Bayesian PINN-based model estimates the spatiotemporal oil temperature of the insulation, $\hat{\Theta}_o(x,t)$, which is coupled with an empirical model, which calculates the transformer winding temperature, $\hat{\Theta}_w(x,t)$, and finally, the estimated thermal stress is used to calculate the insulation ageing model $p(\hat{V}(x,t))$, through an empirical approach.
• Figure 3: Distribution transformer load ($K$), ambient temperature ($\Theta_A$), and top-oil temperature ($\Theta_{TO}$) over four days of operation with one-minute resolution, operated at a floating photovoltaic substation.
• Figure 4: Spatiotemporal transformer oil temperature estimation obtained through a finite element method using the data in Figure \ref{['fig:AvailableTimeSeries']} and the heat-diffusion 1D partial differential equation defined in Appendix \ref{['s:Trafo']}.
• Figure 5: Spatiotemporal transformer insulation oil temperature estimates at different time instants for heteroscedastic B-PINN, heteroscedastic d-PINN, and vanilla PINN models, along with the corresponding measured ground truth values.