Physics-Informed Machine Learning for Transformer Condition Monitoring -- Part II: Physics-Informed Neural Networks and Uncertainty Quantification

TL;DR

This work addresses transformer health monitoring under data scarcity by integrating physics with learning through Physics-Informed Neural Networks (PINNs) and extending to Bayesian PINNs (BPINNs) for uncertainty quantification. PINNs enforce physics by solving a PDE with a residual $r(t,x)=u_t+\mathcal{N}[u]$ and a composite loss, demonstrated on a 1D heat-diffusion model of transformer thermal ageing to yield accurate, data-efficient temperature fields and spatial ageing maps. BPINNs treat network weights as distributions, approximate the posterior $p(\bm{\theta}|\mathcal{D})$ with a variational family $q(\bm{\theta}|\bm{\phi})$ by maximizing the ELBO, and apply this to probabilistic spatiotemporal transformer thermal modelling where the posterior mean tracks a FEM baseline and the posterior variance provides uncertainty maps. The results highlight the value of physics-aware and uncertainty-aware ML for trustworthy prognostics and health management (PHM) of critical power assets, and chart directions for scalable multi-physics PINNs, physics priors, and digital-twin integration.

Abstract

The integration of physics-based knowledge with machine learning models is increasingly shaping the monitoring, diagnostics, and prognostics of electrical transformers. In this two-part series, the first paper introduced the foundations of Neural Networks (NNs) and their variants for health assessment tasks. This second paper focuses on integrating physics and uncertainty into the learning process. We begin with the fundamentals of Physics-Informed Neural Networks (PINNs), applied to spatiotemporal thermal modeling and solid insulation ageing. Building on this, we present Bayesian PINNs as a principled framework to quantify epistemic uncertainty and deliver robust predictions under sparse data. Finally, we outline emerging research directions that highlight the potential of physics-aware and trustworthy machine learning for critical power assets.

Figures (8)

• Figure 1: Generic PINN structure with spatio-temporal coordinates as inputs $x,t$, predictions $\hat{u}$ at the output, temporal and spatial derivatives that define the residual loss $L_r(\bm{\theta})$, and the total loss $L(\bm{\theta})$ with respect to the NN parameters $\bm{\theta}$.
• Figure 2: Schematic of the transformer heat diffusion model, including heat sources and boundary conditions.
• Figure 3: Reference temperature distribution obtained via FEM simulation.
• Figure 4: Prediction error of the PINN model compared to FEM.
• Figure 5: Instantaneous transformer insulation spatial ageing, $V(x,t)$.