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

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A heteroscedastic Bayesian physics-informed neural network that outputs both a predictive mean and an input-dependent variance yields more accurate and better-calibrated probabilistic transformer insulation ageing estimates than determinist

desk verdict Heteroscedastic B-PINN applied to transformer ageing, but the residual loss is undefined and the evaluation is in-sample; useful case study, not a breakthrough. read the letter →

arxiv 2601.03673 v2 pith:FVGGJ2FB submitted 2026-01-07 cs.LG cs.AI

Ibai Ramirez , Jokin Alcibar , Joel Pino , Mikel Sanz , Jose I. Aizpurua This is my paper · ORCID
classification cs.LGcs.AI MSC 68T0762F15
keywords physics-informedneuralnetworksBayesianuncertaintyquantificationheteroscedasticaleatoricepistemictransformerinsulationageingprognosticsandhealthmanagementvariationalinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a physics-informed neural network can predict transformer insulation ageing with a full probability distribution, not just a point estimate, by explicitly separating aleatoric uncertainty (irreducible noise from measurements and operation) from epistemic uncertainty (reducible model uncertainty from limited data). Its core move is a heteroscedastic B-PINN: the network outputs both the predicted oil temperature and an input-dependent variance, trained with a heteroscedastic Gaussian likelihood inside a variational Bayesian objective with physics residuals. On a real distribution transformer feeding a floating solar plant, the model lowers total RMSE from 2.55 for a vanilla PINN to 1.09, and reduces RMSE by 42.6% with 40.2% better CRPS than the best dropout-based competitor. If the claim holds, asset managers gain a principled way to know which part of predictive uncertainty would shrink with more data (epistemic) and which part is inherent to the process (aleatoric), which is exactly what risk-aware maintenance decisions need.

What carries the argument

The load-bearing mechanism is the heteroscedastic B-PINN architecture: the network outputs both a predictive mean and an input-dependent variance, and is trained with a heteroscedastic Gaussian negative-log-likelihood loss for initial, boundary, and residual points inside the variational ELBO. The law of total variance then decomposes total predictive uncertainty into epistemic (variance of the mean across weight samples) and aleatoric (average of the variance output) components.

What would settle it

Hold out the fibre-optic temperature measurements taken at different transformer heights, train the B-PINN without them, and check the empirical coverage of its 90% predictive intervals over the full four-day load cycle; coverage well below 90%, or a lower CRPS for the dropout-PINN on those held-out sensors, would refute the paper's comparative claim.

Watch

Extended reading notes

Core claim

The central claim is that a B-PINN with two outputs—the mean oil temperature and an input-dependent variance—trained by variational inference with a heteroscedastic Gaussian likelihood and physics-residual loss yields the full predictive posterior for spatiotemporal oil temperature, and hence for winding temperature and insulation ageing. Epistemic uncertainty is estimated as the variance across Monte Carlo samples of the Bayesian weights; aleatoric uncertainty is estimated as the averaged variance output; total uncertainty is their sum via the law of total variance. In the transformer case study, this disentangled model achieves the best RMSE, CRPS, and NLL across prediction instants, produ

Load-bearing premise

The load-bearing premise is that the finite-element heat-diffusion solution used to set initial and boundary conditions and to score errors represents the true transformer oil temperature; if that simulation is biased, the reported accuracy and calibration measure fit to a model, not to physical field behaviour.

Editorial extensions

If this is right

  • Transformer operators could use the disentangled uncertainties to decide whether to collect more data (epistemic uncertainty) or to accept irreducible noise (aleatoric uncertainty) when planning maintenance.
  • Dropout-based uncertainty approximations can be over-dispersed in extrapolation, leading to overly conservative ageing estimates and premature asset replacement if used for RUL decisions.
  • Heteroscedastic noise modelling gives better-calibrated predictive distributions than a fixed noise level, at comparable RMSE, in this spatiotemporal heat-transfer setting.
  • Configuration guidance: moderate residual collocation points (around 10,000) and full boundary-condition coverage gave the best probabilistic performance, while initial-condition samples mattered little once the initial state was constrained.
  • The full pipeline—oil temperature, winding temperature, ageing acceleration, loss of life—propagates the predictive temperature distribution into a distribution over insulation ageing, which is the quantity maintenance decisions actually need.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the same dual-output heteroscedastic treatment likely transfers to other degradation prognostics governed by parabolic heat equations, such as battery thermal runaway or bearing heat diffusion, where noise is also load-dependent.
  • Inference: the reported 'true' temperature is a finite-element simulation, not a full set of internal sensor measurements; how much of the comparative advantage holds in the physical transformer is an open question the paper's setup cannot settle.
  • Inference: the learned aleatoric variance could be reused as an acquisition function, sampling more collocation points where variance is high and CRPS is worst, which the paper does not test.
  • Inference: since epistemic uncertainty stayed small while aleatoric dominated, a cheaper surrogate that only models heteroscedastic noise might capture most of the practical benefit; the paper does not test this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a heteroscedastic Bayesian physics-informed neural network (B-PINN) that outputs both a predictive mean and an input-dependent variance for a 1D heat-diffusion model of transformer oil temperature, and couples it with empirical winding-temperature and IEC ageing models to produce probabilistic insulation-ageing estimates. Epistemic uncertainty is represented by variational inference over network weights; aleatoric uncertainty is modeled through a Gaussian likelihood with a learned variance. The method is compared with vanilla PINNs, dropout PINNs, and a homoscedastic B-PINN on a floating-solar-plant transformer case study, using a finite-element solution as the reference. The authors claim improved predictive accuracy and better-calibrated uncertainty estimates, and they report a sensitivity study of initial, boundary, and residual sampling strategies.

Significance. If the headline results held, the paper would provide a practically motivated demonstration that heteroscedastic Bayesian PINNs can disentangle aleatoric and epistemic uncertainty in a PHM application. The strengths are the realistic transformer case study, the comparison with several baselines, and the sensitivity analysis of sampling strategies. However, the central methodological novelty is largely a standard combination of heteroscedastic BNN regression with PINN residuals, and the empirical support is weakened by an underspecified residual likelihood, overlapping confidence intervals, and an evaluation that is effectively interpolation against a simulation rather than forecasting against independent field data. No code or data are provided, which further limits reproducibility. The contribution is therefore better characterized as an application study whose headline claims require substantial revision.

major comments (3)
  1. [Section 2.4, Eq. (21)] The training objective for the heteroscedastic B-PINN is not well-defined. Eq. (20) defines NLL_hetero(x,t;θ) for a point with an observed solution value u. Eq. (21) reuses this NLL for residual points as λ_r NLL_hetero^(r), but no formula for NLL_hetero^(r) is given. At residual collocation points there is no observed u; the quantity being minimized is r(x,t;θ)=L[u_θ]-f, and one cannot simply take σ^2_hat u(x,t;θ), the network's output variance, as the variance of r. The homoscedastic formulation in Eq. (14) used a fixed σ_f for residual points, but the heteroscedastic extension silently changes this. Algorithm 2, the only training algorithm supplied, still refers to Eq. (15) and the homoscedastic residual likelihood, not to Eq. (21). Since the reported accuracy and calibration results are obtained by minimizing this objective, the central claim is not internally supported until the res
  2. [Tables 2 and 3] The headline claim of better-calibrated uncertainty estimates is not supported by the reported statistics. At t=6h, the heteroscedastic B-PINN NLL is 1.61±1.43, worse than the homoscedastic 0.49±0.33. Total NLL values overlap within standard deviations (1.75±0.19 vs 1.98±0.73). Miscalibration areas also overlap (0.1956±0.1616 vs 0.2135±0.0591). With only five runs, these differences are not statistically significant. The text acknowledges the t=6h exception but still claims an overall 11.6% NLL improvement and marginally better calibration; the evidence presented does not establish superiority. The authors should either provide significance tests, confidence intervals, or substantially weaken the comparative claims.
  3. [Sections 3.1–3.2 and Appendix A] The evaluation does not support the 'prognostics' and 'generalization' claims. The FEM solution of Eqs. (22)–(25) is used both to define the initial and boundary conditions for training (Fig. 4, §3.1) and as the reference for computing ageing errors (Fig. 9). The 'prediction instants' reported in Table 2 (t=0, 3, 6, 18, 25, 50 h) all lie inside the four-day training window shown in Fig. 3, so the experiments measure interpolation to simulated data, not forecasting to unseen future conditions. The ageing error in Fig. 9 is computed against the same FEM model, not against independent field measurements. The abstract's 'generalization' and 'prognostics' claims therefore need to be substantially qualified, or the evaluation needs a held-out temporal split and an independent validation data source.
minor comments (4)
  1. [Section 2.2, Eq. (11)] Typo: 'equivalenty implemented' should be 'equivalently implemented'. Also, the notation for variance is inconsistent: Eq. (20) uses σ^2_hat u, while Eq. (35) in Appendix B uses σ(x_i)^2; please unify.
  2. [Table 5] The row 'Loss Function - -ELBO' is confusing for the d-PINN model, since the d-PINN loss is defined via Eq. (35), not the ELBO. The table should state the actual loss used for each configuration.
  3. [Figure 10] The markers and error annotations in the sensitivity figure are difficult to read in print. Please enlarge the figure, use distinct markers, and report the numerical values in a table.
  4. [Abstract and Section 1] The abstract says the framework is 'validated with a finite-element thermal model and field measurements from a solar power plant', but the paper directly validates only the FEM model with field measurements (via ref. [17]); the B-PINN output is compared with the FEM simulation, not with the field measurements. Please clarify this distinction to avoid overstating the validation.

Circularity Check

2 steps flagged · score 6.0 of 10

Heteroscedastic residual loss is self-referential and headline 'prediction' metrics are in-sample, so the claimed improvements are not independently established.

  1. self definitional [Section 2.4, Eqs. (20)-(21); residual definition Eq. (6)]
    "NLL_hetero(x, t; θ) = 1/2 log(2π σ^2_û(x,t;θ)) + (u − µ_û(x,t;θ))^2/(2σ^2_û(x,t;θ)) ... where u is the solution or ground truth ... each NLL^(.)_hetero(θ(k)) term is the sum over the corresponding initial condition, boundary condition and residual points of Eq. (20)."

    Residual points have no observed ground-truth u: the object at those points is r(x,t;θ)=L[û]−f. Applying Eq. (20) at residual points therefore forces the residual likelihood to use the network's solution-variance output σ^2_û as the variance of r. For fixed r, minimizing the resulting term with respect to σ^2_û gives σ^2_û = r², so the physics residual can be made nearly costless by inflating the predicted variance. The 'aleatoric variance' that is later reported as the data-noise component is the same variable that controls the physics penalty; the physics constraint and the fitted uncertainty output are defined in terms of each other, not independently anchored.

  2. fitted input called prediction [Section 3.1 training setup and Table 2 prediction instants]
    "Figure 3 shows the available minutely sampled time series ... with a total of 5760 samples over 4 days of operation. ... All B-PINN models are trained for 15000 epochs, with early stopping ... Table 2 reports the probabilistic performance metrics ... across different prediction instants ... Instant [h] 0 ... 50."

    No train/test split is described: the model is trained on the full 4-day domain, and every reported 'prediction instant' (0, 3, 6, 18, 25, 50 h) lies inside that domain. The RMSE, CRPS, and NLL values in Table 2 are therefore in-sample fit errors, not forecasts from a cutoff time. The headline 57.3% RMSE improvement over the vanilla PINN is a comparison of training-fit quality on the same spatiotemporal window, so the claimed predictive advantage is statistically forced by the training configuration rather than demonstrated on held-out future data.

full rationale

The paper is not wholly self-referential: it uses an external FEM solver (Matlab pdepe), operational field measurements, and standard IEC ageing equations, and the law-of-total-variance decomposition in Eq. (19) is a standard identity. However, the central claimed improvement of the heteroscedastic B-PINN has two load-bearing reductions. First, Eq. (21) reuses the heteroscedastic NLL of Eq. (20) for residual points, where no ground-truth u exists; the only coherent instantiation makes the residual variance equal to the network's learned solution variance, so the physics residual can be satisfied by increasing the output variance rather than by reducing r. This makes the 'aleatoric uncertainty' and the residual penalty the same fitted variable, and Algorithm 2 still refers to the earlier homoscedastic objective, so the actual training objective is both self-referential and unverified. Second, the evaluation instants all fall inside the training time window and no temporal holdout is described, so the reported RMSE/CRPS/NLL gains are in-sample fit metrics, not prognostics predictions. The hyperparameter and baseline configurations inherited from the authors' prior work ([17], [30], [15]) add a self-citation flavor, but the main circularity is the self-referential residual likelihood and the in-sample prediction evaluation. Score 6: partial circularity with the central claim reducing by construction to fitted quantities.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claims rest on a standard physical model (heat diffusion + IEC ageing), a standard Bayesian inference setup, and several hand-chosen hyperparameters. No new physical entities are introduced. The most consequential free parameter is the manual loss weighting, especially the tiny residual weight λr=1e-4, which controls how strongly the physics is enforced. The FEM-as-truth assumption is the primary domain assumption that the evaluation depends on.

free parameters (5)
  • Loss weights λ0, λb, λr = λ0=1, λb=1, λr=1e-4
    Manually weighted to balance IC, BC, and PDE residual losses (§3.1); affects relative physics/data influence and final predictive uncertainty.
  • Homoscedastic noise std σ_i, σ_r = 0.01, 0.01
    Fixed Gaussian noise standard deviations for the homoscedastic B-PINN likelihood terms (Eq. 14); chosen by hand rather than learned.
  • d-PINN dropout rate ρ = 0.1
    Tuned via grid search on the same heat-diffusion PDE (Table 5); affects the baseline's uncertainty estimates.
  • Network architecture = 2 layers × 50 neurons
    Taken from prior work [30] without justification for this dataset (§3.1).
  • Optimizer hyperparameters = Adam lr=0.01, batch=16, 15000 epochs
    Reused/manual configuration; early stopping patience 200 (§3.1).
assumptions (7)
  • domain assumption 1D heat diffusion PDE (Eq. 22) with radiative heat transfer and no convection
    Used as the physical law enforced by the PINN; if convection or 2D/3D effects matter, the residual loss is mis-specified.
  • domain assumption Dirichlet boundary conditions (Eq. 25): ΘO(0,t)=ΘA(t), ΘO(H,t)=ΘTO(t)
    Assumes the oil temperature at the tank bottom equals ambient and at the top equals measured top-oil temperature.
  • domain assumption IEC 60076-7 empirical winding/HST model (Eqs. 26-29)
    Coupled to the oil temperature to estimate winding temperature; standard but empirical.
  • domain assumption Ageing acceleration model V=2^((ΘH−98)/6) (Eq. 31)
    Adopted from IEC standard; no derivation or validation in this paper.
  • ad hoc to paper Gaussian likelihood for aleatoric uncertainty (Eqs. 16, 20)
    Assumes the noise is equivalent Gaussian and captured by a network variance output; not derived from the physical measurement process.
  • standard math Gaussian variational posterior and reparameterization trick
    Standard VI approximation for BNNs; used for tractable inference.
  • standard math Law of total variance decomposition (Eq. 19)
    Used to define total predictive uncertainty as epistemic plus aleatoric; mathematically correct under the assumed Gaussian predictive distribution.

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Cite this review

Pith. "Pith review of Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics." pith.science (2026). https://pith.science/paper/FVGGJ2FB

@misc{pith2026260103673,
  author       = {Pith},
  title        = {Pith review of: Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVGGJ2FB}},
  note         = {Machine review of arXiv:2601.03673}
}
read the original 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, and the influence of measurement noise on aleatoric uncertainty. Overall, the findings highlight the capability of Bayesian physics-informed learning to support uncertainty-aware prognostics and informed decision-making in transformer asset management by tracking aleatoric and epistemic sources of uncertainty.

Figures

Figures reproduced from arXiv: 2601.03673 by the authors.

Figure 1
Figure 1. Proposed B-PINN approach for the probabilistic spatiotemporal transformer thermal model, including spa [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Overall framework for probabilistic transformer insulation ageing estimation. The Bayesian PINN-based [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Distribution transformer load (K), ambient temperature (ΘA), and top-oil temperature (ΘT O) over four days of operation with one-minute resolution, operated at a floating photovoltaic substation [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Spatiotemporal transformer oil temperature estimation obtained through a finite element method using the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Spatiotemporal transformer insulation oil temperature estimates at different time instants for heteroscedastic [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Joint probability density of aleatoric (heteroscedastic) and epistemic uncertainty estimated by the B-PINN [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Spatiotemporal temperature estimates at t = 25 hours with the corresponding decomposition into epistemic and aleatoric uncertainty. (a) B-PINN homoscedastic; (b) B-PINN heteroscedastic configuration. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Probabilistic spatiotemporal ageing predictions for B-PINN and d-PINN models with total UQ: (a) mean [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Spatio-temporal temperature prediction errors for (a) B-PINN and (b) d-PINN. The heatmap shows the mean [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Sensitivity analysis of the CRPS metric with respect to the number of boundary condition samples, initial [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: shows the thermal parameters of the transformer of the heat diffusion model, which considers the heat source, q(x, t), and the convective heat transfer, h(ΘO(x, t) − ΘA(t)). Transformer tank 0 0 H q (x,t) O (0,t)= A (t) O (H,t)= TO (t) h (O (x,t)- A k (t)) Δt x …
Figure 12
Figure 12. Figure 12: Use of dropout at inference stage with MC sampling. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Bayesian-PINN approach with epistemic and homoscedastic uncertainty quantification for the probabilistic [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

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Reviewed August 3, 2026 · model on record in the stance chip above.