Explainable Artificial Intelligence for Thermal Condition Assessment of High-Voltage Power Equipment Using Infrared Imaging

Summary

This study presents a physics-constrained CNN framework for infrared thermographic assessment of high-voltage power equipment, integrating diffusion-based regularization and Monte Carlo dropout for physically consistent temperature prediction and uncertainty-aware thermal monitoring.

Abstract

Infrared thermography is widely used for non-contact thermal monitoring of high-voltage power equipment, where abnormal temperature patterns may indicate developing faults or insulation degradation. However, purely data-driven deep learning models may produce temperature predictions that are not fully consistent with heat-transfer physics. This study investigates a physics-constrained convolutional neural network (CNN) framework for estimating spatial temperature fields from thermographic images. A diffusion-based Laplacian residual loss derived from the heat equation was incorporated to improve physical consistency in the predicted thermal fields. Experimental evaluation on the available dataset showed that the physics-constrained model achieved improved performance compared with the baseline CNN, with the best configuration obtaining an RMSE of 12.013 °C and a CAP R-squared value of 0.4646, indicating moderate predictive capability. A learnable thermal diffusivity parameter was also explored to improve interpretability, although it did not outperform the fixed-parameter formulation. In addition, a source-term-augmented model was evaluated, but it did not provide any further improvement for the snapshot-based thermographic data. Monte Carlo dropout was applied for uncertainty estimation, revealing higher predictive variance near hotspot boundaries and regions with steep thermal gradients. Overall, the findings suggest that physics-based regularization and uncertainty estimation can improve the physical coherence and interpretability of thermographic prediction models, although the results remain limited to the current dataset and validation setting.

Introduction

Reliable thermal monitoring is essential for the safe operation of high-voltage power equipment, because abnormal heating is often an early indicator of insulation degradation, loose connections, or incipient electrical faults. Infrared thermography (IRT) is widely used in this context because it provides non-contact, full-field surface-temperature measurements during routine inspections. However, practical thermographic analysis remains difficult because of emissivity variation, ambient interference, sensor noise, and limited labeled data¹.

Deep learning has improved automatic hotspot detection and temperature-field prediction from thermographic images, but purely data-driven models do not explicitly enforce heat-transfer physics. As a result, predicted thermal fields may be accurate in a statistical sense while remaining only weakly constrained physically. For safety-critical inspection tasks, this limitation reduces confidence in model outputs and weakens their value for maintenance decision-making². A physics-based prior can address this issue by guiding learning toward thermodynamically consistent temperature distributions. For snapshot infrared data, the temporal term in the heat equation is unavailable; therefore, a Laplacian residual serves as a practical regularizer to enforce diffusion-like spatial smoothness. This is particularly relevant for high-voltage equipment, where temperature fields are generally smooth except in localized hotspot regions.

To reconcile data-driven flexibility with mechanistic consistency, physics-informed learning embeds governing partial differential equations (PDEs) or their discrete analogs into training objectives. The physics-informed neural network (PINN) paradigm, where PDE residuals are penalized alongside data loss, has shown that incorporating differential operators as soft constraints improves generalization and can enable simultaneous forward/inverse inference of latent parameters. For the thermal problem, the canonical model is the diffusion (heat) equation,

where T is temperature, α is thermal diffusivity and S a volumetric/source term; enforcing residuals of this operator (or Laplacian-based surrogates) reduces spurious, non-diffusive structure in learned temperature fields³. Uncertainty estimation is also important for operational deployment. A deterministic prediction does not indicate where the model is less reliable, especially near hotspot boundaries and steep thermal gradients⁴. Monte Carlo dropout provides an efficient approximation of epistemic uncertainty and can generate pixel-wise variance maps that support risk-aware interpretation of predicted thermal fields^5,6.

The objective of this study is to develop and evaluate a compact physics-constrained CNN framework for spatial temperature estimation from infrared images of high-voltage power equipment. The proposed model integrates Laplacian-based physics regularization with a baseline CNN, examines adaptive thermal diffusivity as an interpretable calibration mechanism, evaluates a source-term augmented formulation, and applies Monte Carlo dropout for uncertainty mapping. The study is designed to determine whether physics-based regularization improves prediction consistency for snapshot thermographic data and whether uncertainty estimates enhance the practical interpretability of the results.

The main contributions are threefold: first, a diffusion-constrained CNN for temperature-field prediction using a Laplacian residual loss; second, an empirical assessment of learnable thermal diffusivity and source-term augmentation under snapshot-based thermographic conditions; and third, pixel-wise uncertainty estimation using Monte Carlo dropout to identify thermally critical and less certain regions in high-voltage equipment monitoring. According to previous literature, Physics-informed neural networks (PINNs) integrate governing partial differential equations (PDEs) into deep learning, enabling thermal models to respect conservation laws and material physics^7,8. For example, studies demonstrate effective approximation of temperature fields by enforcing boundary and initial conditions through soft PDE-residual losses. Embedding the Laplacian or related differential operators into the loss reduces spurious, physically inconsistent fluctuations in the predicted thermal field. However, standard PINNs often falter when solutions exhibit sharp gradients (e.g., near heat sources or material interfaces). Recent enhancements address this by directly targeting high-gradient regions; for instance, Gradient-Driven PINNs incorporate gradient-based loss terms to better capture sharp thermal features. In summary, PINNs for heat transfer offer data-efficient modeling by embedding conduction physics, but require tailored loss designs (e.g., gradient penalties) when dealing with steep temperature variations.

Infrared thermography (IRT) provides full-field thermal data for electrical equipment in operation, making it ideal for non-invasive fault diagnosis. Convolutional neural networks (CNNs) have been widely used to extract patterns from thermal images for asset health monitoring. For instance, Ullah et al. demonstrated that pretrained CNN features (AlexNet) combined with a random forest classifier can distinguish defective vs. healthy high-voltage equipment with over 96% accuracy⁹. Similarly, transfer-learning with deeper networks (e.g., VGG-16) has achieved nearly perfect hotspot detection on substation thermograms, yielding ≈99.98% classification accuracy¹⁰. Beyond classification, modern object-detection CNNs (Faster R-CNN, YOLO, Mask R-CNN, etc.) have been adapted to localize thermal anomalies in power systems; reported accuracies and mean average precision often exceed 90% on test datasets. In practice, deep IRT models significantly reduce the need for expert interpretation: they can automatically identify heated components and prioritize maintenance actions. Nonetheless, most works to date focus on static image classification or segmentation; integration with temporal or operational context remains an open direction for improving reliability in real-world inspections.

Embedding physics in neural networks also opens the door to inverse modeling: PDE parameters (such as thermal diffusivity or conductivity) can be treated as trainable variables and inferred from data. Recent studies in thermal PINNs exploit this for parameter inversion. For example, coupled-network PINN architectures have been proposed to simultaneously predict temperature fields and recover unknown conduction parameters¹¹. A 2021 study showed that PINNs can invert constant or spatially varying heat conduction coefficients with high accuracy, effectively learning material properties from temperature measurements¹¹. In a related approach, Waseem and Mielle developed an iterative PINN framework that alternates between solving the forward heat equation and optimizing the thermal conductivity of a wall based on observed thermographs¹². Their method successfully estimated in-situ thermal conductivity under varying environmental conditions without prolonged experiments. At the same time, hybrid PINN schemes combine limited sensor data and physics priors to tackle ill-posed inverse problems. A 2026 study introduced a hybrid PINN for 2D transient conduction that identifies an orthotropic conductivity tensor and internal Gaussian heat sources from sparse temperature readings¹³. This framework simultaneously estimates anisotropic material properties and source characteristics (location, amplitude, spread), yielding low errors even with minimal instrumentation. These works collectively demonstrate that incorporating learnable PDE parameters into PINNs or coupling forward/inverse networks enables recovery of latent thermal properties from infrared or sparse thermal data. Such hybrid inverse models offer interpretability (by linking predictions to physical parameters) and are crucial for diagnostics in energy systems, although they require careful balancing of data and physics to ensure robust inference.

Quantifying model uncertainty is critical in safety-sensitive thermal monitoring. A popular approach is Monte Carlo (MC) dropout, which approximates Bayesian inference by applying dropout layers at inference and sampling multiple forward passes. This yields pixel-wise predictive variance maps, effectively highlighting regions where the model is less confident. For example, a Bayesian U-Net employing MC dropout produced informative uncertainty estimates in medical image segmentation: uncertain pixels tended to lie at object boundaries or in atypical regions, and filtering them improved overall accuracy¹⁴. In thermographic monitoring, similar behavior is observed: regions of high predictive variance often coincide with hotspots or sharp thermal gradients, signaling areas requiring caution. Incorporating MC dropout into CNN-based thermal models, therefore, enables risk-aware assessments: operators can visualize uncertainty overlays on temperature predictions, focusing manual inspection or conservative control actions on ambiguous areas¹⁴. Overall, Monte Carlo dropout and related Bayesian techniques have become standard tools for uncertainty estimation in CNN models, improving robustness and interpretability by revealing where the data or model knowledge is insufficient¹⁴.

Protocol

This section describes the complete physics-constrained deep learning pipeline for thermal condition assessment using infrared thermographic imagery. The framework integrates (i) temperature field extraction, (ii) baseline CNN regression, (iii) diffusion-based physics regularization, (iv) adaptive parameter learning, and (v) uncertainty quantification. All the tools used in this study are described in the Table of Materials.

Thermal image acquisition and temperature field construction
Infrared (IR) images of high-voltage power equipment were acquired using a calibrated thermographic camera. Each image was converted into a spatial temperature matrix T(x, y) ∈ ℝ^H×W by mapping pixel intensities to physical temperature values (°C) using camera radiometric calibration parameters. Figure 1A–C shows temperature visualization. The original IR image is converted into a physically meaningful temperature map. The histogram shows a skewed temperature distribution with a concentration between 50–90 °C. Localized high-temperature zones (hotspots) are clearly visible. Figure 2 shows a 3D thermal surface elevation representation of the temperature field. The 3D surface elevation plot reveals steep gradients around hotspot regions. Temperature peaks approach ~120 °C. The thermal field is spatially smooth, except near energized conductors and connection joints. These characteristics justify the use of a diffusion-based regularization term to enforce physical smoothness while preserving localized high-gradient phenomena.

Figure 1: Temperature Visualization. (A) Original infrared (IR) thermographic image of the inspected high-voltage equipment showing the observed thermal pattern. (B) Extracted temperature map obtained from the thermographic image, illustrating the spatial distribution of surface temperatures and associated thermal gradients. (C) Temperature distribution histogram showing the frequency distribution of temperature values within the extracted temperature field. This figure demonstrates the conversion of a thermographic image into a quantitative temperature representation and summarizes the underlying temperature distribution used for subsequent model development and analysis. Abbreviations; IR = infrared. Please click here to view a larger version of this figure.

Figure 2: 3D thermal surface elevation representation of the temperature field. Three-dimensional visualization of the temperature distribution, where surface elevation corresponds to temperature magnitude and highlights localized thermal hotspots. Please click here to view a larger version of this figure.

Problem formulation

Let

 

denote the RGB infrared thermographic image, where H and W represent the spatial dimensions and C = 3 corresponds to the color channels associated with the rainbow thermal palette. Although the infrared image visually represents thermal information, it is not a direct numerical temperature matrix because the displayed colors depend on palette mapping, scaling, interpolation, and imaging conditions.

Let

 

denote the corresponding ground-truth temperature field obtained through radiometric calibration and temperature extraction from the infrared imaging system.

The learning objective is to approximate the mapping

where θ denotes the trainable parameters of the CNN and T̂ represents the predicted temperature field. The purpose of this supervised mapping is not simply to reproduce direct radiometric conversion, but to learn a spatially consistent temperature regression model from rendered thermographic images. Unlike direct pixel-wise color-to-temperature extraction, the CNN can exploit contextual thermal structure across neighbouring regions, reduce noise sensitivity, preserve hotspot characteristics, and generate smoother physically plausible temperature fields under the imposed diffusion constraints. The baseline objective function minimizes the pixel-wise regression error between the predicted and calibrated temperature fields:

where T_ij and T̂_ij denote the ground-truth and predicted temperatures at pixel location (i, j), respectively. Accordingly, the framework should be interpreted as a physics-regularized thermographic regression model rather than a replacement for direct radiometric calibration.

Infrared image acquisition conditions
Infrared thermographic images were acquired using a FLIR thermal imaging camera under outdoor substation operating conditions. The dataset consisted of RGB thermographic images with a spatial resolution of 640 × 480 pixels, recorded using the FLIR rainbow color palette. Each image included an embedded thermal scale bar indicating the temperature range associated with the acquisition frame. The thermographic inspection covered multiple categories of high-voltage equipment, including transformers, bushings, insulators, connectors, and terminal structures. Images were captured during normal operating conditions to include both standard thermal behavior and localized hotspot patterns relevant to condition monitoring applications. Image acquisition was performed from practical inspection viewpoints typically used in field thermography. The camera orientation and viewing angle varied depending on equipment accessibility and substation layout. Thermal images were visually screened to ensure that the target equipment remained clearly observable and that the hotspot regions were distinguishable from the background. The dataset was organized into labeled folders according to equipment category and thermal condition. Prior to model development, the images were segmented and resized into a consistent spatial format for temperature-field extraction and CNN training. The thermographic images contain thermal distributions suitable for computer vision and deep learning studies involving hotspot detection, fault identification, anomaly localization, and temperature-field prediction. In the present work, the thermal profiles extracted from the infrared images were used for physics-constrained temperature estimation and uncertainty-aware thermal assessment.

Thermographic dataset and data partitioning
The dataset used in this study consisted of 174 infrared thermographic images acquired from high-voltage substation equipment using a FLIR thermal imaging system¹⁵. The dataset included multiple types of power-system components, including transformers, bushings, insulators, connectors, and high-voltage terminal structures operating under practical outdoor conditions. Each thermographic image was converted into a spatial temperature field using the radiometric information provided by the infrared camera. The dataset contains thermal variations ranging from normal operating conditions to localized hotspot regions associated with elevated thermal stress. The data were divided into training, validation, and test subsets with a 70:15:15 split, yielding 121, 26, and 27 images, respectively. The partitioning was performed at the image level because the dataset consisted of independently captured thermographic scenes collected under varying operating conditions and viewpoints. Care was taken to avoid near-duplicate frames across subsets in order to reduce data leakage risk and improve evaluation fairness. All images were resized to a uniform spatial resolution prior to training. Temperature normalization was applied to stabilize optimization while preserving relative thermal gradients and hotspot characteristics important for condition assessment.

Baseline CNN architecture
A fully convolutional encoder–decoder regression architecture was adopted to preserve spatial resolution during temperature-field prediction from infrared thermographic images. All input infrared images were resized to a uniform resolution of 256 × 256 × 3 prior to training. The encoder section consisted of four convolutional blocks with progressively increasing feature depth. Each block contained a 3 × 3 convolution layer followed by batch normalization and ReLU activation. Spatial downsampling was performed using strided convolutions rather than max pooling to better preserve localized thermal structures and hotspot information. The encoder channel progression was:

32 → 64 → 128 → 256

The decoder section reconstructed the spatial temperature field using transposed convolution layers for upsampling with the reverse feature progression:

256 → 128 → 64 → 32

Each decoder block similarly employed transposed convolution, batch normalization, and ReLU activation. A final 1 × 1 convolution layer generated the predicted single-channel temperature map:
T̂ ∈ ℝ^256×256

The network therefore outputs a full-resolution spatial temperature field corresponding to the infrared input image. To support uncertainty quantification, dropout layers with probability p = 0.2 were inserted near the bottleneck and decoder stages. During inference, dropout remained active to enable Monte Carlo sampling for predictive uncertainty estimation. All convolutional weights were initialized using He-normal initialization, while bias terms were initialized to zero. The model was trained using the Adam optimizer with an initial learning rate of 1 × 10⁻³, batch size of 8, and weight decay of 1 × 10⁻⁵. Training was performed for 100 epochs using cosine learning-rate decay and early stopping based on validation RMSE.

The baseline objective function is defined as:

     

where L_data denotes the pixel-wise mean squared error between predicted and ground-truth temperature fields.
Performance was evaluated using: R², RMSE, and MAE.

Although the baseline CNN successfully captured the overall thermal structure and hotspot distribution, it occasionally produced non-physical artifacts in low-gradient regions and slight oversmoothing near sharp thermal boundaries. These limitations motivated the introduction of diffusion-based physics regularization in the subsequent model formulation.

Physics-constrained learning via diffusion
Governing heat equation
To enforce thermal consistency in the predicted temperature field, a diffusion-based physics prior was incorporated into the heat equation. Under quasi-steady conditions, the surface temperature approximately satisfies

   

where T denotes temperature, α is thermal diffusivity, and S represents local heat generation. Since the available thermographic samples are single inspection snapshots rather than time-resolved sequences, the temporal derivative cannot be directly evaluated. Accordingly, the model assumes local quasi-steady conduction and enforces

     

away from active heating regions.

The 2D Laplacian was discretized using the standard finite-difference stencil:
      

The physics residual was then computed over the predicted temperature map as

     

To avoid ambiguity at the image boundaries, the Laplacian residual was evaluated only on valid interior pixels, where all neighboring values required by the stencil are available. No explicit masking was applied to hotspot regions, since the objective of the physics term was to regularize the full spatial field rather than suppress thermally active zones. The hotspot structure was therefore learned through the data term, while the Laplacian penalty encouraged diffusion-consistent smoothness across the predicted map.

The final objective function combined the data loss and the physics loss as 

     

where λ_phys controls the strength of the physics regularization. In the ablation study, λ_phys ∈ {0, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹} was examined to assess the effect of increasing physical constraint strength. Moderate values improved generalization, whereas overly strong weighting tended to oversmooth sharp hotspot gradients. This behavior is consistent with the thermal structures observed in the infrared images, where physically smooth regions coexist with localized high-temperature concentrations.

It should be emphasized that the steady-state diffusion formulation used in this work is not intended to represent the complete heat-transfer mechanism governing hotspot formation in high-voltage equipment. In practice, localized overheating arises from internal heat-generation processes such as Joule heating, contact resistance, and insulation degradation. Because the available dataset consists of snapshot thermographic images without temporal information or source measurements, the diffusion constraint is employed as a physics-inspired spatial regularization term. Its role is to encourage locally smooth, physically plausible temperature fields while allowing the hotspot characteristics to be learned from the data. Therefore, the diffusion prior should be interpreted as a spatial smoothness constraint rather than a comprehensive model of the underlying thermal physics.

Learnable thermal diffusivity and source-term augmentation
To improve physical interpretability, the effective thermal diffusivity was treated as a trainable scalar parameter. Let T̂(x, y) denote the predicted temperature field and let α > 0 denote the learnable thermal diffusivity. To enforce positivity, α was parameterized through an unconstrained variable β as

   

In this formulation, the physics prior was written in terms of an effective residual forcing field R(x, y), where R(x, y) represents the local departure from pure diffusion induced by unmodeled heating effects. The adaptive-diffusivity relation is therefore expressed as

     

Here, ∇²T̂(x, y) is the Laplacian of the predicted temperature field, and R(x, y) is the effective residual source term inferred implicitly from the thermographic data. Because the available samples are snapshot images rather than time-resolved thermal sequences, R(x, y) is not separately supervised and serves only as a latent physics quantity during optimization. The corresponding physics loss was defined as

   

For the source-term augmented experiment, an auxiliary CNN head was added to estimate a spatial source map Ŝ(x, y). In this case, Ŝ(x, y) represents the predicted local heat-generation field, and the governing relation becomes

   

The source head was attached to the encoder bottleneck and consisted of a small convolutional branch that mapped shared latent features to a single-channel output map Ŝ(x, y). No ground-truth source labels were available; therefore, Ŝ(x, y) was learned only through the joint objective

   

where L_data is the temperature regression loss, λ_phys controls the diffusion constraint, and λ_s regularizes the source prediction to discourage trivial high-magnitude source maps. In the snapshot-based setting considered here, the source-term formulation remained underdetermined and did not improve prediction accuracy.

Monte Carlo dropout for uncertainty quantification
To estimate epistemic uncertainty in the predicted temperature field, Monte Carlo dropout was applied during inference. Dropout layers with a rate of 0.2 were placed after the bottleneck block and within the decoder stages of the CNN. During testing, these dropout layers were kept active, and 30 stochastic forward passes were performed for each infrared image. For the k-th stochastic pass, the predicted temperature field is denoted by

   

where X is the input infrared image and θ_k represents the sampled network parameters induced by dropout. The predictive mean was computed as

   

and the pixel-wise predictive variance was obtained as

   

with K = 30. The resulting variance map was used to identify spatial regions with lower predictive confidence. Because ground-truth uncertainty labels are not available for thermographic regression, uncertainty was evaluated using pixel-wise variance maps and summary statistics computed over hotspot and background regions, rather than calibration-based metrics. The quantitative analysis showed that the mean predictive standard deviation was 13.046 °C, while the hotspot region exhibited a higher standard deviation of 15.934 °C compared with 12.743 °C in the background region. This corresponds to an approximately 25% increase in uncertainty within hotspot areas relative to the background. These results indicate that the model expresses greater uncertainty near thermally complex regions, particularly around hotspot boundaries and steep temperature gradients, which is consistent with the structure observed in the thermal surface visualizations. The workflow is summarized in Figures 3 and 4 shows the overall CNN-based processing pipeline.

Figure 3: Workflow. Overview of the proposed methodology, including thermographic image acquisition, preprocessing, temperature-field extraction, CNN training, physics-constrained learning, and uncertainty quantification. Abbreviations; CNN = convolutional neural network. Please click here to view a larger version of this figure.

Figure 4: Overall CNN-based processing pipeline. Architecture of the proposed CNN framework showing the transformation of infrared thermographic images into spatial temperature-field predictions. Abbreviations; CNN = convolutional neural network. Please click here to view a larger version of this figure.

Results

This section evaluates the proposed physics-constrained deep learning framework using quantitative metrics and visual analysis. The results demonstrate how integrating diffusion-based physical constraints and uncertainty modeling influences predictive accuracy and thermal interpretability.

Performance comparison of model configurations
Table 1 shows the performance comparison of different model configurations. The baseline CNN achieved an RMSE of 12.345 °C (Figure 5A) and an R² value of 0.4346 (Figure 5B), demonstrating that purely data-driven approaches can capture the general structure of thermal fields but lack strong physical consistency. When weak physics regularization was introduced (λ = 0.01), performance slightly degraded (RMSE = 12.445 °C). This indicates that an insufficient physics weight does not effectively guide the network and may introduce conflicting gradients during optimization. Increasing the physics constraint to λ = 0.1 provided marginal improvements in stability but did not significantly alter predictive performance. However, when the constraint strength was increased to λ = 1, the model achieved its best performance with RMSE = 12.013 °C and R² = 0.4646. The trend is illustrated in Figure 5A–B, the ablation analysis, where increasing the physics weight reduces RMSE and improves R² values until reaching an optimal point. This behavior suggests that diffusion-based regularization improves spatial smoothness, prevents non-physical temperature artifacts, and still allows the model to preserve sharp thermal gradients.

Model Configuration	Physics Constraint (λ)	Thermal Diffusivity (α)	RMSE (°C)	R²	Remarks
Baseline CNN	0	—	12.345	0.4346	Pure data-driven model
Physics CNN	0.01	Fixed (0.1)	12.445	0.4255	Weak physics influence
Physics CNN	0.1	Fixed (0.1)	12.432	0.4266	Moderate physics constraint
Physics CNN	1	Fixed (0.1)	12.013	0.4646	Best performing model
Physics CNN	1	Learnable α	12.309	0.438	Learned α ≈ 0.711
Physics CNN + Source Term	1	Learnable α	12.453	0.4247	Source term not identifiable

Table 1: Performance comparison of different model configurations. Comparison of RMSE and R² values for the baseline CNN, physics-constrained CNN variants, learnable diffusivity model, and source-term augmented model. Abbrevitaions; RMSE = root mean square error; CNN = convolutional neural network.

Figure 5: Effect of physics constraint weight on model performance. (A) Variation of root mean square error (RMSE) with different physics regularization weights (λphys), illustrating the impact of physics-based constraints on prediction error. (B) Variation of the coefficient of determination (R²) with different physics regularization weights (λphys), demonstrating the influence of physics regularization on model explanatory performance. The results indicate that increasing the physics constraint weight improves predictive performance, with λphys = 1.0 achieving the lowest RMSE and highest R² among the evaluated configurations. Abbreviations; RMSE = root mean square error; R² = coefficient of determination. Please click here to view a larger version of this figure.

Ablation study
An ablation study was performed to assess the contribution of the proposed framework components. Starting from the baseline CNN, physics-based regularization was introduced through different values of the physics weight, followed by experiments with a learnable thermal diffusivity parameter and a source-term augmented formulation. As shown in Table 1, weak physics regularization provided limited improvement, whereas stronger regularization yielded the best performance. The fixed-diffusivity physics-constrained model achieved the lowest RMSE (12.013 °C) and highest (R^2) value (0.4646), demonstrating the benefit of diffusion-based regularization. Although the learnable diffusivity parameter converged to a physically meaningful value (Learned α ≈ 0.711), it did not improve predictive accuracy. Similarly, the source-term formulation showed no performance gain, likely due to the snapshot-based nature of the dataset. Overall, the results indicate that diffusion-based physics regularization is the primary contributor to performance improvement, while the additional components mainly enhance physical interpretability.

Visual evaluation of temperature prediction
A qualitative comparison between the ground truth temperature map and the CNN prediction is shown in Figure 6. The qualitative prediction figures correspond to the baseline CNN without physics regularization; therefore, the residual map shown in Figure 6A–C, with R² = 0.409, reflects the data-driven reference model, whereas the improved performance reported in Table 1 (R² = 0.4646) corresponds to the physics-constrained configuration. The figure contains three components: Ground truth thermal image (Figure 6A), Predicted temperature field (Figure 6B), and residual error map (Figure 6C). The predicted temperature field successfully reconstructs the global thermal structure of the equipment, including the major hotspot region. The residual map shows localized prediction errors, particularly near high-temperature boundaries where gradients are steep. The visual comparison confirms that the model preserves large-scale thermal features while maintaining acceptable error levels across most spatial regions.

Figure 6: Qualitative prediction results for the baseline CNN. (A) Ground-truth temperature field derived from the thermographic image, representing the reference temperature distribution used for model evaluation. (B) Temperature field predicted by the baseline CNN, illustrating the model's ability to reconstruct the spatial thermal pattern. (C) Residual error map showing the pixel-wise difference between the predicted and ground-truth temperature fields, with positive and negative values indicating overestimation and underestimation, respectively. The coefficient of determination (R²) for this representative prediction is 0.409. The figure shows that the baseline CNN captures the overall thermal structure of the equipment but exhibits localized prediction errors, particularly in regions with sharp temperature gradients and complex thermal features. Abbreviations; CNN = convolutional neural network; R² = coefficient of determination. Please click here to view a larger version of this figure.

Effect of learnable thermal diffusivity
To enhance physical interpretability, the thermal diffusivity parameter α was treated as a learnable parameter during training. The learned diffusivity converged to approximately α ≈ 0.711, indicating that the model adapted the effective conduction coefficient to best fit the observed thermal distribution. The performance comparison between baseline CNN, fixed-parameter physics model, and learnable diffusivity configuration is illustrated in the model comparison plot (Figure 7A–C). Although the learnable parameter improved physical interpretability, it did not outperform the fixed diffusivity configuration. This result likely arises because the dataset consists of snapshot thermographic images without temporal evolution, making it difficult to reliably infer conduction parameters solely from spatial information.

Figure 7: Performance comparison of baseline, fixed diffusivity, and learnable diffusivity models. (A) Comparison of root mean square error (RMSE) values for the baseline convolutional neural network (CNN), fixed-diffusivity physics-constrained model, and learnable-diffusivity model. Lower RMSE values indicate improved temperature-field prediction accuracy. (B) Comparison of the coefficient of determination (R²) for the three model configurations, illustrating the effect of physics-guided learning on predictive performance. Higher R² values indicate better agreement between predicted and reference temperature fields. (C) Comparison of the fixed thermal diffusivity value and the diffusivity parameter learned during model optimization, demonstrating the model’s ability to estimate physically meaningful thermal properties from data. The results show that the fixed-diffusivity physics-constrained model achieved the best overall predictive performance, whereas the learnable-diffusivity formulation successfully identified a higher diffusivity value but did not yield additional accuracy improvements. Abbreviations; CNN = convolutional neural network; RMSE = root mean square error; R² = coefficient of determination; α = thermal diffusivity. Please click here to view a larger version of this figure.

Evaluation of source-term augmentation
A source-term formulation was explored to capture localized heat generation processes within the thermal field. The predicted heat source map is shown in Figure 8A–C. The estimated source distribution appears spatially diffuse and does not clearly correspond to the observed hotspot regions. This indicates that the model was unable to uniquely identify heat sources from static temperature fields. This limitation is expected because solving the inverse heat conduction problem typically requires temporal temperature measurements or boundary condition information. Consequently, the source-term-augmented model did not improve predictive performance and slightly decreased the R² metric.

Figure 8: Evaluation of source-term augmented physics model. (A) Ground-truth temperature field derived from the thermographic image, serving as the reference distribution for model evaluation. (B) Temperature field predicted by the source-term augmented physics-constrained model, illustrating the reconstructed spatial thermal pattern. (C) Predicted heat-source distribution (S(x,y)), representing the spatially varying source term learned by the model during optimization. Positive and negative values indicate regions of relative heat generation and heat absorption within the learned source representation. The figure illustrates the ability of the source-term augmented formulation to jointly estimate temperature distributions and an auxiliary heat-source field. Although the learned source term provides additional physical interpretability, it did not yield substantial improvements in predictive accuracy compared with the diffusion-based physics-constrained model. Abbreviations; (S(x,y)) = spatial heat-source term. Please click here to view a larger version of this figure.

Spatial thermal profile validation
To further evaluate spatial accuracy, cross-sectional and radial thermal profiles were analyzed. Figure 9 presents three analyses: Cross-sectional temperature profile (Figure 9A), Radial thermal decay from the hotspot center (Figure 9B), and Residual error distribution (Figure 9C). The predicted temperature profile closely follows the ground truth along most spatial locations, particularly in low-gradient regions. Some deviations occur near the hotspot boundaries where steep thermal gradients are present. The residual distribution is roughly symmetric, centered around zero, indicating that the model does not exhibit systematic bias in temperature prediction.

Figure 9: Spatial temperature profile and residual error distribution. (A) Cross-sectional temperature profile comparing the ground-truth and predicted temperature values along a representative spatial line within the temperature field. (B) Radial thermal decay analysis showing the variation of temperature with radial distance from the hotspot region for both the ground-truth and predicted temperature distributions. (C) Distribution of residual prediction errors, illustrating the frequency and magnitude of deviations between predicted and reference temperatures across the evaluated spatial domain. The figure provides a detailed assessment of the model's ability to reproduce spatial temperature variations and thermal decay characteristics. While the predicted temperature profiles generally follow the overall trends of the reference data, residual errors remain present in regions exhibiting sharp thermal gradients and complex temperature patterns. These results help characterize the spatial accuracy and limitations of the proposed temperature-field prediction framework. Please click here to view a larger version of this figure.

Uncertainty quantification analysis
Table 2 shows uncertainty quantification results (Monte Carlo Dropout). Monte Carlo dropout was employed to estimate predictive uncertainty by performing 30 stochastic forward passes at inference time. The resulting predictive mean (Figure 10A), pixel-wise standard deviation map (Figure 10B), and high-uncertainty contours (Figure 10C) are presented in Figure 10. Because ground-truth uncertainty labels are unavailable for thermographic regression, uncertainty was evaluated using spatial variance patterns and region-wise summary statistics. For quantitative reporting, the hotspot region was defined as the set of pixels exhibiting the highest thermal intensities within the equipment region of interest, while the background region corresponded to the remaining non-hotspot pixels, excluding image boundaries. This separation allows uncertainty to be compared between thermally active and thermally stable areas in a reproducible manner. Table 2 reports the mean predictive standard deviation over the full field, the hotspot region, and the background region.

The results show that uncertainty is higher in hotspot areas than in the background, with the largest variance observed near hotspot boundaries and other steep thermal gradients. This behavior is physically consistent, as regions with rapid temperature changes are more sensitive to noise, emissivity variations, and local structural complexity. Accordingly, the uncertainty contours provide a useful visual cue for risk-aware inspection by highlighting areas that may require additional manual verification. The computational workflow was implemented in Python using PyTorch for CNN training, physics-constrained loss computation, and Monte Carlo dropout-based uncertainty estimation, as shown in the Table of Materials. Thermographic image preprocessing and visualization were performed using OpenCV, NumPy, and Matplotlib. Model evaluation was conducted with scikit-learn metrics. The infrared images were acquired with a FLIR thermal camera during field inspections. The training and inference pipeline was executed on a GPU-enabled workstation.

Metric	Value
Mean Predictive Standard Deviation	13.046 °C
Hotspot Region Std	15.934 °C
Background Region Std	12.743 °C
Number of MC Samples	30
Uncertainty Behaviour	Higher uncertainty around hotspot edges

Table 2: Uncertainty quantification results (Monte Carlo Dropout). Summary of predictive uncertainty statistics obtained from 30 stochastic forward passes, including mean, hotspot, and background standard deviation values. Abbreviations; MC = Monte Carlo; Std = standard deviation.

Figure 10: Uncertainty quantification results (Monte Carlo Dropout). (A) Predictive mean temperature field obtained from multiple stochastic forward passes of the physics-constrained model, representing the expected temperature distribution. (B) Predictive standard deviation (Std) map showing the spatial distribution of model uncertainty, where higher values indicate lower prediction confidence. (C) High-uncertainty contour overlay highlighting regions exhibiting elevated predictive variance and increased model uncertainty. The figure illustrates the uncertainty estimates generated using Monte Carlo dropout. Higher uncertainty values are primarily observed in regions with complex thermal patterns and sharp temperature gradients, indicating areas where model predictions should be interpreted with greater caution. These uncertainty maps provide complementary information that can assist in identifying regions requiring additional inspection or verification. Abbreviations; MC = Monte Carlo; Std = standard deviation. Please click here to view a larger version of this figure.

DATA AVAILABILITY:
The infrared thermographic images, processed temperature maps, trained model outputs, and Python-based implementation scripts supporting the findings of this study are publicly available at: https://github.com/KallaKanakaRaju/HV-Equipment-Thermography-CNN.git

Discussion

The present study investigated the integration of physics-based regularization and uncertainty-aware deep learning for thermal condition assessment of high-voltage power equipment using infrared thermographic imagery. The results show that embedding physical priors into CNN-based temperature prediction improves the robustness and physical consistency of the learned thermal fields relative to purely data-driven approaches. One of the most important findings is the role of the diffusion-based physics constraint in shaping model performance. The baseline CNN achieved reasonable predictive capability, confirming that convolutional architectures can learn the dominant spatial structure of thermographic patterns. However, the predicted temperature fields occasionally exhibited localized inconsistencies and instability near steep thermal gradients. Introducing the Laplacian residual constraint improved spatial smoothness and reduced physically implausible fluctuations. The ablation analysis further showed that the effectiveness of the physics prior depends strongly on the weighting coefficient λ_phys. Weak regularization (λ = 0.01) had minimal impact, whereas stronger regularization (λ = 1) provided the best balance between physical consistency and predictive flexibility.

This improvement suggests that diffusion-based regularization functions as an effective inductive bias for thermal learning problems^3,7,8. Since heat conduction is governed by spatial continuity, enforcing a Laplacian prior guides the network toward thermodynamically plausible solutions. In this study, the physics constraint was particularly effective in reducing instability across smooth thermal regions while still preserving localized hotspots, which is essential for reliable infrared-based inspection of power equipment. The learnable thermal diffusivity experiment provided further insight into the trade-off between physical interpretability and predictive accuracy. The adaptive diffusivity parameter converged to a physically meaningful value (α ≈ 0.71), indicating that the model could implicitly estimate effective thermal conduction behavior from image data. Nevertheless, the learnable-parameter configuration did not outperform the fixed-diffusivity model. This is likely due to the snapshot nature of the dataset, which contains independent thermographic frames rather than temporally evolving thermal sequences. Without temporal dynamics, inverse estimation of conduction parameters remains underconstrained^11,12,13, limiting the model’s ability to uniquely infer thermophysical properties.

A similar limitation was observed in the source-term augmented formulation. Although the additional source component was intended to capture localized heat generation mechanisms, the predicted source maps lacked spatial clarity and did not correspond strongly to the observed hotspot regions. Inverse heat conduction problems are inherently ill-posed when only static temperature measurements are available^11,12,13,14, and accurate source reconstruction generally requires transient thermal observations or explicit boundary-condition information. Consequently, the source-term model introduced additional optimization complexity without improving predictive accuracy. The uncertainty quantification results provide an important practical contribution for real-world thermal monitoring. Monte Carlo dropout enabled pixel-wise predictive variance estimation^5,6,14 and revealed that uncertainty was consistently elevated near hotspot boundaries and high-gradient regions. This behavior is physically intuitive because sharp thermal transitions are harder to model accurately and are often more sensitive to noise, emissivity variation, and local environmental effects. The resulting uncertainty maps, therefore, add an operationally meaningful layer of interpretability beyond deterministic temperature prediction. From an application standpoint, jointly estimating temperature and predictive confidence is valuable for condition-based maintenance of power equipment. Conventional thermographic inspection often depends heavily on expert judgment, particularly when distinguishing normal thermal variation from developing faults. By highlighting regions with elevated uncertainty, the proposed framework can support manual inspection prioritization and reduce the risk of overconfident automated decisions in safety-critical environments.

Several limitations should also be acknowledged. First, the dataset consisted of static thermographic frames, which restricted the model’s ability to capture transient heat propagation behavior. Second, the moderate R² values indicate that thermal prediction remains challenging because of environmental variability, surface emissivity effects, and complex heat-transfer mechanisms in real power equipment. Third, the current framework assumes simplified diffusion behavior and does not explicitly model convection or radiation effects, both of which may influence thermal distributions under practical operating conditions.

Despite the observed improvements relative to the baseline CNN, the predictive performance achieved in this study remains modest, with a best-case (R²) value of 0.4646 and an RMSE of 12.013 °C. In addition, localized residual errors remain significant in certain hotspot regions. These results indicate that the proposed framework does not yet satisfy the accuracy requirements necessary for safety-critical deployment in high-voltage equipment monitoring. The limited performance is likely influenced by the relatively small dataset, the use of snapshot thermographic images without temporal information, environmental variability, and the simplified diffusion-based physical model employed in this work. Consequently, the proposed framework should be regarded as an exploratory physics-constrained thermographic analysis approach rather than a standalone operational diagnostic system. Future studies should incorporate larger datasets, spatio-temporal thermal measurements, and more comprehensive heat-transfer models to improve predictive reliability.

Future work should therefore focus on spatio-temporal thermographic sequences to enable dynamic heat-transfer modeling and more reliable estimation of thermal diffusivity and heat-source characteristics. The incorporation of advanced physics-informed neural network architectures, transformer-based thermal representations, and explainable AI techniques could further improve interpretability and predictive performance. In addition, multimodal monitoring frameworks combining thermographic imaging with electrical load, weather, or vibration data may provide a more comprehensive understanding of equipment health in modern smart-grid environments. Overall, the results indicate that combining physics-based regularization with deep learning improves the robustness and physical plausibility of infrared thermographic analysis. The proposed framework offers a practical tool for intelligent thermal monitoring of high-voltage power equipment, supporting early detection of abnormal heating patterns and improving maintenance decision-making in power systems.

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
CUDA / cuDNN	NVIDIA	GPU acceleration for training	GTX 1080 Ti
Infrared thermographic camera	FLIR thermal imaging camera	Acquisition of high-voltage equipment thermograms	portable professional FLIR C5 Infrared camera at different times of the day and load conditions.
Jupyter Notebook / IDE	JupyterLab 4.2.0	Experiment development and analysis	Used in implementation
Matplotlib	Matplotlib Developers	Visualization of predictions, residual maps, and uncertainty maps	Used for figures and plots
NumPy	NumPy Developers	Numerical computing and array operations	Data preprocessing and tensor handling
OpenCV	OpenCV Foundation	Image loading, resizing, and basic preprocessing	Used for thermographic image handling
Python	Python Software Foundation	Main programming environment	Version used for all experiments
PyTorch	PyTorch Foundation	CNN training, physics-constrained learning, and Monte Carlo dropout	Used for implementation of the regression network
Scikit-learn	scikit-learn Developers	Metric computation and train/validation/test evaluation	Used for (R^2), RMSE, and MAE
SciPy	SciPy Developers	Supporting numerical routines	Used in preprocessing or analysis
Thermographic dataset	In-house field inspection dataset	Source of 174 infrared images for model development	Images organized in labeled folders
Workstation / GPU computing system	GTX 1080 Ti	Model training and inference	12 GB RAM,  GTX 1080 Ti , and Windows 11 OS