The temperature distribution T(r,t) across the thickness of the pressurizer wall is governed by the 1D transient heat conduction equation in cylindrical coordinates. The general form of this equation is (2)ρcp∂T∂t=1r∂∂r(k(r)r∂T∂r),r∈[rin,rout]t>0where T(r,t) is the temperature as a function of radius r and time t. k(r) is a piecewise constant function representing the thermal-conductivity of stainless-steel cladding and the carbon-steel base metal. The PINN is designed to minimize the residual of this equation, denoted as f(r,t). The system is subject to the following conditions derived from the RELAP5-3D transient data:

(1) Initial condition T(r,0)=Tsteady(r)

(2) Inner wall boundary condition (fluid-side) −k∂T∂r|r=rin=hin(t)[T(rin,t)−Tfluid(t)]

(3) Outer wall boundary condition (containment-side) ∂T∂r|r=rout=0″

The 1D transient heat conduction equation was solved using Forward Time Central Space (FTCS) explicit finite difference scheme. The wall thickness was spatially discretized into number of radial nodes (Δr). The time domain was discretized into small time steps (Δt). The FTCS scheme was formulated for interior node i at radial position ri. At the interface between the two materials, heat flux conservation was applied using a harmonic mean for thermal conductivity to ensure accuracy across the boundary.

The FDM model was run for each of the three defined accident cases (SBO, LOCA, SBO+LOCA). For each case, the calculations generated a compressive transient temperature profile T(r,t) across the entire wall thickness (1.025–1.175 m) over the specified accident duration. The spatial discretization employed 100 radial nodes across the wall thickness, with a time step of 1 s. The generated time-dependent temperature profiles (middle and intermediate radial positions) along with the corresponding time and radial position temperature data, was organized into a structured format suitable for training of surrogate models.

A Polynomial Regression surrogate model of a third-order degree was constructed based on a general form shown in Eq. (3). The model was designed to establish a baseline for comparison, allowing a clear assessment of the benefit of more, physic-informed approaches. The model was implemented using a standard least-squares approach. The input features (time and boundary temperatures) were standardized to a zero mean and unit variance to improve numerical stability. The general form of the Polynomial Regression model was (3)T(t,r,Tinner(t) and Touter(t))=∑i=0Nt∑j=0Nr∑k=0Nin∑l=0Noutβijkltirj[(Tinner(t))k(Touter(t))l]where T is the predicted temperature, t is the time, r is the radial position, Tinner(t) and Touter(t) are the time dependent boundary temperatures, βijkl are the coefficients, and Nt,Nr,Nin,Nout are the maximum polynomial degrees for each input variable.

A supervised DNN was developed to serve as a purely data-driven surrogate model. The DNN consisted of multiple fully connected (dense) hidden layers with a Rectified Linear Unit (ReLU) activation function for nonlinearity. The feedforward DNN architecture of T^=𝒩(r,t;θ), where θ represents the optimized weights and biases. This was implemented using TensorFlow/Kares. The input features consisted of: time elapsed during the accident scenario, radial position within the pressurizer wall, and temperature at both inner and outer surface of the pressurizer (boundary conditions). The output layer consisted of a single neuron with a linear activation function, corresponding to predicted temperature (Tpred). The model’s performance was optimized by minimizing the MSE loss function as shown in Eq. (4) (4)LMSE=1N∑i=1N((T˙pred,i−TRELAP,i)2)where T˙pred,i is the predicted wall temperature for the data point, and TRELAP,i is the corresponding value yielded from the RELAP and FDM simulations. Adam optimizer was used to minimize the loss, with Mean Absolute Error (MAE), Maximum Absolute Error (MaxAE) and Normalized MAE (NMAE) also monitored as additional evaluation metrics.

The PINN incorporated this physical law into its training through a physical loss term (ℒphys) which is derived from the residual of the governing equation. The PINN model’s NN predicts the temperature, T^(t,r). Its partial derivatives with respect to time and radial position were computed using automatic differentiation, a feature of the Tensor Flow framework. The PINN minimises a loss function that includes the physics residual ℛ(r,t) as shown in Eq. (5)(5)ℛ(r,t)=ρcp∂T^∂t−1r∂∂r(kr∂T^∂r)

The optimization objective is defined as the weighted sum of squares as shown in Eq. (6) (6)minℒ(0)=λdataMSEdata+λphysMSEℛ+λbcMSEbc

To train the PINN, the physical laws and data constraints are unified into a single composite loss function. The total loss, ℒTotal, is minimized using Adam optimizer and is defined by Eq. (7) (7)ℒTotal=wdataℒdata+wPhysℒphys+wbcℒbc+wicℒicwhere ℒdata is the mean squared error between the PINN’s predictions and the RELAP5-3D and FDM-generated temperature data sets. ℒphys is the mean square error of the PDE residual, ℛ(r,t), evaluated at thousands of collocation points throughout the wall thickness. ℒbc and ℒicare the errors at the boundary (cladding/outer wall) and the initial time step (t=0). wi are the weighting coefficients (hyper-parameters) that balance the contribution of physics against the data. In this work, all physics-based weights were equal to 1, to prevent one term form dominating training. To ensure robust training and prevent vanishing gradients, all input features (r,t) and targets (T) were scaled to the range [0, 1] using min-Max Scaling. Xavier (Glorot) initialization was used to maintain variance across layers. 5000 collocation points were sampled using a Latin Hypercube to ensure the physics loss was evaluated uniformly across the (r,t) domain. Table 2 shows a detailed hyper-parameter for the surrogate models for DNN and PINN architectures.

During the early phase (100–2000 s), Tinner dropped from around 690 K to a minimum of around 520 K. Concurrently, Touter dropped from around 590 K to a minimum of around 310 K. This indicates a more gradual cooling effect, as compared to the LOCA cases. During rate phase of the transient (2000 to 10,000 s), both Tinner and Touter reached a state of thermal equilibrium at their lower respective temperatures. The absence of a subsequent temperature rise indicates that the system stabilized at a lower energy state following the initial cooling phase.

Like SBO, both Tinner and Touter underwent a significant initial decrease during early phase (time 100–3000 s). Tinner dropped to minimum of approximately 350 K ∼3000–4000 s. Touter also dropped to a minimum ∼300 K during the same period. During the later phase of the transient (3000–10,000 s), it was observed that both temperatures stable at these reduced levels. The transient behavior was similar to the SBO case, although the magnitude of the temperature drops were more pronounced for Tinner.

The thermal behavior in the SBO+LOCA case shows an intermediate behavior. Both Tinner and Touter showed a similar trend characterized by initial drops, followed by stabilisation at lower temperature levels in the later phases. This suggest that LOCA component might be the dominant factor influencing the thermal response of the pressurizer wall.

The model’s ability to predict the temperature at any radial position as a function of time was evaluated on unseen data. The model achieved a high coefficient of determination (R2 = 0.978), indicating that it captures a significant portion of the variance. The consistent performance across training, validation and test set, suggest that the model is not significantly overfitting. The RMSE and MAE are relatively low, indicating a good average fit, as shown in Fig. 5. However, the MaxAE values ranged up to ∼200.45 K. This suggests that there were specific instances where its prediction deviates highly from the actual data. The residual distribution plots show non-normal distribution with significant errors, as shown in Fig. 6. The predicted against actual plots (as shown in Fig. 7) show some deviations from the ideal 45-degree line. This confirms the model’s low-level ability to accurately capture the nonlinear thermal behavior of the pressurizer wall, making it less reliable for safety-critical analysis.

The results indicate that the polynomial model’s predicted temperature (dashed lines), followed the overall true temperature trend (solid lines) as shown in Fig. 8. However, a few deviations are observed, particularly during phases of rapid temperature changes. For LOCA and SBO+LOCA cases, the model tends to smooth out sharp drops and subsequent fluctuations. The model was more stable for SBO case with its predictions for the middle and outer wall temperatures showing considerable divergence, particularly during the initial transient and later phases.

For LOCA, the model provides a good fit for the radial profiles (Fig. 9). However, SBO+LOCA and mostly SBO, the model shows more significant deviations in the radial profiles, particularly at the inner and outer radial extremes and during the early transient phases (T∼2000 s). It was seen that the model tends to smooth out the curvature of the temperature profile across the wall.

This model was designed to predict the temperature at any radial position as a function of time. From Table 4, the DNN surrogate model showed exceptionally high performance, achieving an overall R2≈0.9996 on the test set. The overall RMSE ∼0.86 K and MAE ∼0.52 K represent a significant reduction in average prediction error, as compared to the Polynomial Regression Model. A MaxAE of 44.4 K with NMAE ∼0.13 are substantially low (as compared to Polynomial model) suggesting that the DNN model is more robust in handling extreme temperature deviations. Training and validation loss curves converged rapidly, and the model performance metrics confirms model’s high accuracy as shown in Fig. 10. Residual distribution plots, show a symmetric bell-shaped tightly clustered. This indicates model’s errors were small and random, as shown in Fig. 11. Fig. 12 shows data points laying on the ideal 45-degree line, indicating that the model provided accurate temperature predictions.

The time history plots for the DNN model show a close agreement with the primary data, across the three accident cases. The model captures initial temperature fluctuations, prolonged plateaus and cooling, and re-stabilization trends. For example, (as shown in Fig. 13) in the LOCA and SBO+LOCA scenarios, the DNN model follows the rapid temperature drops and subsequent complex fluctuations with a few minimal deviations. Even in the challenging SBO case, where the polynomial model struggled, the DNN’s predictions are better aligned with the primary data, capturing the initial transient and later stabilization phases with high fidelity. Fig. 14 shows that the DNN model, near perfectly alignment with the primary data at selected times across the pressurizer wall radius. This indicates the DNN’s robust capability to predict the spatial temperature distribution at any given time, capturing the precise temperature gradients across the pressurizer wall thickness. However, challenges were observed in predicting sharp transients and oscillating temperature trends.

The PINN model provides a compelling trade-off between statistical accuracy and physical fidelity. While its overall test R2∼0.9874 is lower than that of DNN, its performance is physically consistent. The overall test RMSE ∼5.66 K and MAE ∼2.63 K are better than the polynomial regression model. As seen in Fig. 15 (Training and Validation Loss), the model’s loss curves show a steady and consistent decrease. This indicates that the network effectively learned for both the primary temperature data and the physics constraints without overfitting. The performance metrics in Table 4 confirms this, with a higher R2 and moderate error values seen across the training, validation and test sets.

The residual distribution plots (Fig. 16) indicates less clustered, as compared to purely data-driven models, with small and random distributed errors. However, this is still well centered around zero. The predicted against the actual plots (Fig. 17) show that data points closely align with the ideal 45-degree line. This suggest that incorporating physical constraints is highly effective at capturing the dynamics of multi-regime transients, providing a more reliable tool for safety analysis.

The PINN predicted plots for selected radial position (Fig. 18), indicate that the model effectively captures the temperature trends at different depths within the pressurizer wall. The close agreement between the PINN and primary data, indicates that the model accurately learns and simulates the transient thermal behavior over time. Surprisingly, this included the rapid changes during the early phase of a transient and the slower progression to steady state. Lastly, Fig. 19 shows point wise validation of the PINN model’s performance, indicating its reliability and accuracy. Therefore, the use of PINN in pipes such as pressurizer, could be pivotal in understanding the thermal transient behavior, as a substitute of IHPC.

Although the DNN achieves the lowest numerical errors, it occasionally produces non-physical behavior during sharp transients, such as oscillatory temperature predictions and slight violations of the expected monotonic radial gradient. In contrast, the PINN maintains smooth, physically consistent profiles across the pressurizer wall. This is because its solution its constrained by the heat-conduction PDE. For example, during the early rapid-cooling phase of the LOCA case (t = 1500–2500 s), the DNN model slightly overshoots the mid-wall temperature, whereas the PINN preserves the correct curvature and gradient, dictated by the governing physics. The PINN ensures that the temporal and spatial derivatives are coupled according to the material’s thermal diffusivity. This provides a more reliable basis for subsequent thermal stress and structural integrity assessment.