Hybrid feedforward neural network for pressure vessel internal corrosion prediction: integrating chemical models with inspection data for structural integrity assessment

The prediction and detection of corrosion in oil and gas pressure vessels are critical challenges in asset integrity management. Corrosion-induced failures result in significant economic losses, environmental hazards, and potential safety incidents, with repair and replacement costs often exceeding millions of dollars per incident [1]. Pressure vessels operating in harsh industrial environments are continuously exposed to aggressive chemicals, elevated temperatures, and variable pH conditions that progressively degrade material properties, reducing mechanical strength and potentially leading to catastrophic structural failure [2]. Accurate corrosion prediction models are therefore essential for proactive maintenance planning, ensuring operational reliability, and extending asset service life.
Corrosion prediction models have been the most reported approaches for corrosion risk mitigation, with the objective of improving maintenance planning and reducing failure risks [3]. The models utilize record keeping, in situ experimental observations, and inspection data to quantify material degradation in terms of corrosion rates, extents, and locations [4].
Modern corrosion prediction approaches use many different models with different assumptions and strengths. Empirical models based on power-law formulations using historical data are straightforward but not robust to complex conditions [5]. Conventional machine learning (ML) and deep learning (DL) algorithms trained on large datasets can be effective in recognizing patterns, but require substantial computational resources. Physics-informed simulations (i.e., finite element models (FEM)) provide accurate predictions but require high computing power. Hybrid methodologies that integrate machine learning (ML) methods with physics-based models, risk-based inspection (RBI) frameworks, and electrochemical modeling and simulations can yield improved predictive capability and performance [6].
Neural networks (NN) and similar methods are often used to estimate the corrosion rate and assess the integrity of pressure vessels. These hybrid approaches combine data-driven neural networks with physics-based constraints to improve prediction accuracy. Other compatible methods include computational fluid dynamics (CFD), finite element modelling (FEM), and physics-based modelling, which allow for rapid or accurate estimation of corrosion [7]. Hybrid modelling has been demonstrated for all types of corrosive mechanisms, including uniform corrosion, pitting, and stress corrosion cracking, and can adapt to various operational and environmental conditions [8].
Hybrid NN methods offer several advantages, particularly the ability to integrate physics-based models to improve prediction accuracy. Hybrid approaches for modeling ${\mathrm{CO}}_2$${\mathrm{CO}}_2$CO_(2)\mathrm{CO}_{2}${\mathrm{CO}}_2$ and ${\mathrm{O}}_2$${\mathrm{O}}_2$O_(2)\mathrm{O}_{2}${\mathrm{O}}_2$ corrosion rates in pipelines have been developed to account for local phenomena such as flow patterns and mass transfer, and hybrid PCA-MLP models have been applied to subsea pipeline corrosion in heavy multiphase flow conditions [9]. Theory-driven neural network models have also been developed to predict burst pressures of corroded pipelines by incorporating physical constraints into the network architecture [10]. Furthermore, hybrid methods combining ANNs with advanced algorithms have been used to predict corrosion characteristics of subsea pipelines under uncertain operating conditions [11].
In this study, a hybrid model based on chemical corrosion theory and feedforward neural networks (FNNs) was developed. The model is constructed with a physics-informed design using common process data, and corrosion rates and uncertainty estimates are generated via Monte Carlo dropout. The thickness data obtained in these inspections are used to update the prediction and to estimate the remaining useful life (RUL) of the components based on the proposed model.
The purposes of this study:

T
he dataset contains wall thickness measurements taken at 24 ultrasonic measurement points across different pressure vessel sections from 2002 to 2008 [12]. For each measurement, in addition to the historical record data, physicsbased environmental data were generated to represent potential operating conditions. The simulated variables are as follows:

These parameters are then the inputs of a physics-based corrosion model in which the corrosion rate (CR) is estimated.
The model can be expressed as follows:

where R is the universal gas constant, T is the temperature in Kelvin, and $\mathrm{f}\mathrm{\_}\mathrm{pH},\mathrm{f}\mathrm{\_}$$\mathrm{f}\mathrm{\_}\mathrm{pH},\mathrm{f}\mathrm{\_}$f_pH,f_\mathrm{f} \_\mathrm{pH}, \mathrm{f} \_$\mathrm{f}\mathrm{\_}\mathrm{pH},\mathrm{f}\mathrm{\_}$ion, $\mathrm{f}\mathrm{\_}$$\mathrm{f}\mathrm{\_}$f_\mathrm{f} \_$\mathrm{f}\mathrm{\_}$pressure, $\mathrm{f}\mathrm{\_}\mathrm{flow}$$\mathrm{f}\mathrm{\_}\mathrm{flow}$f_flow\mathrm{f} \_\mathrm{flow}$\mathrm{f}\mathrm{\_}\mathrm{flow}$, and $\mathrm{f}\mathrm{\_}$$\mathrm{f}\mathrm{\_}$f_\mathrm{f} \_$\mathrm{f}\mathrm{\_}$MIC are correction factors representing the effects of pH , ionic species, pressure, fluid flow, and microbial activity, respectively [12]. This combined dataset (experimental thickness measurements and physics-based environmental data) was used to train and validate the machine learning model for corrosion rate prediction and remaining life estimation.

Using environmental and inspection data, a Feedforward Neural Network (FNN) model was established to predict the corrosion rates in pressure vessels. The FNN model has 11 input variables: temperature ( T ), pH , chloride ion concentration ( ${\mathrm{Cl}}^{-}$${\mathrm{Cl}}^{-}$Cl^(-)\mathrm{Cl}^{-}${\mathrm{Cl}}^{-}$), oxygen concentration ( ${\mathrm{O}}_2$${\mathrm{O}}_2$O_(2)\mathrm{O}_{2}${\mathrm{O}}_2$ ), sulfate ion concentration ( ${\mathrm{SO}}_4{}^{2-}$${\mathrm{SO}}_4{}^{2-}$SO_(4)^(2-)\mathrm{SO}_{4}{ }^{2-}${\mathrm{SO}}_4{}^{2-}$ ), fluid velocity (v), pressure ( P ), activation energy (Ea), pre-exponential rate constant (K), microbial-induced corrosion (MIC) and historical wall thickness (t_past). The FNN utilizes these explanatory variables in a three-hidden layer neural network architecture with 16, 8, and 4 neurons, followed by a single neuron output layer predicting corrosion rate (CR) in mm/year.
To further improve the generalization and produce estimates of uncertainty, dropout layers were placed following each of the hidden layers during the training and prediction. This provides the capability of Monte Carlo (MC) dropout to estimate the uncertainty in predictions from repeatable stochastic forward passes. The FNN model is illustrated in Fig. 1.

The equations below prescribe the forward propagation through the network:

where $h(l)$$h(l)$h(l)h(l)$h(l)$ the activation of the $l$$l$ll$l$-th hidden layer (a vector of neuron outputs), $\sigma$$\sigma$sigma\sigma$\sigma$ is the ReLU activation function, defined as $\sigma (z)=max(0,z),w(l)$$\sigma (z)=max(0,z),w(l)$sigma(z)=max(0,z),w(l)\sigma(z)=\max (0, z), w(l)$\sigma (z)=max(0,z),w(l)$ the weight matrix for the $l$$l$ll$l$-th layer (connects $h(l-1)$$h(l-1)$h(l-1)h(l-1)$h(l-1)$ to $h(l)),h(l-1)$$h(l)),h(l-1)$h(l)),h(l-1)h(l)), h(l-1)$h(l)),h(l-1)$ the activation of the previous layer (or input variables for $\mathrm{l}=1$$\mathrm{l}=1$l=1\mathrm{l}=1$\mathrm{l}=1$ ), and $\mathrm{b}(\mathrm{l})$$\mathrm{b}(\mathrm{l})$b(l)\mathrm{b}(\mathrm{l})$\mathrm{b}(\mathrm{l})$ the bias vector for the l -th layer [13].

where $z$$z$zz$z$ is the Input to the ReLU function (typically ${\mathrm{w}}^{(1)}{\mathrm{h}}^{(1-1)}+{\mathrm{b}}^{(1)}),\sigma (z)$$\left.{\mathrm{w}}^{(1)}{\mathrm{h}}^{(1-1)}+{\mathrm{b}}^{(1)}\right),\sigma (z)${:w^((1))h^((1-1))+b^((1))),sigma(z)\left.\mathrm{w}^{(1)} \mathrm{h}^{(1-1)}+\mathrm{b}^{(1)}\right), \sigma(z)${\mathrm{w}}^{(1)}{\mathrm{h}}^{(1-1)}+{\mathrm{b}}^{(1)}),\sigma (z)$ is the output of the ReLU function, and all negative inputs are set to zero.

where CR is the output corrosion rate ( mm / year, scalar output), W (4) is the weight matrix for the output layer, $\mathrm{h}{}^{(3)}$$\mathrm{h}{}^{(3)}$h^((3))\mathrm{h}{ }^{(3)}$\mathrm{h}{}^{(3)}$ is the activation of the third hidden layer, and $\mathrm{b}{}^{(4)}$$\mathrm{b}{}^{(4)}$b^((4))\mathrm{b}{ }^{(4)}$\mathrm{b}{}^{(4)}$ is the bias vector for the output layer.

The equation for the exponential model used to predict thickness degradation is presented [14] :

where $\mathrm{T}(\mathrm{t})$$\mathrm{T}(\mathrm{t})$T(t)\mathrm{T}(\mathrm{t})$\mathrm{T}(\mathrm{t})$ is the thickness at time $\mathrm{t},{\mathrm{T}}_0$$\mathrm{t},{\mathrm{T}}_0$t,T_(0)\mathrm{t}, \mathrm{T}_{0}$\mathrm{t},{\mathrm{T}}_0$ is the initial thickness at the reference time (e.g., 2002), k is the decay rate constant fitted using curve fitting, and t is the time elapsed from the reference year (e.g., $\mathrm{t}=$$\mathrm{t}=$t=\mathrm{t}=$\mathrm{t}=$ year - 2002)

The linear model for thickness degradation is expressed as:

where $\mathrm{T}(\mathrm{t})$$\mathrm{T}(\mathrm{t})$T(t)\mathrm{T}(\mathrm{t})$\mathrm{T}(\mathrm{t})$ is the thickness at time $\mathrm{t},{\mathrm{T}}_0$$\mathrm{t},{\mathrm{T}}_0$t,T_(0)\mathrm{t}, \mathrm{T}_{0}$\mathrm{t},{\mathrm{T}}_0$ is the initial thickness at the reference time (e.g., 2002), CR is the constant corrosion rate (mm/year) predicted by the FNN model, and t is the time elapsed from the reference year (e.g., $\mathrm{t}=$$\mathrm{t}=$t=\mathrm{t}=$\mathrm{t}=$ year - 2002).

The mean squared error (MSE) loss function used to train the FNN model is defined as:

where L is the loss value, N is the number of samples in the dataset, ${\mathrm{CR}}_{\mathrm{i}}$${\mathrm{CR}}_{\mathrm{i}}$CR_(i)\mathrm{CR}_{\mathrm{i}}${\mathrm{CR}}_{\mathrm{i}}$ is the true corrosion rate for the i -th sample (mm/year), and $\hat{\mathrm{C}}{\mathrm{R}}_{\mathrm{i}}$$\widehat{\mathrm{C}}{\mathrm{R}}_{\mathrm{i}}$hat(C)R_(i)\hat{\mathrm{C}} \mathrm{R}_{\mathrm{i}}$\hat{\mathrm{C}}{\mathrm{R}}_{\mathrm{i}}$ is the predicted corrosion rate for the i -th sample (mm/year)

To enable robust model training across realistic operating variations, augmented environmental data were generated using a probabilistic sampling approach to augment the available inspection data. The inspection dataset consisted of wall thickness measurements from 24 measurement points across different vessel sections collected between 2002 and 2008, along with single-point measurements of operating conditions (temperature, pressure, pH , etc.). However, since only singlepoint measurements were available and the spatial and temporal variations in operating conditions within the vessel were not available, environmental parameters were sampled from distributions centered around the measured values and spanning typical industrial ranges reported in the corrosion kinetics literature [16] [17]. A total of 2000 scenarios were created to capture realistic operational variability, with each environmental variable sampled independently from either a normal or uniform distribution.
The sampling distributions were defined as follows: temperature ( T ) was sampled from a normal distribution with mean around ${70}^{\circ }\mathrm{C}$${70}^{\circ }\mathrm{C}$70^(@)C70^{\circ} \mathrm{C}${70}^{\circ }\mathrm{C}$ and standard deviation up to ${30}^{\circ }\mathrm{C}$${30}^{\circ }\mathrm{C}$30^(@)C30^{\circ} \mathrm{C}${30}^{\circ }\mathrm{C}$; pH was uniformly sampled between 4 and 7 ; chloride ion concentration $\left({\mathrm{Cl}}^{-}\right)$$\left({\mathrm{Cl}}^{-}\right)$(Cl^(-))\left(\mathrm{Cl}^{-}\right)$\left({\mathrm{Cl}}^{-}\right)$was sampled from a normal distribution with mean around 200 ppm and standard deviation up to 50 ppm ; oxygen content ( ${\mathrm{O}}_2$${\mathrm{O}}_2$O_(2)\mathrm{O}_{2}${\mathrm{O}}_2$ ) was uniformly sampled between 5 and 12 ppm ; sulfate ion concentration ( ${\mathrm{SO}}_4{}^{2-}$${\mathrm{SO}}_4{}^{2-}$SO_(4)^(2-)\mathrm{SO}_{4}{ }^{2-}${\mathrm{SO}}_4{}^{2-}$ ) was sampled from a normal distribution with mean around 50 ppm and standard deviation up to 15 ppm ; fluid velocity (v) was uniformly distributed between 2 and $4\mathrm{m}/\mathrm{s}$$4\mathrm{m}/\mathrm{s}$4m//s4 \mathrm{~m} / \mathrm{s}$4\mathrm{m}/\mathrm{s}$; pressure ( P ) was sampled from a normal distribution with mean around 15 bar and standard deviation up to 5 bar; activation energy (Ea) was sampled from a normal distribution with mean around $75\mathrm{kJ}/\mathrm{mol}$$75\mathrm{kJ}/\mathrm{mol}$75kJ//mol75 \mathrm{~kJ} / \mathrm{mol}$75\mathrm{kJ}/\mathrm{mol}$ and standard deviation up to $15\mathrm{kJ}/\mathrm{mol}$$15\mathrm{kJ}/\mathrm{mol}$15kJ//mol15 \mathrm{~kJ} / \mathrm{mol}$15\mathrm{kJ}/\mathrm{mol}$; and the pre-exponential rate constant (K) was sampled from a log-normal distribution with parameters about $\ln \left({10}^6\right)$$\ln \left({10}^6\right)$ln(10^(6))\ln \left(10^{6}\right)$\ln \left({10}^6\right)$ and standard deviation around 0.5 to reflect the exponential sensitivity of corrosion kinetics. The historical wall thickness (t_past) from the 2002 inspection was included as an additional input feature for each vessel section. This physics-based dataset provides a diverse and statistically representative input space to train the corrosion prediction model under realistic environmental and operating conditions.

Dropout is an important regularization approach for improving the generalization of a Feedforward Neural Network (FNN) that is used in the development of neural network models to predict corrosion rates in industrial components. In each training iteration, dropout reduces the chance of overfitting the training dataset [16] , which contains section-specific environmental parameter values such as temperature, chloride concentration, and historical thickness measurements, by randomly inactivating a fraction of neurons in the model. The underlying stochasticity encourages the model to learn more distributed and fewer neuron-specific features, leading to better generalization performance under new conditions. Dropout was applied in the model architecture immediately after each hidden layer ( 16,8 , and 4 neurons). This configuration balances the noise and variability in the inspection data and produces a stable corrosion rate prediction.
Moreover, MC Dropout is a Dropout enhancement during inference to quantify uncertainty, and it is an important feature desired for applications [17] requiring reliable corrosion rate estimates. By maintaining Dropout during prediction, the model returns multiple outputs for each input, with each prediction generated from a different subset of activated neurons [18]. These predictions yield the expected corrosion rate (mean), and the standard deviation provides the uncertainty in the predictions. By implementing the MC dropout prediction function, this method allows for section-level corrosion rate estimates, with confidence intervals displayed in the validation scatter plots and thickness prediction uncertainty bands. MC

Dropout provides conservative estimates of the corrosion rates, helps interpret prediction uncertainty, and enhances the reliability of the vessel structural integrity assessment to support more informed decisions for corrosion mitigation.

rhe dropout operation randomly deactivates neurons during training by applying a Bernoulli mask to the activations [19] as follows:

where $p$$p$pp$p$ is the probability of retaining a neuron, $a$$a$aa$a$ is the activation vector, $r$$r$rr$r$ is the Bernoulli mask (binary values), and à is the masked activation.
Monte Carlo (MC) equation
MC Dropout prediction is implemented in the prediction function, which performs multiple stochastic forward passes to estimate the predictive mean and standard deviation [19]. The equations used are:

where $\mathrm{f}(\mathrm{x},{\omega }_{\mathrm{t}})$$\mathrm{f}\left(\mathrm{x},{\omega }_{\mathrm{t}}\right)$f(x,omega_(t))\mathrm{f}\left(\mathrm{x}, \omega_{\mathrm{t}}\right)$\mathrm{f}(\mathrm{x},{\omega }_{\mathrm{t}})$ is the output of the neural network at test time with dropout mask ${\omega }_{\mathrm{t}}$${\omega }_{\mathrm{t}}$omega_(t)\omega_{\mathrm{t}}${\omega }_{\mathrm{t}}$, and T is the number of stochastic forward passes.
To facilitate uncertainty quantification, the model integrates Dropout and Monte Carlo (MC) techniques. Dropout layers are applied after each hidden layer ( 16,8 , and 4 neurons), providing regularization during training and enabling Monte Carlo Dropout during inference. The uncertainty is estimated using the MC dropout prediction function, which performs multiple stochastic forward passes of the trained FNN model with dropout enabled. This technique quantifies epistemic uncertainty associated with model parameters. For the exponential thickness model, uncertainty in the fitted parameters ( $t_0$$t_0$t_(0)t_{0}$t_0$ and k ) is propagated to generate uncertainty bands in future thickness predictions. The predictive mean and standard deviation are calculated using Eqns. (10) and (11), producing uncertainty bands in the predicted thickness plots.

The effect of environmental conditions on corrosion plays a significant role in industrial component deterioration because it directs the amount and rate of deterioration of materials over time [20]. Corrosion is highly contingent upon the chemical and physical environments of the substrate, where the temperature, humidity, pH , chloride content, availability of oxygen, and fluid velocity all affect the rates of many processes. For example, the Arrhenius equation demonstrates the dependence of reaction rates on temperature [21], allowing the temperature effects to be captured in the chemical corrosion rate with simulated data [22]. In the model, high chloride levels favor ionic corrosion, especially in aqueous environments, whereas high oxygen content or sulfide concentrations promote oxidation or sulfide corrosion mechanisms. Moreover, pressure, velocity, and pH also affect physical and chemical interactions at the material surface, potentially amplifying erosioncorrosion (E-C) effects [23]. Fig. 2 illustrates the effects of pH and temperature on corrosion rate, showing higher corrosion rates at lower pH values and elevated temperatures from the model.

This case study examines a corroded horizontal pressure vessel made of low-alloy steel (09G2s) operating in an industrial water environment. Wall thickness measurements were collected at 24 inspection points using ultrasonic testing (UT) from 2002 to 2008 . The UT method provided reliable thickness data at designated grid points, as illustrated in Fig. 3.

The corrosion prediction model was developed through a four-stage approach combining chemical fundamentals, machine learning, and inspection data to estimate corrosion rates and remaining useful life (RUL).

Stage 1 - Simulated Data Generation: Environmental parameters (temperature, pH, chloride, oxygen, MIC, pressure, and flow velocity) were input into a modified Arrhenius-based chemical corrosion model (Eqn. 1) to generate training data representing realistic corrosion behavior under various operating conditions.
Stage 2 - FNN Training with MC Dropout: A feedforward neural network with three hidden layers was trained to predict corrosion rates using the 11 input features described earlier. Dropout layers were applied after each hidden layer to provide regularization during training and prevent overfitting. During inference, Monte Carlo dropout was implemented by keeping dropout active and performing multiple stochastic forward passes, generating both mean corrosion rate predictions and uncertainty estimates.
Stage 3 - Calibration with Inspection Data: The FNN predictions were calibrated using historical wall thickness measurements from 2002 to 2008 . Exponential and linear degradation curves were fitted to the measured thickness data for each vessel section. The FNN predictions were adjusted using scaling factors derived from the observed thickness changes to ensure accurate section-specific predictions.
Stage 4 - RUL Estimation: The calibrated corrosion rates were used to project future wall thickness from 2008 to 2040. RUL for each vessel section was calculated as the time until predicted thickness reaches the minimum allowable threshold defined by design standards. Uncertainty bands were generated by propagating MC dropout uncertainties through the exponential thickness model parameters. Fig. 4 illustrates the complete workflow.

Model performance was evaluated using standard error-based metrics for corrosion rate prediction with the FNN model and for thickness prediction using exponential and linear degradation models.

The Mean Squared Error (MSE)

The Mean Squared Error (MSE) is the loss function used to train models using the FNN architecture; it is the average squared difference between the actual values $\left({\mathrm{y}}_{\mathrm{i}}\right)$$\left({\mathrm{y}}_{\mathrm{i}}\right)$(y_(i))\left(\mathrm{y}_{\mathrm{i}}\right)$\left({\mathrm{y}}_{\mathrm{i}}\right)$ and predicted values $\left({\hat{\mathrm{y}}}_{\mathrm{i}}\right)$$\left({\widehat{\mathrm{y}}}_{\mathrm{i}}\right)$( hat(y)_(i))\left(\hat{\mathrm{y}}_{\mathrm{i}}\right)$\left({\hat{\mathrm{y}}}_{\mathrm{i}}\right)$, and N is the number of samples [24]. MSE is the loss function used to train the FNN, calculated as the average squared difference between actual and predicted corrosion rates over N samples. During training, MSE was minimized using the Adam optimizer. As a performance metric, MSE penalizes larger errors more heavily than smaller ones, which improves predictive accuracy for degradation trends. For corrosion rate predictions, the FNN achieved a test MSE of $0.02685(\mathrm{mm}/\text{year}{)}^2$$0.02685(\mathrm{mm}/\text{ year }{)}^2$0.02685(mm//" year ")^(2)0.02685(\mathrm{~mm} / \text { year })^{2}$0.02685(\mathrm{mm}/\text{ year }{)}^2$, demonstrating good convergence with inspection data. By minimizing MSE during optimization, the network learns to predict degradation curves more accurately, enabling improved remaining useful life estimation for each pressure vessel section.

MAE is the mean absolute error between predicted and actual values, providing an interpretable performance measure as it retains the units of the target variable (mm/year for corrosion rates). For corrosion rate predictions, the FNN achieved a test MAE of $0.12039\mathrm{mm}/$$0.12039\mathrm{mm}/$0.12039mm//0.12039 \mathrm{~mm} /$0.12039\mathrm{mm}/$ year . This performance measure is suitable for real-world applications when determining the accuracy of predictions. MAE is important when evaluating whether and when maintenance thresholds are crossed, what sections are at risk, and their associated structural integrity in corrosive environments.

RMSE is the square root of MSE, providing an error measure in the original units while emphasizing larger errors. For corrosion rate predictions, the FNN achieved an RMSE of $0.1639\mathrm{mm}/$$0.1639\mathrm{mm}/$0.1639mm//0.1639 \mathrm{~mm} /$0.1639\mathrm{mm}/$ year. RMSE was also used to compare the exponential and linear thickness degradation models, confirming that the exponential model provided more accurate thickness predictions.

The Coefficient of Determination ( ${\mathrm{R}}^2$${\mathrm{R}}^2$R^(2)\mathrm{R}^{2}${\mathrm{R}}^2$ ) is a measure of how well the model accounts for the variance in the data, as an ${\mathrm{R}}^2$${\mathrm{R}}^2$R^(2)\mathrm{R}^{2}${\mathrm{R}}^2$ value closer to 1 indicates a better fit for the model and explains the variance in the response variable [25]. For corrosion rates, the FNN had a test ${\mathrm{R}}^2$${\mathrm{R}}^2$R^(2)\mathrm{R}^{2}${\mathrm{R}}^2$ of 0.975 , confirming that the proposed model performed well in representing the degradation trends. For thickness predictions, the exponential degradation model achieved an overall ${\mathrm{R}}^2$${\mathrm{R}}^2$R^(2)\mathrm{R}^{2}${\mathrm{R}}^2$ of 0.99 , indicating excellent agreement with the measured wall thickness data across all vessel sections.

Prediction of the corrosion rate is one of the most important aspects when discussing the lifetime and performance of industrial components, while Fig. 6 presents an important comparison between the corrosion rates predicted by the Feedforward Neural Network (FNN) and the corrosion rates based on the inspection data. The FNN also shows a very competitive accuracy, with a Mean Absolute Error (MAE) of $0.12039\mathrm{mm}/$$0.12039\mathrm{mm}/$0.12039mm//0.12039 \mathrm{~mm} /$0.12039\mathrm{mm}/$ year, which confirms its effectiveness for modelling corrosion dynamics. The application of Monte Carlo (MC) dropout even improves the robustness of the model, and bounds for uncertainties are on average one standard deviation from the mean prediction about $0.2\mathrm{mm}/$$0.2\mathrm{mm}/$0.2mm//0.2 \mathrm{~mm} /$0.2\mathrm{mm}/$ year, indicating a high confidence in the predictions. The solid tolerance banding of the data points shown in the figure confirms the robust validity of the model in terms of precision; hence, it is sufficiently sensitive enough in order to adequately mimic the degradation patterns of the sections.

Fig. 5 shows the FNN training history showing the convergence of training loss and validation loss over 500 epochs. Both losses decrease and stabilize, indicating effective model learning and good generalization to unseen data without overfitting. The graph shown is directly from a neural network model trained to predict corrosion rates on equipment used in industry based on past inspection data.

Thickness predictions are crucial for assessing the long-term performance of pressure vessel parts. The thickness predictions are shown in Fig. 7. It shows the predicted minimum wall thicknesses for all sections, historical wall thickness measurements from 2002 to 2008, and predictions to 2040 using both exponential and linear degradation models from the feedforward neural network (FNN). The exponential predicted model uses historical thickness measurements and Monte Carlo (MC) Dropout uncertainty to account for a conservative estimate of maximum loss in thickness ( $0.08\mathrm{mm}/$$0.08\mathrm{mm}/$0.08mm//0.08 \mathrm{~mm} /$0.08\mathrm{mm}/$ year on average). The linear prediction model assumes a constant or average corrosion rate ( $0.10\mathrm{mm}/$$0.10\mathrm{mm}/$0.10mm//0.10 \mathrm{~mm} /$0.10\mathrm{mm}/$ year ) for all uniform reductions. Uncertainty bands and minimum allowable thickness lines were provided for sections E Head 1 and Nozzle N1, allowing engineers to effectively determine where critical sections can potentially monitor, repair, or replace to maintain structural integrity.
The figure shows time-series predictions for critical vessel sections (e.g., head, nozzle, and shell portions) and includes black dots with measured values from the years 2002-2008, red solid lines for the exponential predictions, blue dashed lines for the linear trends, red shaded areas for uncertainty bands ( $95\mathrm{\%}$$95\mathrm{\%}$95%95 \%$95\mathrm{\%}$ confidence), and green dashed lines for minimum thresholds. The FNN-based exponential model predicted accurately with high fidelity to non-linear degradation or empirical corrosionbased observations, providing a more realistic risk-informed approach to predicting structural longevity than purely linear extrapolations that do not take into account the varying corrosion rates across components.

The model accuracy is evident based on the very small absolute error (typically $<0.1\mathrm{mm}$$<0.1\mathrm{mm}$< 0.1mm<0.1 \mathrm{~mm}$<0.1\mathrm{mm}$ ) and relative errors ( $<1\mathrm{\%}$$<1\mathrm{\%}$< 1%<1 \%$<1\mathrm{\%}$ ), even with the highest relative error for Nozzle A1 of $1.25\mathrm{\%}$$1.25\mathrm{\%}$1.25%1.25 \%$1.25\mathrm{\%}$, and all the errors for F Head 2 were very low, below $0.12\mathrm{\%}$$0.12\mathrm{\%}$0.12%0.12 \%$0.12\mathrm{\%}$. These differences indicate that the models performed consistently regardless of section size and thickness, making it reasonable to have confidence in the predictive value of the exponential FNN representation for reliably predicting the long-term performance.
The linear model, although very simple, has the advantage of low complexity and presents an assumption of constant degradation over time, typically underpredicting or showing losses in future thicknesses (i.e., Nozzle N1, Shell 2), thus bringing into question the risks surrounding reliability-centered maintenance, especially at the minimum thresholds, as it did not include the uncertainty of those predictions. Finally, the exponential FNN showed a statistically significant improvement over the linear baseline, conforming more closely to realistic corrosion degradation patterns.

The anticipated minimum wall thickness trends across various sections of the pressure vessel, as shown in Fig. 7, show the historical data from 2002 to 2008 and the projected data to 2040. The exponential feedforward neural network (FNN) model provides a good fit to the measured data in a non-linear decaying pattern, which is consistent with the measured data and reflects the moderated corrosion rate with the later data range. The linear model implies a linear degradation rate and produces a higher decline than the exponential model, particularly in the Nozzle A1 and E Head 1 sections, where the predicted thickness from the linear model falls below the minimum allowable thickness. The uncertainty bands around the exponential predictions based on a Monte Carlo dropout Monte Carlo uncertainty approach show relatively small
uncertainty bands, demonstrating a high degree of confidence in the predictions and robustness of the model. Overall, this analysis is valuable for illustrating the linear versus exponential FNN modelling capabilities with respect to realistic and actual corrosion behaviors.

Tab. 2 provides a structured comparison of the neural network exponential predictive model and the linear model using the summary statistics reported for the minimum thickness predictions over the ten structural sections. The statistics presented include the Mean Absolute Error (MAE), median absolute error, maximum absolute error, mean relative error, median relative error, and maximum relative error, and offer a quantitative measure of model performance at the section level.

Its corresponding section-level predictive performance as a function of time represents the absolute and relative errors separately and proves that the Exponential Feedforward Neural Network (FNN) model performs better in terms of both absolute and relative errors in modeling section-level structural degradation as per Fig. 8. The mean absolute error (MAE) is equal to 0.0389 mm , and the median absolute error is equal to 0.0301 mm , which shows that most of the predicted values differ from the measured values with a gap smaller than 0.04 mm . In most structural monitoring scenarios, such accuracy is important because even a slight underestimation or overestimation can impact maintenance decisions and risk assessment. In addition, the maximum absolute error is bounded by 0.1096 mm , so that all predicted values have the worst deviation within a narrow bound.

The accuracy of the model was even clearer in the relative error metrics, observed a mean relative error of only $0.3480\mathrm{\%}$$0.3480\mathrm{\%}$0.3480%0.3480 \%$0.3480\mathrm{\%}$ and a median relative error of $0.2562\mathrm{\%}$$0.2562\mathrm{\%}$0.2562%0.2562 \%$0.2562\mathrm{\%}$, indicating that the predictions were consistently proportional to the actual thickness measurements. The maximum relative error of $1.2460\mathrm{\%}$$1.2460\mathrm{\%}$1.2460%1.2460 \%$1.2460\mathrm{\%}$ is also low, and the ${\mathrm{R}}^2$${\mathrm{R}}^2$R^(2)\mathrm{R}^{2}${\mathrm{R}}^2$ value is 0.99 , indicating a high degree of explained variance, confirming the model's accuracy.
In comparison with the linear Model, it shows significantly higher errors with a mean absolute error of 0.1350 mm ( 3.5 times higher) and a mean relative error of $1.2458\mathrm{\%}$$1.2458\mathrm{\%}$1.2458%1.2458 \%$1.2458\mathrm{\%}$ ( 3.6 times higher), thus the Exponential FNN model demonstrates better performance with lower error rates across all metrics.

This result demonstrates that the exponential FNN model accurately tracks the actual degradation path of structural elements. The exponential decay formulation is particularly appropriate for structures undergoing erosion or corrosion processes, where degradation rates typically decrease over time as protective oxide layers form or reactive species are depleted. By aligning with observed degradation trends, this approach provides confidence in the model for long-term prediction, remaining useful life estimation, and determining optimal inspection intervals.

To evaluate remaining useful life (RUL) across different vessel sections, both linear and exponential degradation models were applied. The linear model assumes constant corrosion rates over time, representing conservative worstcase scenarios. The exponential model captures nonlinear degradation behavior influenced by environmental factors and protective mechanisms. Both models were used to predict future thickness and assess structural integrity according to ASME Section VIII and API 579-1 standards [26].

The minimum allowable wall thickness ( ${\mathrm{t}}_{\text{min}}$${\mathrm{t}}_{\text{min }}$t_("min ")\mathrm{t}_{\text {min }}${\mathrm{t}}_{\text{min }}$ ) for each vessel section was calculated according to ASME Section VIII and API 579-1/ASME FFS-1 [27], accounting for design pressure, material properties, and safety factors.

where ${\mathrm{t}}_{\mathrm{mm}}$${\mathrm{t}}_{\mathrm{mm}}$t_(mm)\mathrm{t}_{\mathrm{mm}}${\mathrm{t}}_{\mathrm{mm}}$ is the minimum measured thickness, FCA is the future corrosion allowance (corrosion rate × future service period), ${\mathrm{t}}_{\text{min}}$${\mathrm{t}}_{\text{min }}$t_("min ")\mathrm{t}_{\text {min }}${\mathrm{t}}_{\text{min }}$ is the minimum required thickness, and ${\mathrm{t}}_{\text{nom}}$${\mathrm{t}}_{\text{nom }}$t_("nom ")\mathrm{t}_{\text {nom }}${\mathrm{t}}_{\text{nom }}$ is the nominal thickness [27].

where ${\mathrm{t}}_{\text{initial}}$${\mathrm{t}}_{\text{initial }}$t_("initial ")\mathrm{t}_{\text {initial }}${\mathrm{t}}_{\text{initial }}$ is the initial thickness of the material $(\mathrm{mm}),{\mathrm{t}}_{\text{min}}$$(\mathrm{mm}),{\mathrm{t}}_{\text{min }}$(mm),t_("min ")(\mathrm{mm}), \mathrm{t}_{\text {min }}$(\mathrm{mm}),{\mathrm{t}}_{\text{min }}$ minimum allowable thickness $(\mathrm{mm})$$(\mathrm{mm})$(mm)(\mathrm{mm})$(\mathrm{mm})$ determined by fitness-forservice criteria, and CR is the corrosion rate ( $\mathrm{mm}/$$\mathrm{mm}/$mm//\mathrm{mm} /$\mathrm{mm}/$ year ) [28].

The exponential model as per Tab. 4 predicts that all vessel sections will remain above the minimum allowable thickness $\left({\mathrm{t}}_{min}\right)$$\left({\mathrm{t}}_{min}\right)$(t_(min))\left(\mathrm{t}_{\min }\right)$\left({\mathrm{t}}_{min}\right)$ through 2040, indicating SAFE status for all components. This model accounts for decreasing corrosion rates over time due to protective film formation and environmental stabilization. For example, E Head 1 shows a predicted thickness of 6.7 mm compared to $t_{\text{min}}$$t_{\text{min }}$t_("min ")t_{\text {min }}$t_{\text{min }}$ of 4.5 mm , providing adequate safety margin. Thicker sections such as F Head 2 demonstrate minimal relative thickness loss, supporting the validity of the exponential model for long-term predictions.
The linear model as per Tab. 5 applies constant corrosion rates and predicts UNSAFE conditions for four sections: E Head 1 ( $3.9\mathrm{mm}<4.5\mathrm{mm}$$3.9\mathrm{mm}<4.5\mathrm{mm}$3.9mm < 4.5mm3.9 \mathrm{~mm}<4.5 \mathrm{~mm}$3.9\mathrm{mm}<4.5\mathrm{mm}$ ), Nozzle A1 ( $2.6\mathrm{mm}<5.0\mathrm{mm}$$2.6\mathrm{mm}<5.0\mathrm{mm}$2.6mm < 5.0mm2.6 \mathrm{~mm}<5.0 \mathrm{~mm}$2.6\mathrm{mm}<5.0\mathrm{mm}$ ), Nozzle N3 ( $5.8\mathrm{mm}<7.0\mathrm{mm}$$5.8\mathrm{mm}<7.0\mathrm{mm}$5.8mm < 7.0mm5.8 \mathrm{~mm}<7.0 \mathrm{~mm}$5.8\mathrm{mm}<7.0\mathrm{mm}$ ), and Nozzle N4 ( $5.6\mathrm{mm}<7.0\mathrm{mm}$$5.6\mathrm{mm}<7.0\mathrm{mm}$5.6mm < 7.0mm5.6 \mathrm{~mm}<7.0 \mathrm{~mm}$5.6\mathrm{mm}<7.0\mathrm{mm}$ ). This conservative approach is appropriate for worst-case scenario planning and identifies sections requiring priority inspection and potential mitigation measures. Thicker sections such as F Head 2 remain SAFE under both models, confirming adequate structural integrity for these components.

$\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$he following Tab. 6 summarizes the proposed and selected studies, highlighting their methodologies and performance metrics.

The comparison reveals that different models excel at specific prediction tasks. Liu and Meng (2025) developed a physicsinformed neural network for predicting failure pressure of corroded pipelines, incorporating the DNV RP-F101 empirical formula into the loss function. Their model achieved strong accuracy with MAPE $=2.52\mathrm{\%}$$=2.52\mathrm{\%}$=2.52%=2.52 \%$=2.52\mathrm{\%}$ and ${\mathrm{R}}^2=0.9886$${\mathrm{R}}^2=0.9886$R^(2)=0.9886\mathrm{R}^{2}=0.9886${\mathrm{R}}^2=0.9886$ on 60 burst test samples, demonstrating the effectiveness of integrating empirical knowledge with neural networks for failure pressure prediction. The model proposed in this study provides a broader assessment framework by predicting corrosion rates ( ${\mathrm{R}}^2=0.975$${\mathrm{R}}^2=0.975$R^(2)=0.975\mathrm{R}^{2} =0.975${\mathrm{R}}^2=0.975$ ), future thickness degradation (MAE $=0.0389\mathrm{mm}$$=0.0389\mathrm{mm}$=0.0389mm=0.0389 \mathrm{~mm}$=0.0389\mathrm{mm}$ ), and remaining useful life while quantifying uncertainty through Monte Carlo dropout.
The hybrid approach combines a physics-based exponential degradation model with a data-driven feedforward neural network, providing both point estimates and confidence intervals for maintenance planning. Compared to other corrosion rate prediction studies, such as Wang et al. (2022) with ${\mathrm{R}}^2=0.8609$${\mathrm{R}}^2=0.8609$R^(2)=0.8609\mathrm{R}^{2}=0.8609${\mathrm{R}}^2=0.8609$ and Obaseki & Elijah (2021) with ${\mathrm{R}}^2=0.9521$${\mathrm{R}}^2=0.9521$R^(2)=0.9521\mathrm{R}^{2}=0.9521${\mathrm{R}}^2=0.9521$, the proposed model demonstrates acceptable predictive performance. Additionally, the exponential thickness prediction model achieves low errors (MAE $=0.0389\mathrm{mm}$$=0.0389\mathrm{mm}$=0.0389mm=0.0389 \mathrm{~mm}$=0.0389\mathrm{mm}$, relative error $=0.348\mathrm{\%}$$=0.348\mathrm{\%}$=0.348%=0.348 \%$=0.348\mathrm{\%}$ ), outperforming the linear baseline across all metrics. The integrated framework enables comprehensive vessel condition assessment, predicting both current and future degraded states with uncertainty quantification, which supports informed decision-making for maintenance planning and risk management.

This study demonstrates a hybrid feedforward neural network (FNN) model integrated with a physics-based corrosion model for predicting corrosion rates and estimating the remaining useful life (RUL) of pressure vessels. Using nondestructive evaluation (NDE) 24 thickness measurements from different sections (2002-2008) and physics-based training data, the three-layer FNN model with Monte Carlo dropout achieved ${\mathrm{R}}^2=0.975$${\mathrm{R}}^2=0.975$R^(2)=0.975\mathrm{R}^{2}=0.975${\mathrm{R}}^2=0.975$ for corrosion rate prediction. The model achieved MSE $=0.02685(\mathrm{mm}/\text{year}{)}^2$$=0.02685(\mathrm{mm}/\text{ year }{)}^2$=0.02685(mm//" year ")^(2)=0.02685(\mathrm{~mm} / \text { year })^{2}$=0.02685(\mathrm{mm}/\text{ year }{)}^2$ and MAE $=0.120\mathrm{mm}/$$=0.120\mathrm{mm}/$=0.120mm//=0.120 \mathrm{~mm} /$=0.120\mathrm{mm}/$ year for corrosion rate predictions.
For thickness prediction, the exponential degradation model outperformed the linear baseline, achieving over all ${\mathrm{R}}^2=0.99$${\mathrm{R}}^2=0.99$R^(2)=0.99\mathrm{R}^{2}=0.99${\mathrm{R}}^2=0.99$ across all sections, MAE $=0.0389\mathrm{mm}$$=0.0389\mathrm{mm}$=0.0389mm=0.0389 \mathrm{~mm}$=0.0389\mathrm{mm}$, maximum absolute error $=0.1096\mathrm{mm}$$=0.1096\mathrm{mm}$=0.1096mm=0.1096 \mathrm{~mm}$=0.1096\mathrm{mm}$, and mean relative error $=0.3480\mathrm{\%}$$=0.3480\mathrm{\%}$=0.3480%=0.3480 \%$=0.3480\mathrm{\%}$. The exponential model captures the nonlinear nature of corrosion degradation, providing more realistic thickness projections than traditional linear models that assume constant corrosion rates.
The model predictions were integrated into a Fitness-for-Service (FFS) assessment framework based on API 579-1/ASME FFS-1 standards. By comparing predicted future thickness against minimum allowable thickness ( ${\mathrm{t}}_{\text{min}}$${\mathrm{t}}_{\text{min }}$t_("min ")\mathrm{t}_{\text {min }}${\mathrm{t}}_{\text{min }}$ ), components were categorized as "SAFE" or "UNSAFE". The analysis showed that while the linear model flagged several components as potentially requiring mitigation due to its conservative constant-rate assumption, the exponential model provided a less conservative long-term assessment while maintaining structural integrity requirements. This approach translates predictive
analytics into standards-compliant integrity assessments, supporting practical engineering decision-making for conditionbased maintenance planning.

This study demonstrates a hybrid neural network approach for corrosion prediction in pressure vessels using physicsbased training data and historical inspection measurements. Several areas for future investigation are identified. Data Expansion: The current model was validated using 24 thickness measurements from different sections over a six-year period (2002-2008) and physics-based environmental parameters. Future work should incorporate real-time environmental monitoring data and extend validation to longer inspection intervals. Expanding the dataset to include additional vessel geometries and operating conditions would further assess model generalizability.
Model Generalization: The current model is calibrated to a specific pressure vessel asset. Future research could adapt this framework to multiple assets with varying material properties, fluid compositions, and operating environments. This would require developing transfer learning approaches or ensemble methods to maintain accuracy across diverse corrosion scenarios.
Environmental Parameter Refinement: The physics-based corrosion model incorporates temperature, pH, chloride, oxygen, sulfate, MIC, pressure, and flow velocity. Future work could investigate additional parameters such as ${\mathrm{CO}}_2$${\mathrm{CO}}_2$CO_(2)\mathrm{CO}_{2}${\mathrm{CO}}_2$ partial pressure, ${\mathrm{H}}_2\mathrm{S}$${\mathrm{H}}_2\mathrm{S}$H_(2)S\mathrm{H}_{2} \mathrm{~S}${\mathrm{H}}_2\mathrm{S}$ concentration, and stress corrosion cracking factors to improve prediction accuracy in specific industrial applications. Uncertainty Quantification: While Monte Carlo dropout provides prediction uncertainty estimates, future work could explore alternative Bayesian approaches or ensemble methods to better characterize epistemic and aleatoric uncertainty in corrosion predictions.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

The author declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

During the preparation of this manuscript, the author used Google Colab as the computational environment for implementing and training the feedforward neural network model, and Paperpal for language improvement. After using these tools, the author reviewed and edited the content as needed and take full responsibility for the content of the publication.

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