Plate 1 shows visible presence of distortion on mild steel plate sample from welding experiments. The base material, 10 mm thick ASTM A36 mild steel, was cut into plates measuring 300 mm in length and 150 mm in width. A single V-groove butt joint configuration with a 60˚ included angle and a 2 mm root gap was prepared for edge preparation. Welding was performed in autogenous mode for the root pass, followed by two subsequent filler passes using an ER70S-6 filler wire, resulting in a total of three passes. The process utilized Direct Current Electrode Negative (DCEN) polarity. The torch was maintained at a travel angle of 75˚ relative to the workpiece. To minimize excessive out-of-plane deformation while accommodating thermal expansion, the plates were rigidly clamped to a heavy steel backing fixture with a copper backing strip to ensure full root penetration and restrain the workpieces. TIG welding was conducted using a computerized GTAW system with high-purity argon (99.99%) serving as the shielding gas.

Plate 1. 10 mm mild steel plate sample with visible distortion.

Post-weld transverse distortion measurements were conducted after the plates had completely cooled to room temperature, specifically 24 hours after welding, to eliminate any transient thermal expansion effects. Measurements were taken at three distinct locations along the mid-span of the weld length (at the center and 50 mm from both edges). At each location, three readings were recorded and averaged to represent the plate’s distortion. Initial coarse measurements were performed using calibrated digital micrometers, and these were subsequently cross-verified and combined with high-resolution optical profilometry scans at the same locations to maintain a measurement uncertainty of ±0.005 mm.

Four governing process parameters guided the experimental matrix: welding current (A), arc voltage (V), shielding gas flow rate (L/min), and welding speed (cm/min). A Face-Centered Central Composite Design (FCCD) variant was employed for the experimental matrix. The design comprised 16 factorial points (coded as ±1), 8 axial points located at the face centers (coded as α = ±1), and 6 center-point replicates (coded as 0) to estimate pure error and ensure design rotatability. This layout generated the 30 distinct experimental runs presented in Table 1, optimizing parameter space exploration while minimizing resource expenditure.

Table 1. Process parameters and their coded levels for CCD experimental design.

Figure 1. Schematic of the 4-10-1 feedforward ANN architecture for transverse distortion prediction.

A multilayer feedforward neural network presented in Figure 1Figure 1 was developed to map the four input parameters to transverse distortion. The architecture comprises an input layer (4 neurons), a single hidden layer (10 neurons with hyperbolic tangent activation), and an output layer (1 neuron). The 4-10-1 ANN architecture was selected based on a systematic tuning process. The number of hidden neurons was determined using the heuristic formula N h = N i n × N o u t + C (where C is a constant between 1 and 10) and refined via trial-and-error to minimize the Mean Squared Error (MSE) on the validation set without causing overfitting, aligning with established neural network design principles [9].

To establish a baseline and quantify the advantage of the nonlinear mapping capability of the ANN, a simple Multiple Linear Regression (MLR) model was also developed using the same dataset. The MLR model yielded a significantly lower overall R² of 0.784, confirming that the ANN architecture provides a substantial improvement in capturing the complex, nonlinear thermo-mechanical interactions of the welding process.

The Levenberg-Marquardt backpropagation algorithm was selected to ensure rapid convergence on the experimental dataset. Prior to training, min-max scaling normalized all input and output variables to the [0, 1] range, preventing gradient instability. The dataset was partitioned into training (70%), validation (15%), and testing (15%) subsets, preserving balanced parameter distributions across each split. Model training incorporated a maximum of 100 epochs, an early stopping patience of 6 epochs, and a target gradient threshold of 1 × 10⁻⁷. Random weight initialization employed a fixed seed (42) to guarantee reproducibility. All computational procedures were executed in MATLAB R2024b utilizing the Neural Network Toolbox.

Transverse distortion varied between 0.1729 mm and 0.4278 mm across the 30 experimental runs (Table 2). Peak distortion occurred at high current (170 A) and low welding speed (6 cm/min), consistent with elevated net heat input promoting asymmetric thermal expansion and contraction. Conversely, reduced distortion at high welding speed (10 cm/min) correlates with shorter exposure time and lower cumulative thermal strain. These baseline trends establish the physical context for ANN validation.

Table 2. Experimental matrix and measured transverse distortion for 30 CCD runs.

The ANN model achieved optimal performance within 5 training epochs (Figure 2Figure 2). The training state plot confirms gradient descent stabilization, with the final gradient (8.89 × 10⁻⁹) falling well below the convergence threshold (Figure 3Figure 3). Validation checks triggered early stopping at epoch 3, preventing overfitting despite limited dataset size (Figure 4Figure 4). The performance plot demonstrates consistent MSE reduction across training, validation, and test subsets, indicating stable optimization without memorization artifacts.

Figure 2. ANN training convergence plot for transverse distortion prediction (MSE vs. epoch).

Figure 3. Training state plot showing gradient, mu, and validation checks during ANN optimization.

Figure 4. Performance plot illustrating MSE progression across training, validation, and test datasets.

The ANN model predicted transverse distortion with high fidelity, yielding an overall R² of 0.911 and a final MSE of 9.41 × 10⁻⁶ (Table 3). Run-by-run predictions alongside experimental measurements show a mean absolute error (MAE) of 0.018 mm, with 83% of predictions deviating by less than ±0.025 mm from experimental values. Regression analysis (Figure 5Figure 5) confirms a linear correlation slope near unity and minimal intercept bias. The error histogram (Figure 6Figure 6) reveals a Gaussian distribution centered at zero, with ±0.0061 mm encompassing the majority of residuals, confirming unbiased prediction behavior.

Table 3. Experimental versus ANN-predicted transverse distortion with absolute error analysis.

Figure 5. Regression plot comparing experimental versus ANN-predicted transverse distortion (R = 0.911).

Figure 6. Error histogram showing residuals concentrated near zero (±0.0061 mm), confirming unbiased ANN predictions.

Prediction errors concentrated in parameter extremes. Run 11 (130 A, 25 L/min, 8 cm/min) exhibited the largest negative error (−0.063 mm), while Run 20 (170 A, 10 cm/min) showed a positive deviation (+0.051 mm) (Table 3). These discrepancies likely stem from localized cooling rate variations and altered convective heat transfer at high gas flow rates, which modify the effective thermal gradient. The CCD design prioritizes central parameter space coverage, leaving edge regimes underrepresented. Incorporating additional axial points or employing physics-informed constraints could further refine model generalization in these regions. Nevertheless, the ANN consistently captures the dominant thermo-mechanical trends governing transverse distortion. Time series plots indicated that ANN predictions tracked experimental values closely in runs involving extreme parameter combinations, aligning with the theoretical advantage of ANN in capturing complex, higher-order nonlinearities that quadratic polynomials may miss.