Improving the Interpretability of Data-Driven Models for Additive Manufacturing Processes Using Clusterwise Regression

Abstract

Wire Arc Additive Manufacturing (WAAM) represents a disruptive technology in the field of metal additive manufacturing. Understanding the relationship between input factors and layer geometry is crucial for studying the process comprehensively and developing various industrial applications such as slicing software and feedforward controllers. Statistical tools such as clustering and multivariate polynomial regression provide methods for exploring the influence of input factors on the final product. These tools facilitate application development by helping to establish interpretable models that engineers can use to grasp the underlying physical phenomena without resorting to complex physical models. In this study, an experimental campaign was conducted to print steel components using WAAM technology. Advanced statistical methods were employed for mathematical modeling of the process. The results obtained using linear regression, polynomial regression, and a neural network optimized using the Tree-structured Parzen Estimator (TPE) were compared. To enhance performance while maintaining the interpretability of regression models, clusterwise regression was introduced as an alternative modeling technique along with multivariate polynomial regression. The results showed that the proposed approach achieved results comparable to neural network modeling, with a Mean Absolute Error (MAE) of 0.25 mm for layer height and 0.68 mm for layer width compared to 0.23 mm and 0.69 mm with the neural network. Notably, this approach preserves the interpretability of the models; a further discussion on this topic is presented as well.

1. Introduction

Additive Manufacturing (AM) occupies a crucial role in the Industry 4.0 production paradigm (see Figure 1) [1]. AM involves constructing components by incrementally adding material layer-by-layer, enabling the realization of components with complex geometries that conventional methods find challenging to replicate. Furthermore, AM expedites the prototyping process, decreasing time-to-market for new products. Its capacity to minimize waste by utilizing materials only where required promotes sustainability as well.

Among AM technologies, Wire Arc Additive Manufacturing (WAAM) [2,3] (see Figure 2) has garnered significant attention from the scientific community due to its capability to rapidly construct large metal components at reduced costs. This process, also known as Gas Metal Arc Welding Rapid Prototyping (GMAW-RP), is based on depositing metals layer-by-layer in the form of droplets along a path defined by a slicer using the arc welding process [4]. Welding automation systems are utilized in this process, offering flexibility and a large working volume. Typically, an anthropomorphic six-axis robotic motion platform holds a welding torch and a welding machine that generates arc when the wire contacts the deposition substrate. This approach produces near-net-shape components that often require postprocessing, such as machining or thermal treatments, to meet the final tolerances specified by project requirements and mitigate residual stresses, which are typical defects in the parts [5].

Nowadays, the abundance of resources dedicated to the development of AI applications has facilitated its widespread integration across various industrial applications, including modeling [6,7,8,9], control [10,11,12], optimization [13,14,15], and especially monitoring [16,17,18,19,20,21,22,23]. AI has emerged as a prominent tool, particularly in the modeling and monitoring tasks, with supervised learning approaches predominantly favored due to their demonstrated effectiveness and reliability in achieving desired outcomes [24,25,26,27,28]. These approaches leverage labeled datasets to train models, enabling them to accurately identify patterns and make informed decisions based on past observations. As previously mentioned, one of the most promising applications in the field of AM is predictive modeling. This approach allows for the generation of models that are valuable for simulating what-if scenarios and other applications, especially considering the intricate thermomechanical nature of metal transformation that would otherwise require complex physical models [29]. Due to the complexity, high resource demands, and time-consuming nature of modeling physical phenomena using methods such as finite element analysis, AI is emerging as an alternative tool. However, given the “black-box” nature of the data-driven methods they employ, AI models face challenges related to the low interpretability of their results compared to traditional “white-box” approaches. One of the most important application of process modeling in WAAM is the slicer software, as the geometry of the deposited layers (output) heavily depends on various input factors such as the employed process parameters, welding technique, wire diameter, and material composition utilized for component fabrication [11,30]. When a CAD geometry is obtained, the input factors can be used to determine key layer geometry parameters such as width, height, and overlapping distance (d) [31], as illustrated in Figure 3. Additionally, if a part has a height of N mm, the obtained layer height (h) can be used to determine the required number of slices and generate the path planning for the deposition of subsequent layers by adjusting the position of the robot. Selection of the correct process parameters, which depend on the model employed to correlate the layer geometry and input factors, is important; an incorrect parameterization, as may result from using an inaccurate model to estimate layer height, can lead to error accumulation during part production, resulting in defects such as spatters, humping, and porosity. In fact, if a layer height of h is estimated and a $\mathrm{\Delta }E$ exists, it can accumulate during the deposition until the distance between the torch and the workpiece is different from the planned one, leading to an unstable printing process [32].

As discussed above, estimating the layer geometry with the lowest possible error is crucial to reducing the presence of defects within components realized by WAAM technology. Despite this, several process parameters (which represent the input factors in the regression task) are utilized in WAAM, such the wire feed speed (WFS), welding speed (WS), welding voltage (V), gas flow rate (GFR), inter-pass temperature (T), and contact-to-workpiece distance (CTWD). Each of these parameters influences the layer geometry in opposite ways and with different magnitudes, complicating the parameter selection process [15]. A review of the literature on this topic reveals that researchers have explored various methodologies aiming to model this complex input–output relationship, including statistical methods such as polynomial regression [33] and machine learning techniques such as Support Vector Machine (SVM) [34], neural networks [35], and Advanced Regression Tree (ART) methods [36], which represent the state of the-art. However, while these machine learning methods offer promise in capturing complex relationships, their black-box nature often complicates the interpretability of the resulting models [37]. Understanding how each parameter influences the final components becomes challenging, hindering the ability to optimize the WAAM process effectively. Therefore, the present study aims to address these limitations by conducting a statistical analysis of the layer geometry obtained through depositing layers in carbon steel under different process parameters. The research workflow followed in this work is shown in Figure 4.

In particular, in this work we explore whether the combined use of highly interpretable clustering and regression models leveraging process knowledge can enhance modeling performance without compromising accuracy. After collecting and preprocessing data from an experimental campaign, we compare the results of the proposed technique with alternative modeling approaches from the literature, including polynomial regression, linear regression, random forest, and neural networks, allowing for a methodological comparison with other studies. The rest of this paper is organized as follows: Section 2 delves into the data utilized in the study and outlines the methodology employed for statistical analysis; Section 3 presents the key findings; and Section 4 concludes the paper by emphasizing the significance of the findings and suggesting future research directions.

2. Materials and Methods

2.1. Experimental Setup

In the conducted experimental campaign, data were collected while depositing layers through the WAAM process, specifically employing a stainless steel SS308LSi wire with a diameter of 1.2 mm and a constant-voltage welding process. This process, shown in Figure 5, consists of applying a constant voltage to the welding machine; the raw material in wire form is deposited into the melting pool via electromagnetic force applied to the droplet [38].

For the deposition, a six-axis Omron Adept Vipers850 robot was employed as a motion platform, while the welding equipment comprised a Miller Pheonix 456 welding machine. The final bead geometry at each layer was measured using a Lext 5100 laser scanning confocal microscope. The workflow employed to collect the data is reported in Figure 6, and allowed to obtain the necessary labels for the supervised learning task.

The experimental campaign employed a range of process parameters or input factors, including eleven levels of WFS (2.0, 2.5, 2.8, 3.0, 3.6, 4.0, 4.5, 5.2, 6, 6.5, and 7 mm/min), eight levels of WS (90, 130, 180, 250, 450 550, 700, 760 mm/min), four levels of CTWD (9, 12, 15, 20 mm), and eight levels of welding voltage (14.7, 16, 17.5, 18, 20, 22, 26, 30 V), while the interpass temperature and GFR were maintained fixed to the respective values of 100 °C and 18 L/min. However, certain combinations of parameters, such as low wire feed speed and high welding voltage, resulted in defective parts, leading to their exclusion from the conducted analysis, which comprised a total of 174 samples.

2.2. Multivariate Polynomial Regression

Regression analysis is a statistical modeling technique used to describe the relationship between a dependent continuous variable y (output) and one or more independent variables x (input or features) in the form of a mathematical relationship, as reported in Equation (1); different approaches can be used to estimate the parameters $\theta$.

$$y=f(\theta ,x)$$

(1)

The most common form of regression is linear regression, where the relationship between the independent variables and the dependent variable is assumed to be linear, as in Equation (2).

$$y=W·x$$

(2)

Because regression is a supervised learning approach, the weights W (M × N) multiplied by the features × (N) can be estimated using the knowledge of the desired outcome y (M). Ordinary Least Squares (OLS) estimation, Lp estimation [39,40,41], gradient descent optimization [42], or other optimization techniques [43] can be used within this scope. Although linear regression is highly interpretable and simple to train via OLS, it is limited to capturing linear relationships between data and it does not consider the correlation between variables, also known as multicollinearity. Lp-norm regression or advanced gradient-based techniques can be used to avoid this problem.

To address linear relationships between data, polynomial regression can be employed as expressed in Equation (3). Polynomial regression assumes an n-th degree polynomial model between the independent variables and the dependent variables. Unlike linear regression, which assumes a linear relationship, polynomial regression can capture nonlinear patterns, offering a more adaptable model for the data.

$$y={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+\dots +{\beta }_n{x}^n+ϵ=\mathbf{Wx}$$

(3)

In the above equation, y is the dependent variable, x is the independent variable, ${\beta }_0,{\beta }_1,{\beta }_2,\dots ,{\beta }_n$ are the regression coefficients, n is the degree of the polynomial, and $ϵ$ represents the error term. The degree of the polynomial determines the complexity of the relationship that can be captured. Higher-degree polynomials can fit the data more closely but may risk overfitting, especially with limited data. It is essential to strike a balance between model complexity and the risk of overfitting. As in standard linear regression, several methods can be used to estimate the parameters $\beta$. If no correlation between inputs is assumed, the most popular method is OLS estimation; the parameters can be found as reported in Equation (4).

$$\mathbf{W}={\left({\mathbf{x}}^T\mathbf{x}\right)}^{-1}·\mathbf{y}·{\mathbf{x}}^T$$

(4)

2.3. Multicollinearity in Regression

Although polynomial modeling is a statistical tool largely employed to generate simple but effective predictive models, especially from an industrial point of view, as they can be deployed with reduced costs on Programming Logic Controllers (PLCs), it suffers from multicollinearity when the variables exhibit notable interdependence. The OLS method can be applied to find the weight of the regression as the correlation matrix approaches singularity. In a formal context, the diagonal elements of the inverse correlation matrix ${\left(X^{\prime }X\right)}^{-1}$, corresponding to linearly dependent members of X, tend towards infinity. Consequently, accurately distributing the explained variance among variables becomes challenging, resulting in diminished quality of the parameter estimates and difficulty in discerning the independent contributions of explanatory variables. Several methods can be used to evaluate the degree of multicollinearity. Pairwise correlation is a common approach, with correlation coefficients exceeding 0.9 between features often indicating potential issues. However, the Variance Inflation Factor (VIF) is a more reliable metric, with values above 10 generally signaling significant multicollinearity [44]. Additionally, when employing linear regression models, the eccentricity of the confidence region can be used as an indicator of multicollinearity [45]. When detected, multicollinearity can be reduced. To improve model predictions via gradient-based techniques, meaningful and uncorrelated features are then employed to mitigate this effect. The Spearman correlation index ($\rho$) is computed as reported in Equation (5), where $d_i$ represents the difference between the ranks of corresponding variables and n is the number of observations. It is a statistical measure used to assess the strength and direction of association between two ranked variables. Spearman correlation is valuable due to its independence from normality assumptions, making it applicable to diverse data types and serving as a correlation index for nonlinear data. By examining the index’s magnitude, it is possible identify low-correlated (0–0.25) and high-correlated (0.75–1) features within the dataset, which can then be selectively utilized or eliminated. This helps to address multicollinearity issues by ensuring that only uncorrelated features are employed to evaluate regression parameters.

$$\rho =1-{\displaystyle \frac{6\sum d_i^2}{n(n^2-1)}}$$

(5)

This approach is also beneficial for decreasing model complexity and mitigating overfitting while enhancing the ability of the model to generalize.

2.4. Modeling of WAAM Process and Proposal of This Work

In the realm of AM, modeling techniques represent an important tool for both planning and optimization. As discussed in the introduction, the current trend in the literature focuses on using black-box machine learning approaches to address the inherent complexity of AM process modeling [46] in light of the critical role of low estimation errors in planning and optimization tasks. However, while statistical methods can provide valuable insights by elucidating the `black-box’ nature of these models, their performance often falls short of the required standards. This gap has prompted researchers to explore new modeling solutions facilitated by the democratization of machine learning. In the field of WAAM, Xiong et al. [33] introduced a second-order polynomial regression utilizing the wire feed rate, welding speed, arc voltage, and CTWD as inputs, yielding results lower than those obtained by a neural network. Similarly, Naveen Srinivas et al. [47] applied a second-order regression model incorporating the gas flow rate and welding speed along with the wire feed speed for aluminum components. While exhibiting slightly lower performance, these studies have shown that polynomial regression methodologies generate more interpretable models compared to neural networks due to the transparency of their weight parameters. This study aims to demonstrate that combining clustering analysis with polynomial regression (clusterwise regression) can mitigate related problems such as multicollinearity while outperforming standard polynomial models and potentially surpassing the performance of complex black-box neural networks. By enhancing model interpretability, we seek to address a key challenge hindering the industrial adoption of AI [48].

2.5. Proposed Methodology

In a classic regression problem, the estimated parameters are those that minimize the global error. These are unique, and generally describe the general dependencies of the process. However, in complex processes such as WAAM, it is reasonable to expect nonlinear relationships between outputs and variables along with different behaviors in different features spaces. Regarding this particular case, different input factors lead to different metal transfer modes, which introduces changes in the heat input supplied to the material [49]. In Figure 7, it is possible observe the differences in waveform of both the current and voltage signals during the deposition process, which results in different heat inputs, different layer geometries, and the end of deposition.

In the context of a welding-based process such as WAAM, the utilization of cluster-wise regression becomes particularly advantageous due to the diverse impacts that different sets of input factors can have on the heat input delivered to the material and the subsequent heat accumulation. By identifying and accounting for these variations through clusterwise regression [50,51], it is possible to better capture the hidden relationships between input factors and welding outcomes. This not only reduces model complexity and guards against overfitting but also enhances the model’s capacity to generalize across various welding scenarios, leading to more robust and reliable predictive capabilities. Let us first consider a classical linear regression

$$\mathbf{y}=\mathbf{X}\mathbf{b}+\mathbf{e},$$

(6)

with $\mathbf{y}=(y_i;i=1,\dots ,I)$ being the vector of the dependent variable of I units, $\mathbf{X}=(x_{ij};i=1,\dots ,I;j=1,\dots ,J)$ an $I\times J$ matrix of independent regressors, $\mathbf{b}=(b_j;j=1,\dots ,J)$ the vector of J regression coefficients, and $\mathbf{e}=(e_i;i=1,\dots ,I)$ the vector of i.i.d. disturbances. In this particular case, $\mathbf{y}$ is the vector of the layer geometry expressed by the layer width and height, while $\mathbf{X}$ is the vector of the input factors or their combination.

It is well known that under the assumption of the error term having a multivariate normal distribution and $E\left[{\mathbf{e}}^{\prime }\mathbf{e}\right]={\sigma }^2\mathbf{I}$, the parameters of the linear regression can be estimated by maximizing the following likelihood function:

$$L\left(\mathbf{y}\mid \mathbf{b},{\sigma }^2\right)={\left(2\pi {\sigma }^2\right)}^{-1/2}exp\left[-{\displaystyle \frac{{\left(\mathbf{y}-\mathbf{X}\mathbf{b}\right)}^{\prime }\left(\mathbf{y}-\mathbf{X}\mathbf{b}\right)}{2{\sigma }^2}}\right].$$

(7)

Following [52,53], we assume that the relationship between the variables of interest is heterogeneous at the cluster level; in other words, assuming the presence of K clusters, we have $b_{jk}$ as the value of the j-th regression coefficient in the k-th cluster and ${\sigma }_k^2$ as the variance of the error term of the k-th cluster for $j=1,\dots ,J$ and $k=1,\dots ,K$. Given the process knowledge mentioned above, we assume the presence of two clusters associated with the different metal transfer modes.

Moreover, we assume that $y_i$ is distributed as a finite sum or a mixture of conditional normal densities.

$$y_i=\sum _{k=1}^K{\lambda }_k{\left(2\pi {\sigma }_k^2\right)}^{-1/2}exp\left[{\displaystyle \frac{-{\left(y_i-{\mathbf{x}}_i{\mathbf{b}}_k\right)}^2}{2{\sigma }_k^2}}\right],i=1,\dots ,I.$$

(8)

In other words, we assume an independent sample of the observations’ dependent variable $y_1,y_2,\dots ,y_I$ drawn randomly from a mixture of conditional normal densities of underlying clusters in unknown proportions ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_K$.

Given a sample of I independent subjects/observations, it is possible to form a log-likelihood expression

$$lnL=\sum _{i=1}^{I}ln\left[\sum _{k=1}^K{\lambda }_k{\left(2\pi {\sigma }_k^2\right)}^{-1/2}exp\left[{\displaystyle \frac{-{\left(y_i-{\mathbf{X}}_i{\mathbf{b}}_k\right)}^2}{2{\sigma }_k^2}}\right]\right].$$

(9)

Then, we can find the values of ${\lambda }_k,{\sigma }_k^2$, and $b_{jk}$ that maximize the objective function given $K,\mathbf{y}$ and $\mathbf{X}$, and subject to the following constraints:

$$\begin{array}{cc}& 0\le {\lambda }_k\le 1,\hfill \\ & \sum _{k=1}^K{\lambda }_k=1,\hfill \\ & {\sigma }_k^2>0.\hfill \end{array}$$

The algorithm for estimating the parameters and cluster compositions is explained in detail in [52]. In sum, the computation of the maximum likelihood estimates is obtained by means of the EM algorithm. For given starting values of the parameters, the expectation (E phase) and maximization (M phase) steps of this algorithm are alternated until convergence. Another important aspect is that the maximization of the log-likelihood is conditioned to the a priori specification that the number of clusters $K=2$.

2.6. Preprocessing and Workflow

The WAAM process is characterized as a multiple-input multiple-output system where the input parameters represent process variables, namely, the Wire Feed Speed (WFS), Welding Speed (WS), Welding Voltage (V), and Contact-to-Workpiece Distance (CTWD), while the outputs correspond to the dimensions of the printed layers, such as the layer width (w) and layer height (h).

As discussed previously, linear regression is not suitable for modeling such systems due to the inherent nonlinearity related to the underlying complex physical phenomena. Additionally, the physical correlation between output variables further complicates the linear modeling approach, which in state-of-art methods consists of assuming two independent equations. Therefore, given the nonlinear nature and correlated outputs, nonlinear regression methods are more appropriate for modeling the WAAM process. In this work, a multivariate polynomial regression with four inputs and two outputs is used as modeling technique, in which we assume no inherent correlation between the output variables. We consider a second-order combination of the input features without considering the intercept, for a total of fourteen features. These features are represented as $x_i,x_j,x_i·x_j$, where i and j range from 0 to 3, resulting in a comprehensive set of predictors for modeling the WAAM process, obtaining the model in Equation (10)

$$\begin{array}{cc}\hfill y& =\sum _{i=0}^3{\beta }_i{x}_i+\sum _{i=0}^3\sum _{j=i}^3{\beta }_{ij}{x}_i{x}_j,\hfill \end{array}$$

(10)

where:

• ${\beta }_i$ (Linear Coefficients): These coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable $x_i$ while holding all other variables constant.
• ${\beta }_{ij}$ (Interaction Coefficients): These coefficients represent the change in the dependent variable resulting from the interaction between the independent variables $x_i$ and $x_j$ whileholding all other variables constant.

With the model obtained, the OLS was used to estimate the input factors after the feature selection process, for which we used the Spearman correlation index. The final proposed methodology is described in Figure 8. Using the defined set of input factors, the features were extracted using a second-order polynomial. With the aim of mitigating multicollinearity, these features were selected based on the Spearman index to ensure that only low-correlated features were used for regression. The training data were first grouped into clusters using a Gaussian Mixture Model (GMM), then each cluster was individually analyzed using regression models. These regression models represent the relationship between the input factors (inputs) and the target variables (outputs) within the different cluster groups.

The results obtained with the proposed methodology were then compared with other methods mentioned above, such as deep neural networks, linear regression, and polynomial regression. Furthermore, the analysis of both clusters and regression coefficients allows for better analysis of the process and can be easily developed into PLC, as previously discussed.

2.7. Comparison with Optimized Neural Network

Deep Neural Networks (DNNs) represent a powerful class of machine learning models, characterized by their bio-inspired layered architecture of interconnected neurons; these models transform input data through a sequence of nonlinear operations leveraging the gradient descent optimization algorithm, which enables DNNs to generalize complex relationships within continuous functions [42,54]. While they are widely recognized for their performance, especially for supervised models, the training process of DNNs can be challenging due to the significant number of hyperparameters associated with the architecture and training, including:

• Number of layers, which determines the depth of the neural network.
• Number of neurons per layer, which influences the capacity and complexity of the model.
• Learning rate, which controls how much to adjust the model in response to the estimated error each time the model weights are updated.
• Activation functions, which are nonlinear functions applied to each neuron’s output to allow the network to model complex relationships in the data.
• Batch size, referring to the number of samples processed before the model is updated.
• Dropout rate, referring to the probability of randomly ignoring a neuron during training, which helps to prevent overfitting.

Selecting optimal hyperparameters is crucial, as they significantly impact the performance, training time, and generalization ability of DNNs. Manually tuning hyperparameters is a labor-intensive and iterative process that requires domain expertise and considerable computational resources. Traditional methods such as grid search and random search [55,56] are exhaustive and may not efficiently explore the vast hyperparameter space, leading to suboptimal performance. In this work, the Tree-structured Parzen Estimator (TPE) [57] optimization technique is employed to automatically find the optimal hyperparameters, resulting in a reduced validation estimation error when predicting the layer geometry in WAAM. The results suggests that the optimal architecture should contain one hidden layer with 128 neurons and a sigmoid activation function as the hidden layer activation, a learning rate of $\alpha =0.00778$, a batch size of one step per epoch with no dropout, and training for 600 epochs.

3. Results and Discussion

3.1. Data Preprocessing

Starting from the four input factors of Wire Feed Speed ($x_0$), Welding Speed ($x_1$), Contact-To-Workpiece Distance ($x_2$), and Welding Voltage ($x_3$), fourteen features were extracted as described in the previous section to develop the second order multivariate polynomial regression. The Spearman correlation index ($\rho$) was computed by conducting Spearman correlation analysis; the associated color map is shown in Figure 9. From these computed correlations, only those features that presented a low correlation with the others were maintained, thereby avoiding problems related to multicollinearity. At the end of preprocessing, the preserved features were $x_0,x_1,x_2,x_0^2,x_1^2,x_o·x_1,x_0·x_3,x_1·x_2,x_1·x_3$, for a total of nine features. The extracted features were normalized and scaled in the range of 0–1 using Equation (11) for each feature j of the dataset.

$$x[:,j]={\displaystyle \frac{x[:,j]-min\left(x\right[:,j\left]\right)}{max\left(x\right[:,j\left]\right)-min\left(x\right[:,j\left]\right)}}$$

(11)

While the eccentricity of the confidence region for the polynomial regression model decreases by 30% after applying the proposed feature selection, it remains elevated, indicating persistent multicollinearity. This is expected due to the inherent interdependence of the WAAM process parameters. For instance, though theoretically independent, the voltage and wire feed speed are typically adjusted together in order to prevent defects such as burn-through related to poor feeding of the wire in the melting pool. Consequently, certain parameter combinations are underrepresented in the dataset, contributing to high correlation and multicollinearity. In light of our primary focus on prediction accuracy, the Mean Absolute Error (MAE) was used to compare model performance.

3.2. Regression Analysis Results

In this study, we compare different modeling techniques using a dataset where 70% of the samples were used for training and the remaining 30% for validation. The simplest technique employed was linear regression using input factors as inputs. The performance metrics, measured in MAE in millimeters, were 0.4 mm for layer height and 1 mm for layer width on the training dataset. For the test dataset, the MAE was 0.39 mm for layer height and 0.917 mm for layer width. These results indicate poor performance, suggesting that linear regression is insufficient for modeling this system. As shown in

Table 1. Mean Absolute Error (MAE) for different modeling techniques (source: own elaboration).

Model	Training Dataset		Test Dataset
	Height (mm)	Width (mm)	>Height (mm)	Width (mm)
Linear Regression (LR)	0.40	1.00	0.39	0.92
Polynomial Regression (PR)	0.19	0.55	0.24	0.66
Optimized DNN	0.22	0.63	0.24	0.53
Clusterwise Regression	0.21	0.50	0.24	0.47

, introducing a feature extraction procedure to remove uncorrelated features and polynomial regression introduced an increment in modeling performance. The MAE for layer height decreased to 0.19 mm and 0.55 mm for layer width on the training dataset. On the test dataset, the MAE was 0.24 mm for layer height and 0.66 mm for layer width, indicating better overall performance compared to linear regression. The performance obtained by using the optimized DNN was achieved with an MAE of 0.22 mm for layer height and 0.63 mm for layer width on the training dataset. For the test dataset, the MAE was 0.24 mm for layer height and 0.53 mm for layer width. Compared to the previous models, this method shows slight improvement in layer width estimation while maintaining similar performance for layer height.

Finally the proposed clusterwise regression reached MAE values of 0.21 mm and 0.5 mm on the training dataset for layer height and layer width respectively, along with 0.24 mm and 0.47 mm for layer height and layer width on the testing dataset, leading to the best overall performance among all the developed models. The regression plots of the testing dataset results are shown in Figure 10 and Figure 11.

Next, we provide an analysis of how the coefficient of the polynomial regression changes due to introduction of the clustering technique. As discussed in the previous section, GMM is utilized to cluster the data into two groups associated with two different metal deposition modes, each of which exhibits unique relationships with geometric variables, namely, spray transfer and dip transfer. For each cluster, a distinct regression analysis is conducted to ascertain the parameters linked to the weld bead height (${\beta }_h^i$) and width (${\beta }_w^i$), where ’i’ denotes the respective cluster. The final calculation follows Equation (12); it is possible observe the assumption of no correlation between the output variables.

$$\left[\begin{array}{c}h\\ w\end{array}\right]=\left[\begin{array}{cc}{\beta }_h^i& 0\\ 0& {\beta }_w^i\end{array}\right]\left[\begin{array}{c}V\\ WFS\\ WS\\ V^2\\ WF{S}^2\\ V·WFS\\ CTWD·V\\ WFS·WS\\ WFS·CTWD\end{array}\right]$$

(12)

In polynomial regression, as described by Equation (13), the coefficients for predicting the height (h) and width (w) of the layers exhibit specific numerical values for each predictor variable ($V,WFS,WS,V^2,WF{S}^2,V·WFS,CTWD·V$, and $WFS·W$). For instance, the coefficient for V when predicting height is $0.3685$, while when predicting width it is $0.2819$. This means that by increasing the welding voltage, and consequently the heat input, an increment of the both layer width and layer height is reached, with more influence on layer height than layer width. An opposite relationship is obtained for Welding Speed (WS).

Moving to clusterwise regression, which divides the data into distinct groups (Group 0 and Group 1), we observe notable variations in these coefficients. In Group 0’s clusterwise regression model (see Equation (14)) for height prediction, the coefficient for V is $0.2668$, whereas in Group 1 (see Equation (15)) it increases to $0.4181$. This indicates a different emphasis on the influence of V between the two groups, potentially reflecting different operational conditions or underlying data characteristics that the model captures differently. Similarly, for width prediction in clusterwise regression, the coefficient for V in the Group 0 is $-0.0417$, contrasting with $-0.4598$ in Group 1. This substantial shift suggests varying importance or directionality in the effect of V on the width across the two groups.

Upon further analysis, the interaction terms such as $V·WFS,CTWD·V$, and $WFS·WS$ also exhibit distinct coefficients between polynomial regression and clusterwise regression. These interactions capture nuanced relationships between variables that can significantly impact model performance and interpretation. The differences in coefficients between the polynomial and clusterwise regression models underscore how different modeling techniques interpret and utilize the data. Polynomial regression assumes a global relationship across all data points, while clusterwise regression adapts locally within distinct groups, potentially leading to improved predictive accuracy under heterogeneous data conditions. Moreover, the variations in coefficients between Group 0 and Group 1 within clusterwise regression highlight the importance of considering subgroup-specific effects. These differences may arise from operational nuances or varying experimental setups that affect how predictors contribute to the outcome variables (height and width). In conclusion, while polynomial regression provides a comprehensive overview of the entire dataset, clusterwise regression with subgroup analysis (Group 0 and Group 1) offers deeper insights into localized patterns and subgroup-specific variations. Understanding these differences in coefficients enhances our ability to tailor predictive models to specific operational scenarios or experimental conditions, thereby improving model robustness and interpretability in practical applications.

$$\left\{\begin{array}{cc}h\hfill & =0.3685\ast V-0.5574\ast \mathrm{WFS}-0.0015\ast \mathrm{WS}\hfill \\ \hfill & -0.0069\ast V^2-0.005\ast {\mathrm{WFS}}^2+0.0721\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & +0.0104\ast \mathrm{CTWD}\ast \mathrm{V}-0.0002\ast \mathrm{WFS}\ast \mathrm{CTWD},\hfill \\ w\hfill & =0.2819\ast V+2.8982\ast \mathrm{WFS}-0.0291\ast \mathrm{WS}\hfill \\ \hfill & +0.0156\ast V^2-0.1476\ast {\mathrm{WFS}}^2+0.0911\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & -0.0218\ast \mathrm{CTWD}\ast \mathrm{V}+0.0007\ast \mathrm{WFS}\ast \mathrm{CTWD}.\hfill \end{array}\right.$$

(13)

$$\left\{\begin{array}{cc}h\hfill & =0.2668\ast V-0.7124\ast \mathrm{WFS}+0.0092\ast \mathrm{WS}\hfill \\ \hfill & -0.0010\ast V^2-0.0348\ast {\mathrm{WFS}}^2+0.0502\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & +0.0681\ast \mathrm{CTWD}\ast \mathrm{V}-0.0001\ast \mathrm{WFS}\ast \mathrm{CTWD},\hfill \\ w\hfill & =-0.0417\ast V+3.2200\ast \mathrm{WFS}-0.0282\mathrm{WS}\hfill \\ \hfill & +0.0069\ast V^2-0.0327\ast {\mathrm{WFS}}^2-0.0155\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & -0.1145\ast \mathrm{CTWD}\ast \mathrm{V}+0.0021\ast \mathrm{WFS}\ast \mathrm{CTWD}.\hfill \end{array}\right.$$

(14)

$$\left\{\begin{array}{cc}h\hfill & =0.4181\ast V-0.2369\ast \mathrm{WFS}-0.0173\ast \mathrm{WS}\hfill \\ \hfill & -0.0149\ast V^2+0.0491\ast {\mathrm{WFS}}^2-0.0274\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & -0.0079\ast \mathrm{CTWD}\ast \mathrm{V}+0.0002\ast \mathrm{WFS}\ast \mathrm{CTWD},\hfill \\ w\hfill & =-0.4598\ast V+5.4265\ast \mathrm{WFS}-0.0137\ast \mathrm{WS}\hfill \\ \hfill & +0.0277\ast V^2-0.0919\ast {\mathrm{WFS}}^2-0.3467\ast \mathrm{V}\ast \mathrm{WFS}\hfill \\ \hfill & +0.0406\ast \mathrm{CTWD}\ast \mathrm{V}-0.0012\ast \mathrm{WFS}\ast \mathrm{CTWD}.\hfill \end{array}\right.$$

(15)

3.3. Limitations and Future Perspective

While the proposed methodology effectively integrates process knowledge and statistical techniques to create more interpretable models for AM without compromising the quality of the predictions compared to machine learning approaches, a number of limitations introduce aspects in need of further exploration. First, the model overlooks crucial layer geometry factors such as the interpass temperature and overlap distance, which significantly influence subsequent layer formation. Second, despite our feature selection efforts, multicollinearity persists due to the inherent interdependence of AM input factors. For example, the voltage and wire feed speed in WAAM processes are theoretically independent, but must be adjusted in tandem to maintain layer quality. This interdependence also explains data gaps in AM datasets. Finally, the proposed clusterwise regression, while demonstrated to work with the proposed dataset, does not guarantee continuous model behavior across cluster boundaries, particularly when considering different metal transfer modes. Addressing these limitations presents additional avenues for future research.

4. Conclusions

Predictive modeling plays a crucial role in Additive Manufacturing (AM) processes such as Wire Arc Additive Manufacturing (WAAM), allowing for effective process planning and optimization through simulation models. In fact, by accurately estimating geometry based on input factors, it is possible to enhance the precision of WAAM through the use of slicing software. However, given the complexity of AM processes such as WAAM, data-driven techniques such as deep learning models are employed in the current state-of-the-art methods, which provide limited interpretability for engineers who seek to understand the underlying physical phenomena. This represents a critical application of predictive models. In the present study, we compared traditional polynomial regression and an optimized neural network with our novel clustering-based approach. Specifically, we employed a Gaussian Mixture Model to identify two distinct clusters, with each assigned a separate polynomial regression model. This method not only leads to improved performance but also provides better insights into the geometry prediction of the WAAM process. Our findings indicate that “white-box” approaches such as clustering combined with polynomial regression can achieve results comparable to or even surpassing those of complex “black-box” machine learning models. Moreover, our proposed method does not require hyperparameter optimization, unlike many machine learning algorithms, making it more straightforward and easier to implement. Future research will extend the focus from single-layer deposition to the geometry of wall structures, addressing challenges such as neglected physical phenomena, equation dependencies, and model consistency across cluster boundaries. By incorporating complex thermal phenomena and acknowledging the limitations of our current modeling approach, we aim to develop interpretable data-driven models with high performance that are able to reduce the need for feedback control to ensure WAAM process stability.