Prediction of Groove Depth in Femtosecond Laser Ablation via Attention Mechanism and Monotonic Constraint

Abstract

Femtosecond laser ablation (FLA) is efficient for the machining of micro-groove arrays on the surface of ultrahard cutting tools. The depth of the groove determines the precision and efficiency of ablation. In this study, an “Attention-based Monotonic Physics-Guided Neural Network” (AM-PGNN) algorithm is proposed to accurately predict groove depth in the FLA of tungsten carbide (WC). The new algorithm incorporates machining parameters directly governing the energy deposition and thermal accumulation, thereby determining the prediction of the micro-groove depth generation. By embedding the physics-guided monotonic relationships of parameter depth into the learning process, a dedicated physical loss coupled with an attention mechanism to enable adaptive feature weighting is constructed, which strengthens the representation of causal dependencies. Experimental data for training and testing are obtained from the FLA of WC with different machining parameters. Comparison between AM-PGNN and typical algorithms, including a Support Vector Machine (SVM), Deep Neural Network (DNN), Convolutional Neural Network (CNN), Gradient Boosting Decision Tree (GBDT), and a conventional PGNN, demonstrates that the proposed AM-PGNN achieves superior prediction accuracy. Moreover, AM-PGNN attains a physical consistency degree (PCD) of 100%, indicating strict adherence to monotonicity consistent with the actual situation. AM-PGNN also exhibits enhanced robustness to input perturbations, as reflected by reduced standard deviation (Std) and normalized absolute deviation (NAD). Finally, AM-PGNN is shown to be applicable in the FLA of different materials through additional experiments on Cu and SiC, achieving R² values above 0.93 while maintaining a PCD of 100%.

1. Introduction

Tungsten carbide (WC) tools are widely used in industry, due to their high hardness, wear resistance, and strength retention at high temperatures [1,2]. To further enhance the performance, microstructures such as micro-dents or micro-grooves are fabricated on the tool surface, which can significantly improve the friction characteristics and wear resistance and enhance anti-adhesion capabilities [3]. However, the high hardness and brittleness of WC limit the machining efficiency and surface finish when fabricating microstructures via conventional machining methods. With the development of laser technology, femtosecond laser ablation (FLA) is emerging as a promising solution to the high-performance micromachining of ultrahard materials, because of its high precision, minimal surface damage, and non-contact processing characteristics [4,5,6]. The ultrashort pulse duration of FLA enables the near“cold processing” material removal mechanism, which reduces or even eliminates the heat-affected zone (HAZ) [7].

The dimensions of the microstructures determine the precision of FLA, which is governed by the material removal mechanism at different machining parameters. The FLA of WC involves complex processes such as multi-parameter coupling, multi-scale physical evolution, and nonlinear responses; furthermore, the ablation behavior of the WC phase and Co phase is different, which further increases the uncertainty of machining responses [8,9]. Currently, physics-based modeling and machine learning (ML)-based modeling are typical methods for analyzing FLA processes. Physical models elucidate microscopic mechanisms of laser/solid interactions from first principles. For example, Li et al. [10] revealed the evolution of electron temperature and lattice temperature based on the two-temperature model, establishing a theoretical model for FLA of fused SiO₂. Chen et al. [11] developed a multi-physics model with transient material properties to investigate the FLA of Ti6Al4V. Song et al. [12] developed a phenomenological laser ablation model to predict the ablation morphology of fused SiO₂ under single-pulse and multi-pulse FLA processes. Wang et al. [13] developed a numerical model to reveal the influence of focus position on the surface morphology during multi-pulse FLA. These physical models provide profound theoretical insights; however, substantial computational resources are required for the simulation of multi-field coupling, which is not efficient for real-time optimization or online monitoring. Furthermore, most physical models simplify complex factors such as material heterogeneity and plasma shielding effects, which impacts prediction accuracy.

In comparison, ML-based methods are increasingly applied in the monitoring or prediction of laser machining responses [14,15,16]. For example, Campanelli et al. [17] simulated the laser milling process at maximum machining speed via artificial neural networks (ANNs), with actual errors below 5%. Zhang et al. [18] developed an ANN model with laser power, scanning interval, and scanning speed as input, achieving 98% accuracy in the prediction of material removal efficiency during femtosecond laser bone drilling. Li et al. [19] proposed a method that integrates machine learning with multi-objective optimization for predicting and optimizing the surface parameters of CFRP following femtosecond laser processing. However, large amounts of datasets are required for the training of NN models, and it is time-consuming and less cost-effective to obtain high-quality datasets through FLA experiments requiring precision equipment and strict environmental control [20]. Under limited sample conditions, complex models are prone to overfitting, while simple models struggle to capture nonlinear relationships. Moreover, ML-based models can achieve good fitting results on specific datasets, meanwhile completely ignoring considerations of physical processes such as laser energy transfer and phase transformation, leading to the lack of interpretability regarding physical mechanisms [21,22]. The absence of physical information not only increases the model’s dependency on data volume, but also limits the exploration of physical laws during parameter optimization. Therefore, developing ML-based algorithms that incorporate physical prior knowledge and establishing multi-scale coupling models between process parameters and machining responses has become a key breakthrough for enhancing model generalization and mechanistic interpretability. Physics-Informed Neural Networks (PINNs), which integrate mechanistic principles with data-driven modeling, address the limitations of traditional physical models in accurately capturing complex scenarios and those of pure data-driven models that often violate physical constraints [23]. PINNs convert validated physical laws into constraint conditions and embed these constraints into an NN [24], which can effectively capture causal relationships and correlations, thereby achieving interpretable prediction results with high accuracy. Furthermore, the integration of physical laws enhances the model’s generalizability [25,26]. Recently, scholars have embedded various physical models into NNs [27,28]. Masi et al. [29] proposed a Thermodynamics-based Artificial Neural Network (TANN), which can generate physically consistent predictions. Xie et al. [30] embedded heat transfer laws into the loss function of an NN to simulate the temperature field in single-layer and multi-layer DED, achieving a mean relative error of 4.83%.

If physical mechanisms can be expressed analytically, PINN can significantly enhance the accuracy and interpretability of ML models. However, complex mechanisms involved in FLA of WC currently defy a comprehensive mathematical description, limiting the application of typical PINN. To overcome the challenge of the mathematical expression of complex physical processes, introducing physical monotonicity constraints provides an innovative approach. Zhu et al. [31] applied monotonicity constraints to an Extreme Learning Machine (ELM) model, enhancing its generalization ability. Wang et al. [32] introduced the physical relationship between superheated steam temperature (SST) and spray water flow as inequality constraints, and embedded the constraint into the loss function of a Long Short-Term Memory (LSTM) network, improving the prediction accuracy of SST. Ren et al. [33] improved the prediction accuracy of syngas composition by incorporating physical monotonicity constraints during training. Xie et al. [34] established an NN algorithm guided by extreme events and monotonic relationships to simulate the rainfall–runoff process, achieving significantly improved performance compared to a conventional NN. Zhu et al. [35] embedded three typical monotonic relationships in boiler NOx emissions into an NN model; the proposed NN outperformed traditional models in accuracy, interpretability, and generalization. Therefore, the model proposed in this study adopts the Physics-Guided Neural Network (PGNN) method, which was first proposed by Karpatne et al. [36] and fully defined by Karniadakis et al. [37] in Nature Reviews Physics. Integrating physical monotonicity as a loss term into the NN model, rather than relying on massive data to reduce costs, is a strategy to enhance model robustness in sample-limited scenarios and is particularly suitable for laser processing. In FLA, the relationships between key process parameters including laser fluence, repetition frequency, scanning speed and micro-groove depth are governed by fundamental physical mechanisms. Specifically, higher laser fluence increases energy deposition per unit area, leading to higher MRR and deeper grooves; higher repetition frequency promotes thermal accumulation, thereby increasing ablation efficiency; higher scanning speed reduces pulse overlap and energy input, resulting in the reduction in groove depth. The proposed PGNN model is designed to incorporate physical relationships into the learning process. This feature is useful in engineering problems where experimental data are often limited. By embedding such monotonicity constraints, the model can avoid unphysical predictions and improve generalizability.

In this study, a Physics-Guided Neural Network based on monotonicity constraints and an attention mechanism (AM-PGNN) is developed to achieve high-precision prediction of micro-groove depth in FLA. The key innovation of this algorithm lies in incorporating the monotonicity constraints between the principal process parameters and the micro-groove depth into the learning objective, with these constraints being directly embedded in the loss function. This integration ensures the model’s physical consistency, even when training on limited datasets. Furthermore, an attention mechanism is introduced to dynamically identify key input parameters, thereby enhancing the model’s ability to capture parameter interactions. A physical inconsistency loss function is added to the optimization objective, and Particle Swarm Optimization (PSO) is employed for training. The performance of AM-PGNN and other ML algorithms is compared to validate their advantages in prediction accuracy and model interpretability, and validation experiments on Cu and SiC further demonstrate their applicability to the FLA of different materials. The findings of this study underline significant advantages of AM-PGNN in both predictive accuracy and the potential for practical applications.

2. Methodology

2.1. Femtosecond Laser Ablation (FLA) Experiment and Data Processing

The FLA of micro-grooves was performed on a femtosecond laser platform (Figure 1). This system consists of a fiber femtosecond laser source (YSL Photonics Company Ltd., Wuhan, China) and a high-precision three-axis motion table (Precision Industry, Dongguan, China). The table has a positional accuracy of 1 μm in three directions.

Table 1. Relevant parameters of the laser processing platform.

Wavelength (λ)	Pulse Width (${\mathbf{t}}_{\mathbf{p}}$)	Focused Spot Radius (${\mathbf{r}}_0$)	Lens Focal Length (f)	Rayleigh Range (ZR)
nm	fs	μm	mm	mm
1030	300	13	100	0.524

details the laser specifications. Prior to experimentation, all samples (50 mm × 50 mm × 3 mm) were polished. Subsequently, the samples underwent 20 min ultrasonic cleaning. Laser fluence, repetition frequency, and scanning speed are the major machining parameters in FLA, and the range of parameters was optimized in our preliminary experiments. As listed in

Table 2. Design of experimental parameters.

	Variable	Description/Unit	Variation Range	Standard
Inputs	Laser fluence (LF)	Energy transferred by laser per unit area/J/cm2	[0.98, 5.53]	8
	Repetition frequency (RF)	Number of laser pulses emitted per second/MHz	[0.025, 0.3]	5
	Scanning speed (SS)	Moving speed of the laser beam/mm/s	[10, 100]	5
Outputs	Depth (DP)	Micro-groove depth/μm	[0.06, 31]

, laser fluence was set at eight levels (0.98–5.53 J); repetition frequency was set at five levels (0.025–0.3 MHz); and scanning speed was set at five levels (10–100 mm/s). The ranges of the parameters are determined following the criteria that the ablation be implemented with limited effects of plasma shielding, and the proposed monotonic constraints are applicable in such a process window. This is because FLA exhibits more complex behaviors out of the parameter range in Table 2, such as threshold effects, shielding effects, incubation effects, and saturation effects, and these effects could limit the machining efficiency of FLA. The depths of the micro-grooves were measured using a digital microscope system (VHX-7000, Osaka, Japan). The depth of each micro-groove was calculated as the average of ten measurements. The measurement uncertainty is estimated to be ±0.1 μm based on calibration standards. This uncertainty is negligible compared to the depth range of 0.06–31 μm. This ensures the reliability of the data for model training and validation. High measurement accuracy minimizes noise in the dataset. This allows the model to learn physical relationships effectively without significant bias from experimental errors. This study focuses on predicting the depth of micro-grooves from a single laser scan. Target depth in practical machining is often achieved through multiple passes. However, the removal behavior of a single scan is the physical basis for the cumulative effect.

The dataset contains 200 micro-grooves fabricated with different parameter sets. This includes 130 samples for training and 70 samples for validation. Preliminary calculations were conducted using different dataset splits, specifically 140/60, 130/70, and 120/80. For each split, the accuracy and generalization performance after repeated training were compared. The 130/70 split was ultimately selected as it showed the most stable results. Five-fold cross-validation was adopted to ensure robustness and training sufficiency. Normalization was applied to the raw data to enhance training stability. To assess reproducibility, an independent repeated-machining verification experiment was further conducted for six representative parameter groups, as described in Section 4.1. This maintains consistent data distribution and mitigates issues such as gradient explosion. Furthermore, normalization accelerates convergence and improves generalization by standardizing feature scales [38]. Min–Max Normalization is a widely adopted method that linearly maps data to a fixed interval. This ensures all features are on the same standardized scale. The formula is expressed as follows:

$$\begin{array}{c}{\mathrm{x}}^{\prime }={\displaystyle \frac{\mathrm{x}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}\end{array}$$

(1)

where x is the original value, ${\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum value of the variable in the dataset, ${\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum value of the variable in the dataset, and ${\mathrm{x}}^{\prime }$ is the normalized data. In this study, linear mapping was adopted to normalize the data to a fixed interval ([0, 1]). This preprocessed data improves quality and facilitates the modeling of relationships between input and output variables. Consequently, prediction reliability and accuracy are enhanced.

2.2. Micro-Groove Depth Prediction Scheme

Most NN-based models require large volumes of high-quality experimental data. However, acquiring such data is often challenging and costly. With limited data, NN-based models cannot be adequately trained. They become susceptible to overfitting. This leads to substantial prediction deviations during testing. To address these issues, monotonicity constraints between inputs and outputs are incorporated into the NN. Additionally, an attention mechanism is introduced. This mechanism accounts for the varying influence of each input on the output during training. Consequently, a Physics-Guided Neural Network model integrated with both an attention mechanism and monotonicity constraints was developed. It is expressed by the following equation:

$$\begin{array}{c}\widehat{y}=f_{physics}\left(\theta ,\eta ,\lambda \right)\end{array}$$

(2)

where $f_{physics}$ is the proposed NN model; $\widehat{y}$ represents the predicted value; $\theta$ denotes the model parameters; $\eta$ is the physical consistency constraint parameter; $\lambda$ is the weighting factor for each term in the loss function.

The AM-PGNN balances the influence of training data volume and physical consistency constraints on the model’s learning ability. This algorithm addresses the fundamental limitations of two conventional models. First, physical models rely strictly on theoretical assumptions. They often neglect experimental data, resulting in limited expressiveness. These models fail to capture complex nonlinear dynamics in femtosecond laser processing (e.g., plasma shielding effects). This leads to insufficient prediction accuracy. Second, NN-based models possess strong fitting capabilities. However, they depend heavily on massive datasets and lack physical constraints. With limited data, generalizability reduces and prediction errors increase. Outputs often violate basic physical laws. This results in poor physical consistency and a lack of interpretability. AM-PGNN resolves these issues by adaptively correcting theoretical deviations through data integration. This enhances the accuracy of physical models. Furthermore, physical loss functions with enforced monotonicity constraints are embedded. This eliminates physical distortions in data-driven models. Consequently, AM-PGNN achieves high-precision predictions while ensuring physical consistency, interpretability, and strong generalization. The scheme of AM-PGNN is shown in Figure 2.

3. Attention-Based Monotonic Physics-Guided Neural Network (AM-PGNN) Network Architecture

3.1. Monotonicity-Constrained Physics-Guided Neural Network

The key innovation of this study lies in embedding fundamental monotonic relationships—dictated by critical physical mechanisms such as laser energy deposition and material removal—as strong prior knowledge into the model. This approach significantly enhances the accuracy, interpretability, and generalizability of predictions, overcoming the tendency of traditional data-driven models to diverge from actual physical processes.

3.1.1. Basic Architecture

The core of AM-PGNN remains an ANN model, structured with one input layer, one hidden layer, and an output layer. This can be denoted as an $\mathrm{m}-\mathrm{n}-1$ architecture, where $\mathrm{m}$ and $\mathrm{n}$ represent the number of neurons in the input and hidden layers. Define $N$ training samples as $\left\{\right({\mathrm{x}}^{\left(\mathrm{i}\right)},{\mathrm{y}}^{\left(\mathrm{i}\right)}){\}}_{\mathrm{i}=1}^{\mathrm{N}},\mathrm{i}=1,\dots ,\mathrm{N},{\mathrm{x}}^{\left(\mathrm{i}\right)}\in {\mathrm{R}}^{\mathrm{n}}$. Here, $\mathrm{x}$ denotes the input vector of a sample, and $y$ is the corresponding output. The model’s prediction for a given input can be expressed as

$${\widehat{y}}^{\left(i\right)}=f\left(x^{\left(i\right)}\right)=\beta g\left(w{x}^{\left(i\right)}+b\right)$$

(3)

where $y^{\left(i\right)}$ represents the model’s predicted value, $f(·)$ denotes the mapping function embodied by the neural network model, $w$ is the weight matrix connecting the input layer to the hidden layer, $\beta$ is the weight vector connecting the hidden layer to the output layer, $g(·)$ is the sigmoid activation function, and $b$ is the hidden layer’s bias vector.

The training process of an NN involves optimizing the internal parameters ($w$, $\beta$, and $b$) through backpropagation. The goal is to minimize the error $L_r$ between the predicted output ${\widehat{y}}^{\left(i\right)}$ and the actual target value $y^{\left(i\right)}$, defined as

$$L_r={\displaystyle \sum _{i=1}^N}{\left(y^{\left(i\right)}-{\widehat{y}}^{\left(i\right)}\right)}^2$$

(4)

3.1.2. Formulation and Analysis of Physical Monotonic Relationships

During FLA, micro-groove depth exhibits inherent monotonic relationships with the three machining parameters, as determined by underlying physical mechanisms. From a mathematical perspective, the monotonic relationship between the model output $\widehat{\mathrm{y}}$ and an input variable $x_j$ can be formally evaluated through the sign of the corresponding partial derivative ${\displaystyle \frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }{x}_j}}$. Given an observed dataset, a strictly positive value of this derivative indicates that $\widehat{\mathrm{y}}$ increases monotonically with respect to $x_j$, whereas a negative value implies a decreasing monotonic trend. Under the proposed model structure, the analytical form of the partial derivative of $\widehat{\mathrm{y}}$ with respect to the $j$-th input variable is derived as follows:

$$\begin{array}{c}{\displaystyle \frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }{x}_j}}={\displaystyle \frac{\mathit{\partial }\left(\beta g\left(wx+b\right)\right)}{\mathit{\partial }{x}_j}}\\ ={\displaystyle \frac{\mathit{\partial }\left({\beta }_{1}g\left(w_{11}{x}_1+\cdots +w_{1n}{x}_n+b_1\right)\right)+\cdots +\mathit{\partial }\left({\beta }_{\stackrel{\sim }{N}}g\left(w_{\stackrel{\sim }{N}1}{x}_1+\cdots +w_{\stackrel{\sim }{N}n}{x}_n+b_{\stackrel{\sim }{N}}\right)\right)}{\mathit{\partial }{x}_j}}\\ ={\displaystyle \sum _{k=1}^{\stackrel{\sim }{N}}}{\beta }_k{w}_{kj}g\left(w_{k}x+b_k\right)\left(1-g\left(w_{k}x+b_k\right)\right),j=\mathrm{1,2},\dots ,n\end{array}$$

(5)

The physical mechanisms of FLA were analyzed to determine the monotonic relationships between the micro-groove depth and laser parameters. These parameters are laser fluence, repetition frequency, and scanning speed. This analysis establishes specific monotonicity formulations.

Figure 3 illustrates the physical principles of FLA. Laser fluence (LF) determines the energy density (J/cm²) deposited per unit area. Increasing fluence raises the photon density. This enhances multiphoton absorption and avalanche ionization. Consequently, the density of excited free electrons in the material surface layer increases. These electrons transfer energy to the lattice via electron–phonon coupling. The local temperature rises rapidly when the lattice energy density exceeds the ablation threshold. This triggers non-equilibrium phase transitions. Higher fluence heats and removes more material per unit area. Removal occurs through mechanisms such as thermal expansion, phase explosion, or Coulomb explosion. This process expands the molten and vaporized regions and deepens the micro-groove (Figure 3b). This monotonic relationship is mathematically expressed as

$${\displaystyle \frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }LF}}\left(x\right)>0$$

(6)

Repetition frequency (RF) determines the number of pulses irradiating a single point per unit time. High RF results in pulse intervals shorter than the thermal diffusion time. Consequently, heat from a previous pulse does not fully diffuse before the next pulse arrives. This causes continuous heat accumulation in the processing zone. Accumulated heat raises the local temperature significantly. Subsequent pulses reach the ablation threshold with lower energy density. This effectively reduces the apparent ablation threshold. Furthermore, sustained high temperatures maintain the molten region for a longer duration. This expands the molten scope and promotes secondary ablation. Thermal accumulation increases the effective energy deposition per pulse. Therefore, increasing RF enhances material removal efficiency and increases micro-groove depth (Figure 3c). This monotonic relationship is mathematically expressed as

$${\displaystyle \frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }RF}}\left(x\right)>0$$

(7)

Scanning speed (SS) determines the velocity of the laser focus across the surface. It directly affects the spatial overlap of adjacent pulses and the total laser interaction time. Lower scanning speed results in longer laser dwell time per unit length. This significantly increases the number of pulses at the same spatial location:

$$\mathrm{N}={\displaystyle \frac{\mathrm{D}·\mathrm{f}}{\mathrm{v}}}$$

(8)

where $\mathrm{D}$ is the beam spot diameter, $\mathrm{f}$ is the repetition frequency, and $\mathrm{v}$ is the scanning speed.

More pulses result in a higher total energy deposition. At low scanning speeds, pulses overlap significantly. Energy is deposited multiple times in the same area. This intensifies thermal accumulation and material removal. Extended interaction time allows molten material to be expelled from the ablation crater. Surface tension and vapor pressure drive this expulsion. Consequently, recasting and redeposition are reduced. Subsequent pulses act effectively on fresh material layers. This deepens the micro-groove (Figure 3d). Conversely, high scanning speeds result in low pulse overlapping. The energy received per unit area decreases drastically. The interaction time is insufficient for effective material expulsion before resolidification. This increases the recast layer thickness and limits ablation depth. Thermal accumulation is also weakened. Therefore, a monotonically decreasing relationship exists between scanning speed and micro-groove depth:

$${\displaystyle \frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }SS}}\left(x\right)<0$$

(9)

To verify the monotonic relationships among laser fluence, repetition frequency, scanning speed, and groove depth within the selected process window, a controlled variable analysis was conducted for LF, RF, and SS, respectively. Specifically, when examining the effect of a single variable on groove depth, the remaining variables were held constant. The data were then partitioned into several subsequences, and the corresponding output sequence ({$y_i$}) was arranged in ascending order. The first-order differences were subsequently computed as

$$∆y_i=y_{i+1}-y_i,\hspace{1em}\hspace{1em}i=1,2,\dots ,n-1$$

(10)

The consistency of the signs of the differences is used as the criterion for monotonicity. When all $∆y_i>0$ (or $∆y_i<0$), the relationship between the variable and the output is considered to be monotonically increasing (or decreasing). In this context, $\uparrow$ indicates a positive correlation between the variable and the groove depth, whereas $\downarrow$ indicates a negative correlation. As shown in

Table 3. Monotonic relationships between each variable and groove depth.

Variable	Total Number of Groups	Monotonic Group Number	Monotonic Direction
LF	25	25	$\uparrow$
RF	40	40	$\uparrow$
SS	40	40	$\downarrow$

, all groups exhibit monotonicity, indicating that each variable exhibits a stable monotonic relationship with the groove depth ($y$).

In addition, several groups were selected for visualization. As shown in Figure 4, the results are consistent with the conclusions drawn from the overall analysis.

The AM-PGNN model incorporates these physical relationships into the monotonicity formulation (Equation (5)).

3.1.3. Physical Information Loss Function

Building upon conventional neural networks, this study fully incorporates prior monotonic relationships between the input parameters and the output. Two additional constraints are introduced into the loss function during training: structural loss $L_s$ and physical inconsistency loss $L_p$, which enhance the model’s performance and interpretability when predicting results for unknown samples, as described by Equations (11) and (12):

$$L_s=\sum _{j=1}^n\sum _{k=1}^{\stackrel{\sim }{N}}(w_{jk}{)}^2+\sum _{k=1}^{\stackrel{\sim }{N}}({\beta }_k{)}^2$$

(11)

$$L_p=\sum _{j=1}^n\sum _{i=1}^{T_j}{\displaystyle \frac{1-m_j^{\left(i\right)}\times \mathrm{sign}\left(\frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }{x}_j}\left(x^{\left(i\right)}\right)\right)}{2}}$$

(12)

where $\mathrm{sign}\left(\theta \right)=-1$ if $\theta <0$, and $\mathrm{sign}\left(\theta \right)=1$ if $\theta \ge 0$. ${\mathrm{x}}^{\left(\mathrm{i}\right)}$ denotes the $i$-th training sample; ${\mathrm{m}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}\in \left\{1,-1\right\}$ is the monotonicity factor indicating the expected direction between the $j$-th input and output: 1 for positive monotonic relationships and −1 for the negative; ${\mathrm{T}}_{\mathrm{j}}$ is the number of samples considered for the monotonic constraint associated with input $x_j$. Intuitively, if $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}{\displaystyle \frac{\mathit{\partial }\widehat{\mathrm{y}}}{\mathit{\partial }{\mathrm{x}}_{\mathrm{j}}}}$ agrees with the prescribed direction ${\mathrm{m}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}$, the corresponding penalty is suppressed; otherwise, the penalty is activated, thereby discouraging violations of the physical monotonic trend. The total loss function for the AM-PGNN can be expressed as Equation (13):

$$\begin{array}{c}L=L_{data}+{\lambda }_s{L}_s+{\lambda }_p{L}_p\\ ={\displaystyle \sum _{i=1}^N}({\widehat{y}}^{\left(i\right)}-y^{\left(i\right)}{)}^2+{\lambda }_s\left\{{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^{\stackrel{\sim }{N}}}(w_{jk}{)}^2+{\displaystyle \sum _{k=1}^{\stackrel{\sim }{N}}}({\beta }_k{)}^2\right\}+{\lambda }_p{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{i=1}^{T_j}}{\displaystyle \frac{1-m_j^{\left(i\right)}\times \mathrm{sign}\left(\frac{\mathit{\partial }\widehat{y}}{\mathit{\partial }{x}_j}\left(x^{\left(i\right)}\right)\right)}{2}}\end{array}$$

(13)

where ${\lambda }_s$ and ${\lambda }_p$ denote the weightings associated with structural loss and physical inconsistency loss, respectively. Data-driven regression loss $L_{data}$ quantifies the discrepancy between observed and predicted outcomes. Structural loss $L_s$ functions as a regularization component, aiming to mitigate overfitting while improving the model’s generalizability. In addition, physical inconsistency loss $L_p$ quantifies the extent to which the learned model violates the prescribed physical monotonicity constraints.

3.2. Attention Mechanism

In many operating regimes, the input variables contribute unequally to the target response. To account for such heterogeneous feature contributions, we incorporate a feature-wise attention gating module inspired by Cai et al. [39] into the physics-guided monotonic neural network. Specifically, given the input vector $\mathrm{X}=[{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{\mathrm{M}}]$, the gate produces non-negative attention coefficients $\alpha \in {\left(0,1\right)}^M$ and yields a reweighted representation ${\mathrm{X}}^{\prime }=\alpha \odot \mathrm{X}$, which is subsequently fed into the predictive network. In contrast to monotonicity-based constraints, which enforce a consistent input–output trend without assigning weights to individual input features, the proposed gating mechanism enables adaptive assignment of feature weights, whereas monotonicity constraints maintain directionality consistent with the underlying physical meaning. The combination improves predictive accuracy and robustness, especially under limited-data settings.

It is important to note that, while the weighted feature vector $X^{\prime }$ serves as the input for predicting $\widehat{y}$, physical inconsistency constraints are still based on the partial derivatives of original inputs. Although $X^{\prime }$ undergoes numerical changes, the model maintains three independent channels for LF, RF, and SS. For example, when calculating ${\displaystyle \frac{\mathit{\partial }\widehat{\mathrm{y}}}{\mathit{\partial }\mathrm{L}\mathrm{F}}}$, the network gradient can still backpropagate to the original LF through ${\alpha }_1\odot LF$.

3.3. Training the AM-PGNN Model

Preliminary calculations demonstrated that the non-continuous and non-differentiable nature of the loss function compromised the stability of gradient-based optimizers such as Adam. As shown in Figure 5, the gradient-free optimization algorithm (PSO) provided more stable convergence. Similar strategies have been applied to optimization problems with non-differentiable constraints [40]. The final configuration determined through five-fold cross-validation consists of a network structure of 3-22-1, with a structural loss weight ${\lambda }_s$ of 0.001, a physical inconsistency loss weight ${\lambda }_p$ of 0.06, a population size of 100 particles, and a maximum of 1000 iterations. An early-stopping strategy was applied when the objective function converged, which may terminate the optimization before reaching the maximum number of iterations. This approach allows the model to converge to a solution that balances data fitting accuracy with physical constraint requirements.

In summary, physically monotonic relationships between laser fluence, repetition frequency, scanning speed, and micro-groove depth are embedded as physical consistency constraints in the proposed prediction model, together with an adaptive feature-weighting attention mechanism. The overall AM-PGNN architecture is shown in Figure 6.

3.4. Metrics for the Evaluation of the Model’s Performance

To evaluate the performance of the AM-PGNN, this study benchmarked typical algorithms: SVM, DNN, CNN, GBDT, and PGNN. The hyperparameters for SVM, GBDT, PGNN, and AM-PGNN were optimized using PSO, while DNN and CNN were optimized using the Adam optimizer. Furthermore, DNN, PGNN, and AM-PGNN share the same underlying network architecture. Each model was optimized using the corresponding strategy to achieve its best performance. To ensure a fair and reliable comparison, each model was trained and evaluated ten times with different random initializations. The average values of the evaluation metrics were reported. In addition, the Std of the results from the ten runs was calculated to reflect the stability of the models. Numerical computations were carried out in MATLAB R2023b, and prediction accuracy is assessed using three standard statistical indicators:

Coefficient of determination ($R^2$) reflects the model’s fitting accuracy to experimental data. A higher $R^2$ indicates better alignment between the model and experimental data. $R^2$ is defined as

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left(y^{\left(i\right)}-{\widehat{y}}^{\left(i\right)}\right)}^2}{{\displaystyle {\sum }_{i=1}^N}{\left(y^{\left(i\right)}-\overline{y}\right)}^2}}$$

(14)

where $N$ represents the total number of samples; $y^{\left(i\right)}$ and ${\widehat{y}}^{\left(i\right)}$ correspond to the measured and predicted responses of the $i$-th sample, respectively; and $\overline{\mathrm{y}}$ denotes the mean value of the measured observations.

Root Mean Square Error ($RMSE$) represents the absolute fit between the model and the data and possesses the property of using the same units as the response variable. $RMSE$ is defined as follows

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\left(y^{\left(i\right)}-{\widehat{y}}^{\left(i\right)}\right)}^2}$$

(15)

The standard deviation ($Std$) is used to evaluate the stability of each model. It is defined as follows

$$Std=\sqrt{{\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{i=1}^n}{\left(y_i-\overline{y}\right)}^2}$$

(16)

The physical consistency degree (PCD) evaluates the agreement between the model’s predictions and the actual physical monotonicity rules. PCD can be calculated using the following equation:

$$\mathrm{P}\mathrm{C}\mathrm{D}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}}{\displaystyle \frac{1-{\beta }_{\mathrm{j}}\times \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(\frac{\mathit{\partial }\mathrm{o}}{\mathit{\partial }{\mathrm{x}}_{\mathrm{j}}}\left({\mathrm{x}}^{\left(\mathrm{i}\right)}\right)\right)}{2}}\times 100\mathrm{\%}$$

(17)

where $x^{\left(i\right)}$ refers to the $i$-th data sample and $T$ denotes the number of samples that satisfy the monotonic relationship between the $j$-th input variable and output. ${\beta }_j$ is defined to encode the direction of physical monotonicity: −1 for positive monotonic relationships and 1 for the negative.

For computational convenience, the partial differential equation is solved using finite difference approximation:

$${\displaystyle \frac{\mathit{\partial }o}{\mathit{\partial }{x}_j}}\approx \underset{\Delta x_j\to 0}{lim}{\displaystyle \frac{f\left(x_1,x_2,\cdots ,x_{j-1},x_j+\Delta x_j,x_{j+1},\cdots ,x_n\right)-f\left(x_1,x_2,\cdots ,x_{j-1},x_j,x_{j+1},\cdots ,x_n\right)}{\Delta x_j}}$$

(18)

where $\Delta x_j$ denotes a small perturbation approaching zero.

4. Results and Discussion

This section first evaluates the reproducibility of the experimental data and the predictability of the model. Subsequently, the performance of the developed model is comprehensively evaluated from four perspectives: (1) three metrics are employed to analyze models’ predictive ability; (2) the interpretability and generalization ability of the AM-PGNN model are investigated, focusing on the monotonic behavior; (3) a robustness analysis is performed to assess model’s sensitivity to input perturbations and parameter uncertainty, where the stability of the predictions is quantitatively characterized; (4) additional FLA experiments are conducted to further validate the generalization ability of AM-PGNN under different machining scenarios.

4.1. Statistical Analysis of the Measurement of Micro-Groove Depth

To rigorously validate the reproducibility of the experimental data and the predictability of the model, repeated-machining verification was performed using representative process parameters (

Table 4. Experimental parameter design for repeated-machining verification.

	Param Group	LF (J/cm2)	RF (MHz)	SS (mm/s)
Minor-parameter regime	P1	3.98	0.025	0.02
	P2	5.25	0.05	0.01
	P3	3.8	0.1	0.02
High-parameter regime	P4	4.33	0.2	0.01
	P5	5.45	0.2	0.01
	P6	4.2	0.3	0.01

) [41]. Under identical laser parameter combinations, three micro-grooves were machined for each set, designated G1–G3. For each groove, 30 measurement points ($i\in \left\{1, 2, \dots , 30\right\}$) were selected at equal intervals from the starting point to the endpoint. A measurement strategy consistent with the dataset construction was employed to minimize stochastic reading fluctuations.

For each groove, the longitudinal coordinate is normalized as $\mathrm{s}\in \left[0,1\right]$. The depth of the $i$th measurement is denoted as $D_k\left(s_i\right)$ (where $k=1,2,3$ corresponds to G1–G3). The representative overall depth for each groove is defined as the arithmetic mean along the groove ${\overline{\mathrm{D}}}_{\mathrm{k}}$:

$${\overline{\mathrm{D}}}_{\mathrm{k}}={\displaystyle \frac{1}{30}}{\displaystyle \sum _{\mathrm{i}=1}^{30}}{\mathrm{D}}_{\mathrm{k}}\left({\mathrm{s}}_{\mathrm{i}}\right)$$

(19)

Depth variations along the groove are governed by both the average MRR and factors including thermal accumulation, material heterogeneity (e.g., WC/Co phase differences), molten recast, and spatter redeposition, all of which induce local fluctuations. To determine whether these fluctuations constitute “random noise” or “systematic patterns,” a first-order difference analysis is performed on the depth sequence ${\mathrm{D}}_{\mathrm{k}}\left({\mathrm{s}}_{\mathrm{i}}\right)$ for each groove:

$$\Delta {\mathrm{D}}_{\mathrm{k}}\left(\mathrm{i}\right)={\mathrm{D}}_{\mathrm{k}}\left({\mathrm{s}}_{\mathrm{i}+1}\right)-{\mathrm{D}}_{\mathrm{k}}\left({\mathrm{s}}_{\mathrm{i}}\right),\hspace{0.33em}\mathrm{i}=\mathrm{1,2},\dots ,29$$

(20)

Figure 7 illustrates the results for P1, P3, and P5, which are selected based on their representative nature in covering the range of process parameters across shallow and deep grooves. Results show that, for all three grooves under the same parameters, $\Delta D_k$ exhibits zero-centered random fluctuations. The mean values are close to zero, and the difference distributions are approximately symmetric. Furthermore, the Autocorrelation Function (ACF) shows no significant peaks at non-zero lags, indicating that the depth fluctuations along the groove are primarily zero-mean random perturbations, with no evidence of systematic drift or periodic regularity. These findings suggest that local depth variations are mainly caused by microscopic disturbances and do not constantly increase or decrease along the scanning direction. Consequently, ${\overline{D}}_k$ serves as a stable, representative scalar response, and the fluctuations in depth can be considered zero-mean random errors superimposed on the actual removal depth, providing a statistical basis for subsequent reproducibility assessment and model predictability verification.

To unify the comparison standards between the repeated experimental results and outputs, this study defines the experimental benchmark depth as the grand mean of the three grooves’ depths:

$${\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{\mathrm{k}=1}^3}{\overline{\mathrm{D}}}_{\mathrm{k}}$$

(21)

Machining reproducibility is quantified by the inter-groove standard deviation:

$${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}=\sqrt{{\displaystyle \frac{1}{2}}{\displaystyle \sum _{\mathrm{k}=1}^3}{\left({\overline{\mathrm{D}}}_{\mathrm{k}}-{\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}^2}$$

(22)

Fluctuation along the groove is characterized by the intra-groove average standard deviation:

$${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{\mathrm{k}=1}^3}\sqrt{{\displaystyle \frac{1}{29}}{\displaystyle \sum _{\mathrm{i}=1}^{30}}{\left({\mathrm{D}}_{\mathrm{k}}\left({\mathrm{s}}_{\mathrm{i}}\right)-{\overline{\mathrm{D}}}_{\mathrm{k}}\right)}^2}$$

(23)

To facilitate comparisons across different depth magnitudes, the coefficient of variation ($\mathrm{C}\mathrm{V}$) is introduced:

$$\mathrm{C}\mathrm{V}={\displaystyle \frac{\sigma }{{\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}$$

(24)

Statistical analysis is performed on the three repeated micro-grooves G1–G3. As shown in

Table 5. Statistical analysis of depth, reproducibility, and fluctuation for different parameter sets.

Parameter Set	${\mathit{D}}_{\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{\sigma }}_{\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}}$	${\mathit{\sigma }}_{\mathit{i}\mathit{n}\mathit{t}\mathit{r}\mathit{a}}$	${\mathit{C}\mathit{V}}_{\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}}$	${\mathit{C}\mathit{V}}_{\mathit{i}\mathit{n}\mathit{t}\mathit{r}\mathit{a}}$
P1	1.067	0.046	0.106	0.043	0.099
P2	4.12	0.074	0.129	0.018	0.031
P3	4.921	0.098	0.133	0.02	0.027
P4	21.211	0.275	0.597	0.013	0.028
P5	22.898	0.279	0.75	0.012	0.03
P6	30.575	0.663	0.916	0.022	0.03

, the ${\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ values for the six parameters sets are 1.067, 4.12, 4.921, 21.211, 22.898, and 30.575 µm. ${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}$ ranges from 0.046 to 0.663 µm, and ${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}$, ranges from 0.106 to 0.916 µm. Normalizing these values using the coefficient of variation, the ${\mathrm{C}\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}$ values are between 0.012 and 0.043, and the ${\mathrm{C}\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}$ values are between 0.027 and 0.099. With the exception of the shallow groove condition (P1), ${\mathrm{C}\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}$ generally remains within the 0.012–0.022 range, indicating excellent consistency in average depth among repeated grooves under identical parameters. The relatively large ${\mathrm{C}\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}$ for the P1 group (0.099) is primarily attributed to the increased relative impact of measurement uncertainty at smaller depths. In contrast, for grooves with larger depth, although the absolute fluctuation amplitude (${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}$) increases with depth, the normalized ${\mathrm{C}\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}$ remains stable at approximately 0.028–0.033, suggesting that random fluctuations along the path remain proportional to the depth magnitude.

In the predictability verification, the benchmark depth ${\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ derived from three repeated grooves serves as the ground truth, and the model output is denoted as ${\mathrm{D}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$. The prediction error is defined as

$$\epsilon ={\mathrm{D}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-{\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$$

(25)

Both the absolute error $|\epsilon |$ and the relative error $\epsilon \mathrm{\%}$ are reported:

$$\epsilon \mathrm{\%}={\displaystyle \frac{|\epsilon |}{{\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}\times 100\mathrm{\%}$$

(26)

An engineering depth control tolerance of ±2 µm is adopted as the acceptance criterion [42]; predictions are considered satisfactory if $\left|\epsilon \right|<2$ µm.

The results in

Table 6. Model prediction error analysis for different parameter sets.

Param Group	${\mathit{D}}_{\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{D}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}$	$\mathit{\epsilon }$	$\left|\mathit{\epsilon }\right|$	$\mathit{\epsilon }\mathbf{\%}$	Within $\pm 2$ μm
P1	1.067	1.098	0.031	0.031	2.89	Yes
P2	4.12	4.085	−0.034	0.034	0.834	Yes
P3	4.921	5.003	0.082	0.082	1.671	Yes
P4	21.211	22.677	1.466	1.466	6.913	Yes
P5	22.898	23.486	0.588	0.588	2.57	Yes
P6	30.575	31.52	0.945	0.945	3.09	Yes

demonstrate that model predictions for all six parameter groups satisfy the ±2 µm tolerance. For P1–P3, the absolute errors $\left|\epsilon \right|$ are 0.031, 0.034, and 0.082 µm, indicating that the model maintains high absolute accuracy at small depth magnitudes. It is worth noting that the groove with small depth (e.g., P1) exhibits higher sensitivity in terms of relative error (2.89%), which is mainly due to the amplification of absolute deviations of the same order of magnitude when divided by a small benchmark depth. Therefore, $\epsilon \mathrm{\%}$ serves to quantify the degree of relative deviation across different depth magnitudes, aiding in assessing model sensitivity under shallow conditions. For P4–P6, errors range from 0.588 to 1.466 µm, which are significantly below the ±2 µm tolerance, with relative errors between 2.57% and 6.91%. This means that the model possesses good predictability even under conditions of intensive energy input. In summary, the comparison between the experimentally defined $D_{ref}$ and the model-predicted $D_{pred}$ validates the model’s capability to achieve depth prediction within engineering tolerances across the full parameter range. Furthermore, the analysis of relative error $\epsilon \mathrm{\%}$ provides a sensitivity assessment for different depth magnitudes, facilitating refined judgment for engineering design and experimental control.

The rigorous experimental verification of reproducibility and model predictability confirms the high reliability and precision of the proposed machining process. The results of repeated machining under identical parameter sets demonstrate excellent consistency in groove depths. This indicates that the machining process produces highly reproducible results under varying process parameters. Additionally, the model’s predictability is verified through comparison with the experimental benchmark depth. The model successfully predicts groove depths within the engineering tolerance of ±2 µm across all parameter groups, demonstrating its robust accuracy and ability to maintain precision even under extreme parameter conditions. The relative error analysis further highlights the model’s capability to handle both shallow and deep groove conditions, ensuring reliability for diverse applications.

4.2. Performance Analysis and Comparison of Models

Figure 8 shows a comparison of the prediction performance of SVM, DNN, CNN, GBDT, PGNN, and AM-PGNN during both training and testing. Specifically, the prediction accuracy of SVM and CNN was significantly lower than that of other algorithms during training, with most predictions falling outside the deviation lines, while DNN, GBDT, PGNN, and AM-PGNN all demonstrated strong fitting capabilities. In testing, AM-PGNN showed significant advantages, with the vast majority of its prediction points located within the deviation lines. In contrast, the prediction accuracy of other models was relatively lower, with their prediction points showing varying degrees of dispersion and some exceeding the deviation lines (Figure 8b–d).

Table 7. Comparison of performance among various models.

Algorithm	R2		RMSE (μm)
	Train	Test	Train	Test
SVM	0.878 ± 0.098	0.841 ± 0.104	2.412 ± 0.567	2.495 ± 1.051
DNN	0.938 ± 0.045	0.920 ± 0.049	1.714 ± 0.332	1.775 ± 0.699
CNN	0.926 ± 0.018	0.910 ± 0.029	1.823 ± 0.133	1.941 ± 0.580
GBDT	0.896 ± 0.059	0.829 ± 0.078	1.801 ± 0.387	2.068 ± 0.667
PGNN	0.923 ± 0.023	0.915 ± 0.031	1.909 ± 0.109	1.822 ± 0.358
AM-PGNN	0.940 ± 0.014	0.937 ± 0.020	1.695 ± 0.088	1.573 ± 0.165

provides detailed information on R² and RMSE for all six models, demonstrating that the AM-PGNN model achieves excellent prediction accuracy for both the training and test sets.

By incorporating an attention mechanism, AM-PGNN achieves adaptive weighting of critical information within the input features, thereby enhancing its capability to identify and model causal relationships between inputs and outputs. Figure 9 presents the attention weights of the three input parameters as computed by the attention mechanism. The results indicate that SS has the highest weight at 40.1%, followed by RF at 34.5%, and LF at 25.4%. Furthermore, as listed in Table 7, AM-PGNN achieves an R² of 93.7% and an RMSE of $1.573 \mu \mathrm{m}$ on the test set, significantly outperforming the PGNN model without the attention mechanism (R² of 91.5% and RMSE of $1.822 \mu \mathrm{m}$). This means that the introduction of an attention mechanism substantially enhances model accuracy and predictive performance.

To further demonstrate the prediction performance of PGNN, Figure 10 comprehensively evaluates the prediction performance of each model on both training and test using the Taylor diagram. This analytical method simultaneously displays three metrics through trigonometric geometry: correlation coefficient, standard deviation, and root mean square deviation (RMSD). These metrics share a key geometric relationship derived from the cosine theorem.

The correlation coefficient R corresponds to the cosine of angle $\theta$:

$$\mathrm{R}=\cos \theta$$

(27)

where $\theta$ is the angle of the model point relative to the observation point (in polar coordinates). Through the cosine theorem, the relationship between RMSD, R², and Std is given by

$${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}}^2={\sigma }_{\mathrm{f}}^2+{\sigma }_{\mathrm{o}}^2-2{\sigma }_{\mathrm{f}}{\sigma }_{\mathrm{o}}\mathrm{R}$$

(28)

where ${\sigma }_f$ is the standard deviation of the model predictions, and ${\sigma }_o$ is the standard deviation of the observed data. In the Taylor diagram, the observation point is located at (${\sigma }_o$,0), while the model point is located at (${\sigma }_f\cos \theta$, ${\sigma }_f\sin \theta$). RMSD is the straight-line distance between these two points, and the cosine theorem describes the relationship between this distance and $\sigma \&\theta$. An ideal model point should be close to the reference point (small RMSD), with a high R and an appropriate $\sigma$ (close to ${\sigma }_o$).

The results for the training are shown in Figure 10a. The standard deviation of the observed data is $6.90 \mu \mathrm{m}$. The standard deviation of DNN is closest to the observed value of 6.51 μm, while GBDT shows the smallest standard deviation of 4.68 μm. In terms of correlation, CNN, GBDT, and AM-PGNN show the highest correlation of 0.97, while SVM shows the lowest correlation of 0.94. Also, the RMSDs of DNN and AM-PGNN are the lowest, $1.69 \mu \mathrm{m}$ and $1.68 \mu \mathrm{m}$, respectively, while SVM and GBDT have the highest RMSDs $2.40 \mu \mathrm{m}$. For testing (Figure 10b), AM-PGNN’s three metrics are closest to the observation point $\mathrm{R}=0.97$, $\mathrm{S}\mathrm{t}\mathrm{d}=5.71 \mu \mathrm{m}$, $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}=1.52 \mu \mathrm{m}$. SVM performs the worst due to its significantly lower R² and the largest RMSD. Overall, AM-PGNN model achieves the highest prediction accuracy. By incorporating the attention mechanism, its accuracy shows a slight improvement compared to PGNN, confirming the initial expectation that AM-PGNN can effectively predict the depth of micro-grooves processed by femtosecond laser.

4.3. Analysis of Interpretability and Generalizability

Section 3.4 details the quantification of input–output monotonicity using PCD. Figure 11 presents the results. Both PGNN and AM-PGNN achieve a PCD value of 100% during training and testing. These models fully satisfy the prescribed monotonic constraints over the evaluated samples. For SVM, the PCD values for the three inputs in the test results are 67.14%, 81.43%, and 81.43%. The overall PCD is 76.67%. Approximately one quarter of the predictions violate the actual physical monotonicity rules. For the DNN model, the PCD for SS is 100%. However, the overall PCD is 94.76%. A small number of samples fail to satisfy the monotonicity rules. Both CNN and GBDT models exhibit an approximate 17% of data points that violate the actual physical monotonicity rules. The prediction accuracy of DNN and CNN is comparable to PGNN and AM-PGNN. However, their interpretability and generalization are inferior. This reveals a drawback of conventional models. They often sacrifice physical consistency for high-precision fitting. The results in Figure 11 demonstrate the limitations of relying solely on statistical metrics. The PCD method can effectively determine whether the model is consistent with the true physical priors, thereby enabling an assessment of its interpretability and generalization capability.

To further validate the physical consistency of the proposed model in capturing input–output relationships, this study employs partial dependence plots (PDPs) to analyze the significance of target features on prediction outcomes during testing. PDPs isolate the independent effect of feature $x_s$ on micro-groove depth predictions by fixing $x_s$ and marginalizing the output over the joint distribution of other features $x_c$. PDPs consider the target response as a function of $x_s$ [43]:

$${\widehat{\mathrm{f}}}_{\mathrm{s}}\left({\mathrm{x}}_{\mathrm{s}}\right)={\mathrm{E}}_{{\mathrm{x}}_{\mathrm{c}}}\left[\widehat{\mathrm{f}}\left({\mathrm{x}}_{\mathrm{s}},{\mathrm{x}}_{\mathrm{c}}\right)\right]=\int \widehat{\mathrm{f}}\left({\mathrm{x}}_{\mathrm{s}},{\mathrm{x}}_{\mathrm{c}}\right)\mathrm{d}\mathrm{P}\left({\mathrm{x}}_{\mathrm{c}}\right)$$

(29)

Analysis of the monotonic relationships between micro-groove depth and laser parameters using univariate PDPs (Figure 12) reveals significant differences in the ability of different models to represent physical mechanisms. The prediction results of SVM (Figure 12a) show that micro-groove depth first increases and then decreases with increasing laser fluence, which contradicts the physical mechanism of FLA. Higher laser fluence enhances energy per unit area, thereby intensifying melting and vaporization effects and ultimately increasing ablation depth. Although the overall trend of DNN, CNN, and GBDT (Figure 12b–d) reflects the monotonic relationship of micro-groove depth and laser fluence, some data points still deviate from physical monotonicity, indicating that these data points fail to strictly adhere to physical rules. In contrast, due to the incorporation of monotonicity constraints, the prediction results of PGNN and AM-PGNN (Figure 12e,f) demonstrate consistency with the actual physical monotonicity rules: micro-groove depth increases with laser fluence, but the incremental depth gradually decreases as fluence rises. This aligns with the effect of plasma shielding. The increased laser fluence leads to higher plasma density, causing partial energy to be shielded by the plasma and not fully reach the bottom of the groove, thereby reducing the incremental depth [8]. Additionally, micro-groove depth increases regularly with a higher repetition frequency and lower scanning speed. These observations prove that the proposed model can extrapolate learned monotonic relationships, thereby enhancing its generalizability.

As shown in Figure 13, a comparison of the training processes between the two models reveals that although the total loss of DNN gradually decreases during training, the loss induced by physical inconsistency persists. This means that DNN struggles to achieve physical consistency. In contrast, while the total loss of AM-PGNN converges, its physical loss remains consistently zero. This strictly ensures physical consistency and makes its predictions fully compliant with the constraints imposed by physical laws.

To further investigate the synergistic effects of the laser parameters on micro-groove depth, this study employs bivariate partial dependence plots for visualizing parameter interactions. Figure 14 illustrates the performance differences among the six models through contour plots. The predictions of SVM, DNN, and CNN (Figure 14a–c) show that micro-groove depth increases monotonically with higher repetition frequency. However, in high-laser-fluence regions, the depth exhibits non-monotonic fluctuations, and the predictions of both models deviate from the physical mechanisms of FLA. The GBDT model exhibits non-monotonic fluctuations in both low- and high-laser-fluence regions. In contrast, the predictions of the PGNN and AM-PGNN models (Figure 14e,f) demonstrate that micro-groove depth increases with both higher laser fluence and repetition frequency.

Further analysis of the parameter interactions (Figure 15) reveals that both PGNN and AM-PGNN accurately capture the synergistic enhancement effect of increasing laser fluence and decreasing scanning speed on micro-groove depth. This aligns with the physical mechanism that higher laser fluence and lower scanning speed collectively contribute to larger micro-groove depth (Figure 15e,f). In contrast, SVM fails to demonstrate a significant correlation between laser fluence and micro-groove depth (Figure 15a), while the contour distributions predicted by DNN, CNN, and GBDT exhibit non-monotonic characteristics in specific regions (Figure 15b–d). Experimental results confirm that, by incorporating physical monotonicity, both PGNN and AM-PGNN enhance interpretability and generalizability, making the predictions consistent with the actual physical monotonicity rules.

4.4. Analysis of Sensitivity and Robustness

In practical laser processing, the nominal input parameters inevitably deviate from their setpoints due to equipment uncertainty and process fluctuations. To evaluate the sensitivity and robustness of the proposed AM-PGNN with respect to such perturbations, we perform a Monte Carlo perturbation test on the held-out dataset. Specifically, for each test sample $X$, we generate perturbed inputs $\stackrel{\sim }{X}$ by injecting small zero-mean Gaussian noise to each parameter, as expressed by

$${\stackrel{\sim }{\mathrm{x}}}_{\mathrm{j}}={\mathrm{x}}_{\mathrm{j}}\left(1+{\mathsf{ϵ}}_{\mathrm{j}}\right)$$

(30)

where ${ϵ}_j\sim \mathcal{N}\left(0,{\sigma }^2\right)$ and $\sigma$ is chosen such that the perturbation magnitude approximately corresponds to a $\pm 1\%$ fluctuation. For each original sample, $K$ perturbed realizations are evaluated to obtain an output distribution ${\left\{{\widehat{y}}^{\left(k\right)}\right\}}_{k=1}^K$. The robustness of the model is quantified using two metrics: (i) the prediction standard deviation $Std\left(\widehat{y}\right)$ across perturbations and (ii) the normalized absolute deviation ($\mathrm{N}\mathrm{A}\mathrm{D}$), given by

$$\mathrm{N}\mathrm{A}\mathrm{D}={\displaystyle \frac{1}{\mathrm{K}}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{K}}}{\displaystyle \frac{\left|{\widehat{\mathrm{y}}}^{\left(\mathrm{k}\right)}-\widehat{\mathrm{y}}\right|}{\left|\widehat{\mathrm{y}}\right|+\epsilon }}$$

(31)

where $\widehat{y}$ is the baseline prediction and $\epsilon$ is a small stabilizer. A lower $Std\left(\widehat{y}\right)$ and $NAD$ indicate reduced noise amplification and improved robustness.

As shown in Figure 16, when subjected to the same $\pm 1\%$ input perturbation, AM-PGNN demonstrates noticeably smaller fluctuations in its predictions than both DNN, CNN, GBDT, and PGNN without an attention mechanism. In particular, AM-PGNN maintains low values of both the $Std$ and the $NAD$ under perturbation, and the corresponding variation ranges remain narrow, indicating a stable response to input uncertainties. The SVM model is not included in this comparison because its robustness was insufficient for meaningful evaluation under the same perturbation setting. Collectively, these results confirm the value of introducing attention, as it mitigates sensitivity to perturbations and improves overall robustness. More specifically, the attention-based reweighting strategy, together with the imposed monotonicity constraints, appears to enhance the model’s tolerance to parameter uncertainty encountered in practical applications.

4.5. Multi-Scenario Ablation Experiments

To further validate that AM-PGNN is applicable to different materials, this study fabricated micro-grooves on Cu and SiC using FLA. The datasets for the ablation of Cu and SiC were obtained from the experiment, and the sample size and parameter ranges are consistent with those used in the WC experiments. Because the physical properties of Cu and SiC are different from those of the WC, the model was retrained separately for each workpiece material. However, within the studied parameter range, the monotonic relationships between the processing parameters and the ablation depth remain consistent across the studied materials. Figure 17 shows the prediction of the micro-groove depth in the FLA of Cu. Specifically, the majority of prediction points of AM-PGNN fall within the deviation lines and are close to the 1:1 line, while the other five models exhibit varying degrees of prediction points exceeding the deviation lines. For a more intuitive comparison of model performance, quantitative analysis was conducted using R², RMSE, and PCD metrics (Figure 18). AM-PGNN achieved the highest R² of 93.2% and 91.8% on training and testing, and the lowest RMSE values of 2.83 and 2.76, with PCD consistently at 100%, indicating that the predictions align with physical mechanisms. R² values of SVM, DNN, CNN, and GBDT algorithms on both the training and test sets are significantly lower than those of the AM-PGNN, and some predictions violated physical monotonicity. As for the PGNN model, although its PCD is 100% and the predictions satisfy physical monotonicity, the R² values on training and testing are 90.3% and 90.1%, respectively, which are lower than those of AM-PGNN.

In the ablation of SiC (Figure 19), SVM, DNN, CNN, GBDT, and PGNN exhibited varying degrees of prediction points exceeding the deviation lines, while most of AM-PGNN’s predictions remained within the deviation lines and closely aligned with the 1:1 line. In terms of evaluation metrics (Figure 20), SVM performed the worst with the lowest R² and highest RMSE. In contrast, AM-PGNN achieved the highest R², the lowest RMSE, and maintained a PCD of 100% on both the training and test sets, demonstrating the best overall performance.

5. Conclusions

This study proposes an effective ensemble method for the field of femtosecond laser ablation, integrating physics-guided constraints with neural network learning. The proposed algorithm integrates monotonic constraints and an attention mechanism, achieving accurate predictions based on the available experimental dataset. The main conclusions are summarized as follows:

(1)

Integration of domain knowledge and attention mechanism: a physics-guided architecture was established by translating key domain knowledge into physical constraints embedded into the NN, which incorporates the positive monotonic relationships of LF and RF, and the negative monotonic relationship of SS, with micro-groove depth. Additionally, an attention mechanism allows for the adaptive weighting of input features, prioritizing critical physical parameters during training.

(2)

Improved accuracy and physical consistency: AM-PGNN significantly outperforms other ML methods in prediction accuracy. The embedded monotonicity constraints ensure that the predictions strictly follow the prescribed monotonicity relationships, thereby preserving the physical consistency of the overall trend, which is essential for the accurate modeling, control, and optimization of FLA.

(3)

Enhanced robustness: the proposed model exhibits superior stability compared to conventional models. Under identical ±1% input perturbations, the predictions of AM-PGNN demonstrate smaller fluctuations than those models.

(4)

Applicable to different materials: in the FLA experiments on Cu and SiC, the proposed AM-PGNN framework can be applied to different materials after training on a specific material. This result shows that the framework is applicable in the FLA of different materials and can achieve accurate predictions by incorporating physical monotonicity rules rather than relying solely on statistical data. The method offers a solution with low data requirements and high interpretability, making it particularly suitable for high-cost experiments.

The proposed AM-PGNN model achieves high-accuracy groove depth prediction under limited sample conditions and improves physical consistency through monotonicity constraints. Additional results on other materials indicate that, after retraining for a specific material, the framework can be extended to different material systems. This indicates the algorithm has potential in the application of the digital twin architecture for FLA. On this basis, future efforts can be directed toward integrating the model with in situ real-time sensing modalities, such as laser-induced plasma diagnostics and optical profilometry. This development is expected to facilitate real-time optimization and online monitoring of FLA. Furthermore, the current study is confined to the prediction of micro-groove depth under single-scan conditions, while the cumulative material removal behavior in multi-pass machining and the characterization of complex three-dimensional surface topographies remain to be investigated. Therefore, future work will extend the AM-PGNN framework by describing multi-pass ablation dynamics based on physical principles, developing a strongly differentiable optimization strategy, and expanding its predictive scope from one-dimensional depth estimation to full three-dimensional surface topography prediction.