For vulnerable road users, the configuration of a vehicle’s frontal structure is pivotal in reducing both the likelihood and severity of pedestrian injuries. Within leg-injury assessment protocols, tibial acceleration, knee-joint bending angle, and knee shear displacement provide direct and quantitative indications of trauma severity. Accordingly, the present work adopts these three metrics as the output responses for evaluating the pedestrian-protection performance of alternative front-end structural designs.

On the input side, eight design-variable categories—comprising a total of 45 distinct parameters—are defined. Owing to space constraints, Table 8 reports only the 10 most influential variables together with their admissible ranges. Once the output responses yi and design variables xi were established, a series of (xi, yi) pairs was generated via finite-element simulations, thereby furnishing the training corpus for the surrogate machine-learning models.

Because data-driven algorithms benefit from large sample sizes whereas each finite-element run is computationally intensive, a multidimensional, space-filling Latin hypercube sampling (LHS) strategy was adopted to explore the design space efficiently. The definitions and upper–lower bounds of the sampled variables are summarised in the same (Table 8). All simulations were executed automatically on an integrated numerical co-simulation platform, which returned the corresponding output responses for every sampled design point. The dataset information is shown in Table 9.

Data preprocessing is a critical step in constructing machine learning models [26], directly impacting the model’s performance, convergence speed, and final predictive accuracy. This process involves extracting useful features from raw data and transforming them into a format suitable for machine learning algorithms [27]. The feature variables selected from this study’s dataset encompass continuous, discrete, and textual types, with considerable differences in their magnitudes. Employing preprocessing techniques enables the model to better adapt to the features and accelerates convergence.

For evident outliers, such as excessive acceleration, knee shear displacement, or knee bending angles that exceed normal ranges, or situations where finite element calculations do not converge or yield negative volumes, these data points are either removed or necessitate a reconstruction of the finite element model and recalculation. Efforts are made to ensure that the dataset remains within a reasonable range. In cases with minimal missing values, the missing data are filled using the column mean.

Continuous features, which constitute the most prevalent type of data collected—such as speed, collision location, and material properties—are processed through methods like normalization and standardization. In this study, due to the differing units used for various continuous features leading to substantial discrepancies in data magnitudes, normalization is employed. The continuous features in the dataset are normalized column-wise to yield values distributed within the range of [0, 1]. For discrete features, such as vehicle types and the connection relationships between the impact beam and the bumper, one-hot encoding is utilized. After performing feature selection, the matrix of correlation coefficients for some of the variables is shown in Fig. 10. To validate the reliability of our collected dataset, the predicted peak values were compared against the EEVC regulatory thresholds. The comparison results are presented in the Table 10 below.

Machine learning—an essential methodology within contemporary data analytics—aims to automate the construction of predictive and descriptive models [28]. As a sub-discipline of artificial intelligence, it is premised on the capacity of computational systems to glean knowledge autonomously from data, uncover latent patterns, and make informed decisions with minimal human supervision. Depending on the availability of labelled outcomes or feedback, machine-learning techniques are typically grouped into supervised, unsupervised, and reinforcement learning.

This study constructs surrogate regression models to predict three continuous indicators of pedestrian leg injury severity—tibial acceleration (A), knee flexion angle (ANG), and knee shear displacement (DSTR). Because exhaustive hyperparameter tuning for all candidate algorithms incurs prohibitive computational and time costs, each method is initially trained using the default settings listed in Table 11 to establish a baseline performance profile. Rigorous model selection at this stage is essential to ensure a high-fidelity surrogate. The preliminary evaluation encompasses a broad spectrum of regression approaches: linear models (i.e., Linear Regression), kernel methods (Support Vector Regression), tree-based ensembles (Random Forest, CatBoost, LightGBM, AdaBoost), and deep-learning architectures (CNN–LSTM and TabNet). Baseline models are ranked by their coefficient of determination (R2), with the top performers subsequently advanced to targeted hyperparameter optimization.

The training corpus is drawn from a finite-element crash-simulation dataset containing roughly 3000 impact scenarios across ten vehicle types. Of these, 200 cases are reserved as an external test set, while the remainder are split into training and validation subsets in a 5:1 ratio. Each record provides 46 explanatory variables describing vehicle and impact conditions, together with the three response variables noted above. As all responses are continuous, the analysis considers only regression algorithms. Candidate models are ranked on the basis of the coefficient of determination, R2, and root-mean-square error (RMSE), computed according to Eqs. (4) and (5). Whereas R2 quantifies the proportion of variance explained by the model, RMSE gauges the average deviation between predicted and observed values, thereby providing complementary assessments of explanatory power and predictive accuracy. (4)R2=1−∑i=1n(yi−y^i)2∑i=1n(yi−y¯i)2 (5)RMSE=1n∑i=1n(yi−y^i)2

The overall methodological workflow is depicted in Fig. 11. The average score and average error of the machine learning model on the three target variables (all of which have been normalized) were chosen as the final metrics for evaluating the model’s performance. The machine learning models chosen are common machine learning models for processing tabular data and deep learning combinatorial model. Below provided Fig. 12 provides insights into the performance of different machine learning algorithms (using default parameters). An R2 value closer to 1 indicates a stronger model fit to the data, while a lower RMSE reflects smaller prediction errors across the dataset.

Inspection of Fig. 12 shows that TabNet attains the highest coefficient of determination (R2) while sharing the lowest root-mean-square error (RMSE = 0.14) with CatBoost, and thus delivers the best overall accuracy. CatBoost matches Random Forest in R2 (0.83) but exhibits a lower RMSE, ranking second only to TabNet. By contrast, conventional linear regression performs poorly, displaying both the largest RMSE (0.36) and the smallest R2 (0.57), and therefore fails to satisfy the required precision. The remaining algorithms—Random Forest, LightGBM, SVM, CNN-LSTM, and AdaBoost—form a middle tier, with R2 values between 0.78 and 0.83 and RMSE values between 0.15 and 0.21. On the basis of these baseline results obtained under default hyper-parameters, TabNet, CatBoost, Random Forest, and LightGBM were selected for subsequent hyper-parameter optimisation.

Four high-performing models—TabNet, CatBoost, Random Forest, and LightGBM—were identified in our preliminary screening and advanced to automated hyper-parameter tuning. Given the substantial computational cost of this task, we employed Optuna’s Bayesian-optimization framework, configuring up to 50 parallel trials and a maximum of 200 total evaluations. For each injury metric (tibial acceleration, knee-flexion angle, shear displacement), all four algorithms were fit in turn under identical conditions, and the configuration yielding the highest cross-validated accuracy was selected as that metric’s surrogate. To guard against overfitting from a static train_test split, the Bayesian optimizer was integrated with a K-fold cross-validation scheme (train:test = 3:1). In this arrangement, the dataset is partitioned into K subsets; each trial evaluates a given hyper-parameter vector by averaging the model’s performance—measured via R2 and RMSE—across the K validation folds. Fig. 13 illustrates this procedure: each horizontal segment corresponds to one sampled hyper-parameter set, colored by the resulting validation score (deeper red indicates higher accuracy). By embedding K-fold cross-validation directly within the Bayesian-optimization loop, we ensure that every candidate configuration is assessed on multiple splits, yielding robust, low-bias hyper-parameter estimates and ultimately enhancing the surrogate models’ generalizability.

The optimal model parameter settings are shown in the table below. This section describes the hyperparameter optimization process for the TabNet model using Bayesian optimization, while the hyperparameter optimization of other models is not further discussed in this section. The key parameters of each model and the search range are shown in the Table 12.

Bayesian optimization significantly accelerates convergence to optimal hyperparameter configurations compared with exhaustive and grid-search methods. As shown in Table 13, all candidate models exhibit substantial gains in both R2 and RMSE following tuning. Among them, TabNet achieves the best trade-off between explanatory power and predictive accuracy, recording the highest R2 and the lowest RMSE. CatBoost also attains strong accuracy but slightly underperforms TabNet in variance explanation. Although Random Forest reports a high overall R2 of 0.93, its RMSE remains relatively large, indicating uneven performance across targets. A more granular analysis reveals that Random Forest predicts knee-flexion angle and shear displacement with excellent precision—achieving R2 values of approximately 0.94 and 0.95 and RMSEs of roughly 0.10 and 0.08—yet struggles with tibial acceleration, where its RMSE jumps to about 0.27. Consequently, its aggregate RMSE of 0.15 exceeds LightGBM’s 0.11, despite LightGBM’s lower R2 of 0.90. This discrepancy underscores that a higher R2 does not inherently guarantee lower error: models may explain variance well while still producing large absolute deviations for certain outcomes. Therefore, R2 and RMSE should be considered jointly when evaluating regression performance, as relying on either metric in isolation can be misleading—particularly for challenging targets such as tibial acceleration.

For this application, the TabNet regressor [29] was selected as the final surrogate model, having delivered the highest R² and the lowest root-mean-square error (RMSE) among all candidates. Fig. 14 presents its learning curves for the three injury targets—(a) tibial acceleration (pred_A), (b) knee-flexion angle (pred_ANG), and (c) shear displacement (pred_DSTR)—with validation RMSE shown in blue and test RMSE in red. In each case, RMSE decreases sharply within the first 10–15 epochs, indicating rapid capture of the underlying data patterns, and begins to plateau by around epoch 20, signifying convergence. The near-perfect overlap of validation and test RMSE throughout training confirms that the model generalizes well, with no evidence of over- or underfitting. Minor, synchronized fluctuations (e.g., around epoch 25 in pred_ANG) reflect stochastic variations in the optimization process and do not materially affect overall performance. Collectively, these learning curves demonstrate that the TabNet surrogate converges efficiently and exhibits strong generalization across all three injury metrics, achieving low and stable error within relatively few training epochs. Fig. 15 below shows the prediction results of the complex machine learning model on the test set. It is evident that most of the scatter points are concentrated around the line, with the errors falling within the ±10% error margin. This indicates that the model’s predictions are nearly identical to the true values in the test set, further validating the feasibility and effectiveness of using machine learning methods to construct high-precision surrogate models. Therefore, complex models not only provide more accurate fitting for pedestrian safety performance prediction, but also offer more precise references for subsequent vehicle front-end structure optimization.

In this work, TabNet was adopted as the principal machine-learning framework. TabNet (Tabular Neural Network) is a specialized deep-learning architecture for tabular data that combines the interpretability of decision-tree models with the representational capacity of neural networks [29]. Its sequential attention mechanism highlights the most relevant features at each decision step, thereby alleviating the curse of dimensionality. Moreover, TabNet natively accommodates both continuous and categorical variables, obviating complex preprocessing pipelines. The model’s use of unbiased, distributed representations and its decision-step structure enable efficient learning of sparse, high-dimensional patterns, which improves generalization while preserving computational efficiency and robustness. The overall architecture of TabNet is illustrated in Fig. 16. To ensure the stability of TabNet’s predictions, we evaluated its performance under the finalized hyperparameter configuration using five repeats of four-fold cross-validation (for a total of 20 independent trials). The aggregated results—mean ± standard deviation for both R2 and RMSE—are reported in Table 14.

TabNet utilizes a built-in attention mechanism and dynamic feature masking, which enables it to perform feature importance analysis in a manner like tree-based models. At each decision step, TabNet selectively prioritizes different features, assigning importance based on attention weights. By aggregating these weights across all decision steps, TabNet calculates the final importance score for each feature. Additionally, the multi-step decision structure of TabNet ensures the reevaluation of previously unused features, leading to a more comprehensive and unbiased assessment of feature importance. As illustrated in Fig. 17, the importance of selected feature categories in predicting target variables is clearly shown. Notably, the feature with the highest total share across the three target variables was the bumper architecture, followed by impact point distance and material properties.

The horizontal stacked-bar chart illustrates the relative importance of six feature groups—bumper architecture, impact-point distance, material properties, vehicle category, under-ride clearance, and legform impactor type—across three injury metrics: tibial acceleration (A), knee flexion angle (ANG), and shear displacement (DSTR). The x-axis quantifies each group’s percentage contribution to the surrogate model’s predictions. As shown, front-end geometric parameters (bumper architecture and impact-point distance) exert an overwhelmingly dominant influence on injury outcomes, particularly for knee flexion and shear displacement. Material properties significantly affect shear displacement but have only a minor effect on knee flexion. Vehicle category and under-ride clearance exhibit moderate influence, indicating that overall vehicle size and chassis height should also inform design decisions. Although legform impactor type remains a standard testing parameter, its importance in this multivariate surrogate model is comparatively low, underscoring the primacy of structural features in real-world crash risk. In summary, pedestrian-protection design should prioritize optimization of bumper-architecture parameters, followed by adjustments to impact-point positioning and material stiffness, with secondary refinements based on vehicle class and ground clearance.

In addition to the feature importance integrated within the model, it is crucial to conduct model interpretability analyses using external methods. Among the various model interpretation techniques, SHAP (Shapley Additive Explanations) [30] has gained considerable attention due to its solid theoretical foundation and broad applicability. SHAP is rooted in Shapley value theory from game theory, and it quantitatively assesses the marginal impact of adding or removing a specific feature on the model’s predicted value. By calculating this marginal effect, SHAP evaluates each feature’s contribution to the model’s prediction. Specifically, the SHAP method approximates the complex black-box model f(x) using a set of simple binary linear functions, leveraging the Shapley value and the additivity principle, as shown in Eq. (6). (6)f(x)≈g(z′)=ϕ0+∑i=1Mϕizi′

ϕ0 denotes the baseline prediction value, zi′∈{0,1}M represents whether the ith feature can be observed or not, M denotes the number of input features, and ∑ϕi denotes the SHAP contribution of each feature, and the sum of the contribution values is equal to the final prediction value, whose value is calculated in Eq. (7): (7)ϕi=∑S⊆X{xi}|S|!(|X|−|S|−1)!|X|![f(S∪{xi})−f(S)]where S denotes the subset of features (without feature xi), f(S) denotes the predicted value obtained using only the subset S, and f(S∪{xi}) denotes the predicted value after adding xi. The method not only provides a global interpretation, but also a detailed analysis of individual predictions, thus allowing a fairer and more consistent interpretation of the internal mechanisms of complex models.

In the local interpretation method of SHAP, the influence of each sample on the output response—along with its positive and negative contributions—can be visualized using a force plot. This section presents the effect of each design variable on the output response, exemplified by a randomly selected data sample from the test set of the pedestrian crash safety performance dataset.

In Fig. 18, we analyze three individual samples from the prediction process. Blue arrows on the right side of the output indicate feature contributions that push the predicted value downward, while red arrows on the left indicate contributions that push it upward. The length of each arrow reflects the magnitude of the feature’s effect.

Sample 1 (baseline normalized acceleration = 0.23): features D (collision location) and A_I (vent position) have the largest impact. Specifically, an increase in D substantially lowers the predicted tibial acceleration, whereas a higher A_I raises it.

Sample 2 (baseline normalized knee-flexion angle = 0.68): features M_bf (skin Young’s modulus), M_br (bumper-beam modulus), and F_B (bumper fascia presence) all contribute positively to the predicted flexion angle, while BD_R (crush-box position ratio) and D reduce it.

Sample 3 (baseline normalized shear displacement = 0.84): both D (collision location) and H (impact height) strongly increase the predicted shear displacement.

The range and distribution of the top 10 design variables influencing safety performance are shown in the figure below. The y-axis represents the top 10 design variables, sorted by their overall contribution ∑|ϕi(j)|, while the x-axis indicates their corresponding SHAP values ϕi(j). Each point on the Fig. 19 represents a sample, with color gradation from red to blue indicating the SHAP value of the design variable, from high to low. A higher SHAP value reflects a greater impact of the design variable on the target performance.

From the summary plot above, the horizontal width represents the range of SHAP values for each feature across all samples, indicating the magnitude of its impact on the model’s predictions. For the acceleration target in particular, feature D (longitudinal collision location) exhibits the widest distribution, making it the most influential factor with the largest positive and negative “push” on the predicted outcome. The next most important features are A_I (vent position), A_I_L (vent length), Tbr (bumper-beam thickness), and V (impact speed). Points are colored from blue (low feature value) to red (high feature value) to reflect each feature’s magnitude. For D, red points (high values) cluster on the right side of the plot, indicating that a larger D increases the predicted injury severity, whereas blue points (low values) tend to produce negative contributions that lower the prediction. Similarly, for A_I, lower vent positions (blue) reduce the predicted severity, while higher positions (red) increase it. Features near the bottom—such as Tbf (outer-skin thickness)—show much narrower distributions, signifying a comparatively minor overall effect. To illustrate the cumulative effect of feature interactions and the combined impact of multiple features on prediction results, this paper employs a decision diagram to display the changes in model predictions based on a set of design variables, as shown in Fig. 20. In the diagram, the prediction trajectory of each sample is represented by a colored curve. Each curve intersects the horizontal axis at the corresponding prediction point, with its color reflecting the observed prediction value at that point. As the curve progresses from bottom to top, the SHAP value of each feature accumulates towards the model’s baseline value, visually demonstrating the contribution of each feature to the overall prediction.

From the decision plot above, each trajectory traces the cumulative effect of features—ordered by their importance—from the model’s baseline output to the final prediction. Line color encodes the predicted severity: blue lines indicate lower injury predictions, and red lines indicate higher ones. For the knee-flexion angle outcome, the earliest features (e.g., H, impact height; BD_R, crush-box position ratio) introduce the largest initial shifts: high impact heights (red lines) rapidly elevate the baseline. Subsequent mid-sequence features (such as F_B, bumper fascia presence; D, collision location) further separate the trajectories, creating a clear divergence between high- and low-severity predictions. Although later, lower-importance features (such as V, impact speed; C_T, vehicle category) continue to adjust the output, the prediction paths have by that point already been largely determined.

The error is calculated according to Eq. (8). A positive error sign indicates overprediction, i.e., the predicted target value is higher than the actual target value. On the other hand, a negative sign of error indicates under prediction i.e., the predicted target value is lower than the actual target value. (8)error=Ypred−YtrueYtrue×100% Fig. 21 illustrates the distribution of prediction errors for three lower-limb injury metrics—tibial acceleration (ERROR_A, yellow), knee-flexion angle (ERROR_ANG, light blue), and shear displacement (ERROR_DSTR, dark blue)—expressed as a percentage of the true response. The two horizontal red lines denote ±20\%  error bounds. Over 80% of validation samples for all three metrics fall within this ±20% range, and the dense clustering of points around 0% indicates no systematic bias in the surrogate model’s predictions.

Among the metrics, acceleration errors exhibit the greatest variability, with a small number of outliers exceeding ±30\% . Flexion-angle errors are the most tightly distributed—most lie within ±15%—and shear-displacement errors mirror this pattern, although slightly more points approach the +20% threshold. The overall symmetry of the error distributions confirms an absence of consistent over- or under-prediction. The presence of outliers in ERROR_A suggests that extreme impact scenarios remain challenging for the model. To enhance predictive robustness, particularly for acceleration, we recommend targeted hyper-parameter tuning and the inclusion of additional high-severity collision cases in the training set. A detailed breakdown of these errors is provided in Table 15.

As can be seen in Table 15, most of the absolute values of the errors are below 20 per cent, with 84.5 percent, 87.5 percent and 90 percent of the collision points having absolute error values of less than 20 percent for the three target variables. Taking the simplified vehicle front-end model described in Section 3 as an example, conducting finite-element simulations for 100 pedestrian-leg impact locations on a 12-core, 20-thread CPU requires approximately 84 h for a single set of parameter configurations. By contrast, a machine-learning surrogate can generate predictions for the same scenario in a fraction of that time while maintaining low error levels. Leveraging such a surrogate enables rapid evaluation of numerous front-end design variants, with predictive deviations kept within acceptable engineering tolerances. The prediction time of the finite element method and machine learning surrogate model is shown in Table 16.