TPMS structures are defined by implicit functions. Common TPMS types include I-WP, Neovius, F-RD, Gyroid, Primitive, and Diamond surfaces. Among them, the Gyroid structure possesses a smooth and continuous surface with excellent self-supporting characteristics; the Primitive structure exhibits cubic symmetry; and the Diamond structure features a high specific surface area and complex connectivity. Owing to these distinctive advantages, these three TPMS types are more widely applied in engineering and biomedical fields. Their mathematical representations are given in Eqs. (1)–(3) [13], where c denotes a constant used to control the level-set threshold. By thickening or filling the isosurface defined by the implicit function, corresponding TPMS lattice structures can be generated. (1)sin⁡x⋅cos⁡y+sin⁡y⋅cos⁡z+sin⁡z⋅cos⁡x=c, (2)cos⁡x+cos⁡y+cos⁡z=c, (3)cos⁡x⋅cos⁡y⋅cos⁡z−sin⁡x⋅sin⁡y⋅sin⁡z=c.

The geometry of a TPMS model can be controlled through its implicit function; however, due to the structural complexity of TPMS surfaces, conventional modeling methods are generally inapplicable. Common software tools for TPMS modeling include MATLAB, nTop, MSLattice [14], Flatt Pack [15], and RegionTPMS [16]. In this study, the focus is placed on the wall thickness parameters of homogeneous TPMS lattice models. Therefore, due to its high efficiency in model construction, the Uniform TPMS Lattices–Cuboid module in the MSLattice software was selected, and the corresponding structural parameters were obtained through direct geometric measurement of the generated models.

A total of 152 TPMS models were constructed in this study. The selected TPMS types included Gyroid, Primitive, and Diamond structures. The filling type of the TPMS lattices was categorized into two forms: Sheet-type and Skeletal-type. The primary modeling parameters of the TPMS models consisted of the volume fraction (VF) and the cell length (CellLength).

All TPMS lattices in this study were modeled as cubic unit cells, and the collected data included modeling parameters such as wall thickness, pore size, volume fraction, and cell length. The TPMS samples were categorized and labeled according to their TPMS type, filling type, and cell length. For instance, a Gyroid-type lattice with Sheet filling and a cell length of 1 mm was denoted as GS1. Similarly, a Gyroid Skeletal lattice with the same cell length was labeled as GN1. The relationship between volume fraction and wall thickness for several groups is illustrated in Fig. 1.

At present, the wall thickness parameter of TPMS lattices can be calculated using an analytical expression, as given in Eq. (4) [17]. In this equation, ρ* represents the volume fraction, V_cell denotes the volume of a single unit cell, and A_TPMS refers to the surface area of the TPMS isosurface when the control parameter c in the implicit function is set to zero. Owing to the geometric complexity of TPMS structures, the value of A_TPMS cannot be determined analytically and must instead be obtained from the modeling software after surface generation. It is also noteworthy that A_TPMS varies slightly among different implicit function formulations. (4)Wall=ρ∗VcellATPMS.

From the variables in Eq. (4), it can be observed that the wall thickness parameter is essentially related to the volume fraction, the unit cell volume, and the TPMS structural type. Therefore, in this study, the TPMS type, TPMS lattice filling type, volume fraction, and cell length were selected as the input parameters for predicting the wall thickness parameters of TPMS lattices.

In the construction of ML models, raw data often contain noise and missing values, making them unsuitable for direct use in model training. Therefore, data preprocessing is necessary to enhance data validity and reliability. Matrix-form datasets exhibit greater adaptability and effectiveness when processing 3D point cloud data with spatial structural features. In this study, a preprocessing framework is proposed to convert STL point cloud files into a matrix format. The process involves discretizing the point cloud to obtain regularized data, expanding the dataset through voxel-based downsampling, and enhancing data recognizability via rasterization, after which the processed data are output as a matrix representation suitable for machine learning model training.

The CNN model is a deep learning architecture designed for feature extraction and pattern recognition, capable of efficiently capturing spatial correlations within structured data. The SVM model, on the other hand, is well suited for small-sample learning and nonlinear regression problems, providing excellent generalization capability through optimal hyperplane separation in high-dimensional feature space.

The CNN module is employed as a hierarchical feature extractor, taking preprocessed 48 × 48 matrices as input. These matrices integrate geometric projections and numerical design parameters through an early data-fusion strategy. The CNN architecture consists of multiple convolutional blocks with 3 × 3 kernels, each followed by batch normalization, ReLU activation, and max-pooling layers, enabling progressive abstraction of local-to-global structural features. The network is trained using the Adam optimizer with an initial learning rate of 3 × 10⁻⁴, a mini-batch size of 32, and a maximum of 100 epochs, while early stopping based on validation performance is adopted to mitigate overfitting.

After feature extraction, the high-level representations obtained from the fully connected layer are fed into an SVM regression module, which replaces the conventional softmax-based output layer. By performing regression in the learned feature space, the SVM enhances prediction robustness and stability, particularly under limited data conditions. As illustrated in Fig. 5, this architecture allows the CNN to effectively capture representative TPMS structural features, while the SVM ensures accurate and stable prediction performance, resulting in improved overall model reliability.

To ensure scientific and reliable model training, the dataset was randomly divided into a training set and a testing set with a ratio of 7:3. The training set was used to establish the prediction model, tune the hyperparameters, and obtain an optimized regression framework, while the testing set was employed to evaluate the model’s generalization performance and predictive accuracy.

Since the traditional accuracy metric for classification tasks is not applicable to regression-based parameter prediction, this study adopts the proportion of samples with a prediction error less than 20% as an alternative accuracy measure. In addition, the coefficient of determination (R²) and mean square error (MSE) were selected as the core evaluation metrics to quantify the fitting degree and prediction precision between the predicted and actual values. The value of R² ranges from 0 to 1, where a higher value indicates better goodness of fit, while MSE is always greater than zero, with smaller values representing higher prediction accuracy. The calculation formulas of both metrics are shown below. (6)R2=1−∑i(YTrue−YPredict)2∑i(Y¯True−YTrue)2, (7)MSE=1m∑i=1m(YTrue−YPredict)2,where Y_True represents the true values of the testing set, Y_Predict denotes the predicted values obtained from the model, and Y_True is the mean value of the true data in the testing set.

Multiple Linear Regression (MLR) is a widely used statistical method for analyzing the relationship between multiple independent variables and a dependent variable. The general form of the MLR equation is given in Eq. (8). In this study, MLR was employed to quantify the influence of different design parameters on the wall thickness parameter. Each regression coefficient in the model represents the contribution weight of a specific design parameter to wall thickness. (8)Y=b0+b1X1+b2X2+…+bmXm+ε,where Y is the dependent variable (wall thickness); X₁, …, X_m denote the independent variables; b₀, b₁, …, b_m are the regression coefficients to be estimated; and ε represents the random error term.

In MLR modeling, the input matrix constructed in Section 2.1 is not directly applicable due to the categorical nature of some parameters. Therefore, one-hot encoding was applied to process categorical variables. The TPMS Type, a three-class variable (Gyroid, Primitive, and Diamond), was represented by a three-dimensional binary vector: [1, 0, 0], [0, 1, 0], and [0, 0, 1], respectively. The Filling Type, a two-class variable (Sheet and Skeletal), was represented by a two-dimensional vector: [1, 0] and [0, 1], respectively.

To verify the effectiveness of the processed point cloud matrices in representing the geometric characteristics of different structures, Fig. 6 presents the grayscale images corresponding to various TPMS structures after point cloud processing.

For different TPMS structures, the grayscale images exhibit significant variations in pattern and intensity distribution. In contrast, TPMS structures with identical design parameters—such as similar volume fractions or cell lengths—show comparable grayscale patterns, although subtle differences can still be observed in the blank regions. These results demonstrate that the processed matrices can effectively capture the geometric characteristics of various TPMS structures, thereby providing high-quality input features for the machine learning model.

The training process of the model is illustrated in Fig. 7, where Fig. 7a shows the model performance metrics during training, and Fig. 7b presents the loss decay curve. As observed from the figures, the performance indicators tend to stabilize around the 144th iteration, while the loss function gradually flattens after approximately 154 iterations. These results indicate that the CNN-SVM model achieves rapid convergence with a relatively small number of iterations, demonstrating stable training behavior and strong learning capability.

In this experiment, not only the CNN-SVM model but also the MLR, SVM, and CNN models were constructed for performance comparison. The established dataset was imported into the MLR, SVM, and CNN models for training and testing, and the corresponding prediction results are presented in Fig. 8. The key evaluation metrics of different prediction models are summarized in Table 2.

Fig. 8a–d shows the comparison between the predicted and actual wall thickness values in the testing set for the MLR, CNN, SVM, and CNN-SVM models, respectively. Fig. 8e–h illustrates the distribution of testing data within the ±10% and ±20% error bands for the same models. As shown in Fig. 8a–d, both the CNN and CNN-SVM models exhibit smaller deviations between predicted and actual values, indicating their superior predictive performance for wall thickness parameters. Further examination of Fig. 8e–h reveals that the data points are well distributed along both sides of the diagonal line, particularly for the CNN and CNN-SVM models, demonstrating their strong generalization capability.

In the testing set, the samples with prediction errors exceeding 20% were primarily concentrated in TPMS lattices with actual wall thickness values below 0.2 mm. This indicates that the CNN-SVM model performs better in predicting TPMS lattices with larger wall thicknesses. Among these samples, the lattice PS1_V0.13 exhibited the largest absolute error of 0.11 mm. Two main factors contribute to this phenomenon. First, TPMS lattices with extremely thin walls exhibit high geometric sensitivity, where local perturbations or subtle deviations introduced during rasterization can be amplified, leading to relatively large prediction errors. In addition, when the wall thickness approaches the voxel resolution limit in the rasterization process, partial geometric details may be lost or blurred, further enlarging the deviation between the extracted features and the true structure. Second, the limited number of samples within this thickness range causes the model to rely excessively on the general trends of wall thickness variation derived from other design parameters, thereby reducing prediction accuracy. This data imbalance becomes more pronounced in the ultra-thin region, where the model struggles to capture the distinct geometric behaviors associated with extremely small wall thicknesses. This finding suggests that further optimization of data balance and sample selection in the training dataset is necessary to enhance the model’s predictive accuracy and robustness across different wall thickness ranges.

According to the experimental results summarized in Table 2, the CNN-SVM model achieved the best overall performance in wall thickness prediction. Although its R² and MSE metrics were not the highest among all models, it demonstrated outstanding control of prediction errors: compared with the conventional MLR, CNN, and SVM models, the proportion of testing samples with prediction errors less than 20% increased by 15.8%, 10.6%, and 18.5%, respectively. From an additive manufacturing perspective, maintaining prediction errors within a bounded range is more critical than marginal improvements in global accuracy metrics such as R² or MSE. The CNN-SVM model effectively integrates the strengths of CNN and SVM architectures—using convolutional operations to efficiently extract key structural features from TPMS lattices and employing the SVM module for regression-based decision making. The synergistic combination of these two components significantly enhances the overall prediction capability and stability of the model.

The multiple linear regression (MLR) model not only enables prediction of wall thickness parameters but also provides insight into the correlations between design variables and wall thickness through the obtained regression coefficients. The influence weights of each parameter, represented by the regression coefficients, are illustrated in Fig. 9.

According to the regression coefficients derived from the MLR equation, among the two numerical input variables—VF and Cell length—the latter exerts a more pronounced influence on wall thickness. Therefore, in cases where insufficient wall thickness occurs in a TPMS lattice with a fixed structural type, increasing the cell length is a relatively effective approach to improve manufacturability. For the two categorical variables, TPMS Type and Filling Type, the regression coefficients of TPMS Type are all negative, indicating that different TPMS geometries reduce wall thickness to varying degrees. Among them, the Diamond type shows the most significant negative effect, suggesting that this geometry should be selected with caution, as it often requires larger VF and cell length values to satisfy the minimum wall thickness requirements of additive manufacturing. In contrast, within the Filling Type variable, the Sheet structure exhibits a stronger positive influence on wall thickness than the Skeletal type, implying that Sheet-type configurations should be prioritized in design considerations.