Intelligent material design technology has significantly improved the efficiency of new material research and development, from the microscopic to the production level. Noh et al. (2020) proposed reverse design to accelerate traditional material design by leveraging hidden knowledge in material data to predict properties, and developed an image-based generator framework named iMatGen Zeng et al. (2020). Han et al. (2023) created a generative model using a crystal diffusion variational auto-encoder to customize crystal structures based on desired compositions. The model employs a deep neural network to extract global features from the crystal’s physical properties and optimizes structures using density functional theory. Hu et al. (2022) added a formation energy predictor to improve the model’s potential space, ensuring that the generated structures are morphologically reasonable and energetically stable. These approaches highlight the potential of machine learning in designing and reverse engineering stable crystalline materials. The above methods highlight the potential of machine learning in designing stable new crystalline materials, especially for prediction and design.

Data-driven technology in absorber design improves performance prediction and the discovery of efficient absorbers. Nadell et al. (2019) used deep learning to predict transmittance spectra for all-dielectric surfaces based on ADM parameters. Hou et al. (2020) developed a deep neural network for on-demand meta-material design, calculating split ring resonator parameters from reflectivity. On et al. (2024) created an electromagnetic absorber using deep learning, integrating a variational auto-encoder with CMA-ES optimization for efficient meta-structure design in a specific frequency band.

Multi-objective regression is a key area of machine learning that predicts multiple output variables from given input variables. The challenges of multi-objective regression can be addressed through algorithm-level and ensemble-level methods.

At the algorithmic level, single-objective regression methods are optimized for multi-objective scenarios. M-SVR Tuia et al. (2011) and MLS-SVR Xu et al. (2013) enhance support vector machine (SVM) techniques to optimize multiple outputs while considering their interrelationships and nonlinear correlations. Tran et al. (2024) found that SVM methods like ELS-SVR are ineffective for target component issues and suggested using artificial neural networks. They proposed a multi-output regression technique with gradient boosting and deep neural networks, training each layer on the residuals of the previous iteration’s squared loss function. Zheng et al. (2023) introduced a multi-objective prediction method using an adaptive dynamic genetic algorithm and adaptive moment estimation (ADGA-AM-ANN), which adds noise to the output and globally optimizes ANN.

At the ensemble level, regression chains sequentially concatenate multiple regression problems to predict target variables. Spyromitros-Xioufis et al. (2016) introduced the ensemble regression chain method, which incorporates previous target predictions as additional inputs. The maximum correlation chain model (SVRCC) Melki et al. (2017) builds on this concept, leveraging target correlations to enhance prediction performance and reduce computational complexity. Geiß et al. (2022) developed a regression chain ensemble method using repeated permutations to address insufficient multi-task objectives. This approach enhances the model’s ability to learn inter-task dependencies by propagating each target variable’s predicted values to subsequent models, thereby improving the accuracy of multi-variable predictions. The regression chain enhances prediction accuracy through multi-task concatenation and task relevance, and is widely used in energy materials estimation Yu et al. (2023), production process design Turetskyy et al. (2021), and material surface design Akhtar et al. (2024) in intelligent material design.

We selected EWA materials from 1 GHz to 18 GHz working frequency and acquired the performance. By varying the components in carbon materials, we tested their dielectric constant and permeability. The sample preparation process is shown in Figure 1.

Permittivity and permeability as input X to predict the values and categories of carbon materials. The properties of absorbing agents like carbon nanotubes, carbon black, and carbon fiber vary with different thicknesses and mass fractions at specific operating frequencies. We denote the predicted numerical variables as y 1 , y 2 , and y 3 . Carbon nanotubes are categorized into two types, which differ in outer diameter, pile density, and other characteristics. This is treated as a binary classification problem with variable value y 4 .

There is a complex nonlinear relationship between properties and composition. Figure 2 shows the dielectric constant and permeability changes of T N I M 8 carbon nanotubes at mass fractions of 1 % and 7.7 % across different operating frequencies. Gradient Boosting Decision Trees (GBDT) effectively capture this complexity by integrating multiple decision tree models, and managing feature interactions to improve material composition predictions. The performance in Figure 2 shows significant fluctuations in several local frequency ranges, and GBDT is robust against noise and outliers from experimental data. Thus, we apply GBDT as the base learner to enhance prediction performance based on local features and fitting errors.

GBDT employs classification and regression trees as weak learners, where errors from each learner optimize subsequent predictions of EWA materials. Taking the composition of the EWA material as an example, the data composition is as follows: { ( x i , y i ) } n , i = 1,2 , … , n , where n represents the number of samples, x i represents the material performance of the i -th sample, while y i denotes the material component to be predicted. The loss function is denoted as L ( y , F m ( x ) ) , which can be defined as the mean squared error in regression and exponential loss in classification. Additionally, F m ( x ) represents the m -th weak classifier. During the initialization phase, c can be set as the mean of the material indicators from the training set, and is expressed as Equation 1: F 0 x = arg min γ ∑ i = 1 n L y i , c , (1) where γ minimize the loss function and can be the mean or majority vote in the first iteration. In the M iteration stage, for m = 1,2 , … , M , the input optimization of the tree is performed. For the i -th sample, the negative gradient of the m -th tree is expressed as Equation 2: γ m , i = − ∂ L y i , F x i ∂ F x i F x = F m − 1 x . (2)

The current decision tree h m ( x ) utilizes to train the weak classifier, thereby obtaining the corresponding leaf stage area R m , j , where j = 1,2 , … , J m , J m represents the number of child nodes of the m -th regression leaf. For each leaf node, its fitted value is calculated as Equation 3: c m , j = arg min c ∑ x i ∈ R m , j L y i , F m − 1 x i + c . (3)

Obtain the expression for the strong learner F M ( x ) : F M x = F 0 x + ∑ m = 1 M ∑ j = 1 J c m , j I x ∈ R m , j . (4)

Among them, I represents the indicator function, which indicates whether the sample x is at the leaf node j . The attenuation coefficient can be added to the last term of Equation 4 to gradually increase the influence of subsequent tree models.

The cumulative strategy is a relearning procedure that increases data dimensions linearly with each iteration in the regression chain. According to Geiß et al. (2022), both non-cumulative and cumulative enhancements of the feature vector yield competitive predictions. Our accumulation strategy employs single variable prediction as the data dimension, with the q -th dimension’s prediction depending on the arrangement of the ( q − 1 ) dimensions and the prediction result of the single variable y ̂ . While assuming y ̂ q − 1 that represents the prediction from the ( q − 1 ) -th dimension regression chain, and y ̂ q − 1 s denotes the prediction result of this variable that solely depends on x , then the q -th dimension predictor variable is: y ̂ q = f q x 0 , … , x d , y ̂ 0 s , y ̂ 1 s , y ̂ 1 , … , y ̂ q − 1 s , y ̂ q − 1 , (5) where f q identifies the GBDT in the q dimension and d is the dimension of x . Due to limitations in absorbent data acquisition, we use a separate model to predict results in the cumulative regression chain. This method is also widely utilized in the field of computer vision, particularly in image pyramids, which employ feature maps of varying scales. These maps are integrated into the model system to enhance data dimensionality, allowing the model to focus on different perspectives. The cumulative design identifies the relationship between y q and y q − 1 and analyzes the predicted value based on the independent variable x . As learning progresses, the target variable’s estimation is added to subsequent models as features, enhancing the training dataset in the regression chain order.

Various strategies exist for constructing regression chains. Geiß et al. (2022) expanded single-variable regression to multiple variables, creating chains that were alternately combined to find the optimal link. Wahid et al. (2023) utilized three different regression links for combined predictions, while Melki et al. (2017) based link ordering on the correlation coefficient between variables. However, correlation links may not align with regression performance, and accumulating errors can reduce the accuracy of subsequent predictors. For example, our tests showed that the predicted result for y 1 ̂ − y 2 ̂ − y 3 ̂ was lower than that for y 1 ̂ − y 3 ̂ − y 2 ̂ . To achieve the optimal regression link, we predict variables based on performance. For instance, after constructing a link with Equation 5, we calculate and store the coefficient of determination ( R 2 ) for certain sequence in the training set. If R 2 exceeds a preset top N , we record the predicted results from the test set, averaging these for the composition prediction. The same approach applies when using an indicator like mean squared error (MSE) to predict the smallest top N values. The pseudocode is shown in Algorithm 1.

Algorithm 1

EWA Material Multi-object Regression.

Input: Training dataset X ⊆ R n , d and Y ⊆ R n , q , optimal performance retain N , testing dataset X ′ ⊆ R m , d .

In material composition design, we analyze the training set’s performance and composition to derive numerical compositions and material types. We use observed material properties with y 1 , y 2 , and y 3 as input to predict carbon nanotube types with a classifier. X is constructed from the required material properties and numerical composition, enabling the trained model to predict y 4 . This method optimizes the use of existing numerical components and meets practical needs for numerical components and types based on material properties. The pseudocode is shown in Algorithm 2.

Algorithm 2

EWA material Classification.

Input: Training dataset X ( n , d ) , Y ( n , q ) and y 4 . Testing dataset X ( m , d ) ′ and multi-regression prediction Y ̂ ( m , q ) .

The overall process flow is illustrated in Figure 3.

Data for EWA material batches were collected: carbon nanotube T N I M 8 and M 8130317 were 22022, carbon black was 20020, and carbon fiber was 25025. Split the training set and the test set in a 4:1. The detail of the material dataset with components and properties is shown in Table 1.

Variations in feature dimensions and numerical ranges can affect their influence during model training, impacting performance and accuracy. Thus, data normalization for the experimental data, as shown in Equation 6. X norm = X − X min X max − X min (6)

In addition, the regression values y 1 , y 2 , y 3 can also be normalized from Equation 6 to accurately evaluate the changes in MSE corresponding to different dimensions of the dependent variable.

The experiment aims to analyze the impact of chain sorting and compare it with other five GBDT-based or multi-regression methods.

• GPCC: Our proposed method, the parameters for the base leaner keep the same with GBDT.

We use R 2 and MSE as evaluation metrics for the RQ1. The calculation equations are shown in Equations 7, 8: R 2 = 1 − S S res S S tol , (7) M S E = 1 n ∑ i = 1 n y i − y ̂ i 2 . (8)

S S res is the sum of squares of the residuals, which is the sum of the squares of the differences between the predicted values and the actual values. The total sum of squares S S tol is the sum of the squares of the differences between the actual values and the mean of those values. A higher R 2 value, closer to 1, indicates a stronger explanatory power of the model. MSE is the average of the squares of the differences between predicted values and actual values.

For RQ 2, we used GBDT as a classifier and used Accuracy (ACC) and Matthews Correlation Coefficient (MCC) to evaluate the test set. The calculation equations are shown in Equations 9, 10: A C C = T P + T N T P + T N + F P + F N , (9) M C C = T P ∗ T N + F P ∗ F N T P + F P ∗ T P + F N ∗ T N + F P ∗ T N + F N , (10) where TP indicates true positives, FN denotes false negatives, FP represents false positives, and TN signifies true negatives. ACC measures the proportion of correct predictions, while the MCC evaluates misclassifications, reducing the impact of sample imbalance on performance metrics.

We use R 2 and MSE corresponding to three types of EWA materials, as shown in Table 1, 2.

The proposed GPCC demonstrates a significant improvement over alternative methodologies, achieving an average increase in R 2 of 0.1 and a reduction in MSE of 0.008 when compared to GBDT. The consistent predictive performance of GPCC suggests that our optimal link search methodology effectively identifies the most accurate predicted values. Furthermore, we observed that the link mining sequence that yields the highest R 2 and the lowest MSE remains consistent, indicating the reusability of the optimal link and predicted location. In contrast, GBDT chain exhibits comparable thickness and operating frequency to GBDT but experiences an average decrease in R 2 of 0.4. This observation implies that, despite the correlation among variables, errors tend to accumulate following the construction of links, thereby diminishing predictive performance. Although certain studies, such as those by Melki et al (2017), have introduced correlation coefficients, these do not directly correlate with performance outcomes, as evidenced by the results of SVRCC. Additionally, GBNN exhibits instability in predictions, particularly with significant errors in the operating frequency of carbon black. When predicting the properties of EWA materials, models that utilize extensive feature sets, such as neural networks, perform less effectively than GPCC, further highlighting the latter’s efficacy.

We examine the predictive capabilities of various methodologies for assessing the absorption characteristics of materials, specifically focusing on carbon black, carbon nanotubes, and carbon fibers. The performance of carbon black stabilizes at a mass fraction of 18 % , exhibiting negligible variations beyond this threshold. The GPCC demonstrates the most effective regression performance for carbon nanotubes, yielding R 2 and MSE of 65.26 % and 0.0223, respectively. Conversely, the GBDT chain reveals a decline in predictive accuracy for both carbon nanotubes and carbon fibers, suggesting a lack of stability in its predictions. Notably, carbon nanotubes and fibers outperform carbon black in predicting operational frequency, although carbon black is associated with a higher MSE. Therefore, a thorough analysis during the design of compositions necessitates an evaluation of the predicted properties of various components across different material types.

Numerical predictions from five comparison algorithms were used to predict carbon nanotube types by combining material performance data. ACC and MCC evaluated their performance, with M 8130317 as the positive class and T N I M 8 as the negative. Performance is shown in Table 4.

Table 4 shows the ACC and MCC results from the five-fold cross-validation to improve the stability of the prediction. In the parameter search, we use actual properties and components as the training set and those predicted in Section 5.2 as the test set. We apply five-fold cross-validation to find optimal parameters and evaluate them with the test set. The parameter search candidates are: number of trees 10, 20, 30, learning rate 0.01, 0.1, 0.2, and maximum depth 3, 5, 7. Unlike other methods, GPCC does not have more misjudgments for positive classes. SVRCC is ineffective for regression but achieves about 0.93 accuracy in classification, emphasizing its performance importance. GPCC improved ACC and MCC by 6.46 % and 12.9 % , respectively, compared to SVRCC, while maintaining high accuracy. By combining regression and classification for EWA materials, GPCC proved its superiority and practicality in the test set against various related algorithms.

In Figure 4, we list the importance of each feature when GBDT is used as a classifier. It can be seen that the features related to magnetic permeability are of relatively high importance for the classification of carbon nanotubes. The predicted values of y ̂ 3 and y ̂ 2 by the regression method corresponding to the GPCC that we proposed also play a significant role in the classification, which is greater than the importance of the corresponding dielectric constant. This indicates that the prediction results of the regression chain we proposed contribute well to the classification performance.

To verify the influence of cumulative strategy on the prediction ability, we used GPCC nonC , which represents the method without using single prediction as a feature, and used the regression index of 5.1 for comparison. The prediction ability of GPCC in materials was compared, as shown in Figures 5, 6.

The mean R 2 of GPCC nonC was 5.38 % lower than GPCC, which optimizes GPCC nonC ’s MSE by 38 % . G P C C nonC outperforms other methods, only lagging behind GPCC in carbon nanotubes and carbon black, underscoring the effectiveness of indicators for link selection. Carbon fiber also shows strong results in mass fraction and operating frequency, and the performance can be improved from GPCC nonC using cumulative features.

To verify the influence of model parameters on the fitting effect, we used GPCC 1 , GPCC 2 and GPCC 3 to represent the increase of the number of individual trees by 300–500, respectively, and the results were shown in Figures 7, 8. Overall, the complexity of the model is increased, and the performance is improved in each component. The degree of enhancement is inconsistent in different materials. MSE decreases significantly in carbon black at operating frequency, but MSE optimization is low in carbon nanotubes and fibers. In the process of model construction, there is no overfitting phenomenon, indicating the robustness of our GBDT as a base learner. In practical applications, it is necessary to balance model complexity and evaluation metrics to complete the prediction within the inference time.