Introduction

Power modules based on wide-bandgap semiconductors typically operate at high-power and high-temperature conditions. To ensure their long-term performance and reliability, the die-attach layer, which provides both mechanical and electrical connection between the chip and the frame, should maintain thermal stability under harsh conditions -something traditional alloy solders struggle to achieve1,2. In the past decade, low-temperature sintering of nano-sized metal particles (NMPs), usually Cu3,4,5 and Ag6,7,8, has emerged as a promising alternative for die-attach bonding. This method is based on the principle that reducing particle size increases surface energy, which enhance surface diffusion and allows sintering at temperatures that are far below the bulk melting point9. The low-temperature sintering usually generates a porous microstructure, the morphology of which is generally considered to be significantly influenced by bonding conditions, eventually affecting the properties of the bonding layer, such as strength10, conductivity11,12, creep resistance13, and fatigue14. Therefore, the prediction of structure properties from a given microstructure and its reverse engineering is always a critical focus in the field15.

Presently, most studies have focused on the characteristics of pores (or voids) dispersed in the sintered microstructures to evaluate properties, as the pores act as the second phase. The effective properties of this heterogeneous mixture structure of pores and sintered metals are generally considered to be related to the properties and spatial distribution of their constituents. Additionally, pores have become a critical concern as they tend to segregate and grow under the electrothermal effect during service16,17, propagating into cracks and delamination18,19,20,21, causing serious degradation or failure. As a result, pore characteristics such as porosity22,23,24, pore shape11,24, and sintering neck size25 are of significant interest. Many studies have reported the effects of pore features on structural properties, with some even employing finite element method (FEM) simulations to analyze observed cross-sectional microstructures to predict bonding performance24,26. Typically, most studies assess local microstructures as a representation of the entire bonding layer, calculating its ‘effective properties’ based on microstructure features extracted from limited SEM images. This approach may exhibit reasonable accuracy in uniform microstructures and allows for comparisons between samples with distinctly different morphologies. However, microstructure formation naturally involves randomness, which raises two critical challenges to the microstructure-property assessment: accuracy (to what extent can local microstructures accurately represent the overall structure?) and pertinence (to what extent are selected features observed from morphology relevant to properties?).

Recently, artificial intelligence (AI) tools have been widely employed to solve problems in underlying correlations through data-driven approaches27,28, and are even expected to synthesize the findings and deliver previously unknown discoveries for various fields without human intervention29,30. In the material engineering aspect, establishing correlations between microstructure, macroscale properties, and processing conditions becomes attractive. Li et al.31 proposed a CNN method for determining the relationship between mechanical properties and microstructure for shale, to exploit the implicit mappings between the microstructure and the effective moduli. Given that the performance of porous structures is mainly dictated by distribution pattern of pores, a similar approach should be possible in sintering microstructures. In fact, many studies have explored this direction. Najjar et al.32 predicted the mechanical properties of Cu-Al2O2 nanocomposites using a combination of micromechanical modeling, FEM, and machine learning, where the training data was derived from computed using the FEM model rather than experiments. Du et al.33 built a generative AI model which generates random cross-sectional morphologies showing microstructures that meet the input effective thermal conductivity, to help them design their desired structure based on a specific conductivity. However, achieving this still requires linking process parameters to the generated morphology. In an inverse way, Tang et al.34 built a model to predict alumina sintered microstructure generated through various laser sintering processes. Wijaya et al.15 argued a single mathematical formulation based on porosity and material properties is insufficient. Instead, they utilized a multi-method machine learning approach, including feature extraction and synthetic microstructure reconstruction, to study the correlation of the extracted microstructure features and electrical conductivity.

Obviously, since machine learning training, especially deep learning, requires a large dataset for training, data collection and experimental workload are highly demanded. To address this challenge, researchers sometimes employed algorithms like generative adversarial networks (GANs) to generate a great number of “virtual samples”. These samples, combined with their property data calculated by FEM simulations, construct a training dataset. Only a reasonable amount of experimental data is necessary to validate the FEM results and ensure the accuracy of predicted performance values33. With this approach, various machine learning algorithms including k-means35, artificial neural network (ANN)36 have been successfully employed to various tasks. However, most current models are trained to imitate microstructural morphology without embedding physical knowledge into the learning process, which limits the accuracy and physical consistency of the generated microstructures. Methods that combine physical constraints, e.g. physics-informed neural networks (PINNs)37, may help address these limitations, if the relevant physical descriptors are correctly identified. Moreover, it is important to note that data-driven models have long been criticized for their lack of interpretability, as the way in which a model represents scientific laws rarely aligns with conventional scientific frameworks, making them difficult for researchers to analyze28.

In this study, we used machine learning as an analytical tool to enhance our understanding of pore features in sintered porous structures. The features that are commonly used to characterize cross-sectional morphologies were manually extracted, with their correlations analyzed and derived. According to correlations, related features were grouped and transformed into representative descriptors through principal component analysis (PCA). Different classification models were developed and trained to reveal the pertinence and significance of these descriptors in microstructure-property assessment.

Methods

Sample preparation

In this study, a series of SEM images captured from cross-sections of Cu-sintered bonding samples, as a representative porous structure, were used as training materials. The nano-sized Cu particle paste, as the bonding material, is a product provided by Daicel Co. The bonded sample in a sandwich way is composed of a SiC chip (5 × 5 × 0.36 mm) with a Cu electroplated bottom layer as the upper component and a direct-bonded copper (DBC) substrate (32 × 43 × 0.72 mm) as the lower component. Before bonding, both the chips and the substrates were cleaned using dilute hydrochloric acid, deionized water, and alcohol. The Cu paste was then applied to the substrate using a mask with a printing size of 5 × 5 × 0.1 mm. After the substrates with printed paste were preheated in a muffle furnace (Budatech VS160) under N2, and the chips were assembled on the printed paste, followed by a bonding process using a large-area pressure bonding machine (RB-100D, Ayumi Co.) with different temperatures and pressures. The sample preparation process is illustrated in Supplementary 1a. The fabricated samples underwent thermal cycling tests (TES-11-A, ESPEC), alternating between − 50 °C and 200 °C with 30-min dwell time at each extreme. The thermal cycling profile is shown in Supplementary 1b. Four groups of samples were prepared under different sintering temperatures and assist pressures, and each sample underwent thermal cycling to increase microstructural heterogeneity of the microstructure within the same sample. Detailed experimental conditions are presented in Table 1.

Table 1 Bonding process parameters and thermal cycles for samples involved in this study.
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Feature extraction

The samples were embedded in resin, then cut and polished to examine their cross sections. To minimize the influence of polishing on the pore morphology, ion milling was applied for final polishing. SEM images were captured at a consistent magnification (10,000×) from randomly selected areas within each sample, thus constructing a database of over 120 cross-sectional images. The SEM datasets used in this work are available in the figshare repository (DOI: https://doi.org/10.6084/m9.figshare.29252036). This study focuses on pore features that are widely studied in current research. Instead of using computer vision techniques like convolutional neural networks (CNNs), pores were identified and measured automatically using the CV2 module in Python. This approach significantly reduces training costs and maintains the physical meaning of the extracted features, compared with computer vision methods such as convolution. The definitions of extracted pore features are illustrated in Fig. 1. It is noted that the pores smaller than 2 pixels (20 nm) were ignored from being collected, to filter out the noises that inevitably exist in the SEM images. From the viewpoint of the properties considered in this study, such nanoscale pores are considered to effectively vanish. However, it is noted that the nanoscale pores may still contribute to some properties and behaviors, such as fatigue crack initiation, which are not addressed in this study.

Fig. 1
Fig. 1
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Definition of the pore features extracted from a cross-sectional SEM image.

The measurements, as mentioned, were performed on the cross-sectional SEM images. In practice, the pores in sintered microstructures are typically tiny and densely distributed, which makes large-scale three-dimensional (3D) characterization difficult. Although FIB–SEM serial sectioning can provide precise 3D microstructural10,15, its low efficiency limits the applicability for large-area evaluation. As a result, 2D cross section has been widely adopted to estimate the quality of sintered microstructures3,4,5,6,7,8,9, under the assumption that these sections reflect the 3D characteristics. It is noted that if anisotropic effects exist, such as temperature gradients or pressure, cross-sections captured normal to these gradients may exhibit differences according to section position. In this study, the samples were prepared with anisotropies of heat propagation and pressure, and these two directions were aligned. Therefore, we carefully prepared the cross-sectional samples at comparable positions along this common direction to minimize the influence of anisotropy. The extracted pore features and their calculation methods are summarized as follows and in Fig. 1:

Dispersion (d): the average distance between a pore and its four nearest neighbors, as detailed in Supplementary 2.

Aspect ratio of pores (δ): the ratio of the width to height of a pore’s minimum enclosing rectangle.

Area of pore (A): the total area of an individual pore.

Circularity (c): a measure of how closely a pore resembles an ideal circle, with 1 representing an ideal circle.

Radius (r): the radius of the minimum circumscribed circle of the pore.

Neck Length (Li): calculated under the assumption of ideal spherical particles sintering based on the distances between pores. See identification strategy in Supplementary 2.

angle (α): The angle between the line connecting a pore’s center to the image center and the X-axis. This parameter is used to monitor feature extraction but is not included in model training or calculations.

Additionally, statistical descriptors were derived from these features for each image:

Number of pores (n): the total number of pores in the image;

Average dispersion (\(\:\stackrel{-}{d}\)): the mean dispersion of all pores in the image;

Average area of pores (\(\:\stackrel{-}{A}\)): the mean area of all pores in the image;

Average aspect ratio of pores (\(\:\stackrel{-}{\delta\:}\)): the mean aspect ratio of all pores in the image;

Porosity (p): the ratio of pore pixels to the total image pixels;

Average neck length (\(\:\stackrel{-}{L}\)): the mean neck length of all pores in the image;

Proportion of pores whose δ value range in 0.5 to 2 (nδ): the proportion of pores with an aspect ratio between 0.5 and 2 relative to the total pore number;

Proportion of pores whose circularity value is larger than 0.69 (nc): the proportion of pores with circularity exceeding 0.69, corresponding to an ellipse whose minimum enclosing rectangle has an aspect ratio between 0.5 and 2.

It is important to note that the features mentioned above may be referred to differently in other literature. For instance, “circularity” in this study is also referred to as “elongation"14 and as “shape factor"38. Therefore, careful attention should be given to each feature’s definition and calculation method. Further details on feature extraction strategies, physical interpretations, and calculation methods are provided in Supplementary 2, and extracted dataset structure is provided in Supplementary 3.

Learning models

The extracted pore features from each SEM image were used as feature vectors, while the corresponding image group served as the label for training classification models. In this study, four machine learning algorithms were employed, covering weak to strong learners: K-nearest neighbor (KNN), support vector machine (SVM), random forest (RF), and artificial neural networks (ANN). All the classification models were trained on CPU with fixed random seeds (10) to ensure reproducibility. As this study adopted PCA to reduce the dimensionality of the input features, relatively simple model architectures were sufficient to achieve the target classification accuracy (~ 85%). After model optimization, the selected hyperparameters were as follows: KNN with k = 3; SVM with an “RBF” kernel using the one-versus-one scheme; RF with max_features = 4 and n_estimator = 40; and ANN with hidden layers of (3, 7, 9). Hyperparameter tuning was performed using a grid search strategy; an example of the tuning for ANN is shown in Supplementary 4b,c. For larger datasets or industrial-scale applications, more advanced tuning strategies and greater expertise may be required.

In addition, to compare the physically defined features with those extracted by unsupervised deep learning, a simple VAE–ANN framework was also constructed for complementary analysis in this study. A variational autoencoder (VAE) was employed to encode SEM images into latent feature vectors, which were subsequently used in ANN classification. The comparison between the original SEM images and the reconstructed images generated by the trained VAE model is shown in Supplementary 4a. The similarity between the two images indicates that the encoder of VAE captures the main features of the SEM images. The VAE adopts a U-Net-based architecture and was trained with GPU (N6000 ada) with CUDA (12.6), by minimizing the sum of the reconstruction loss and the Kullback–Leibler divergence. To reduce computational cost, the original SEM images were downsampled to one-quarter of their original resolution (222 × 320 pixels), and were then encoded to a 64-dimensional latent vector for the next ANN classification. For classification model training, the dataset was randomly split into a training set (70% of the data) and a test set using the validation dataset method. All analyses and models were implemented in Python (version 3.11.7). The main Python libraries included CV2, NumPy (1.26.4), Sklearn (1.2.2), Pytorch (2.5.1).

Results and discussions

Feature correlation

Features were extracted from a total of 124 cross-sectional SEM images, capturing over 26,000 pores per group. For data inspection, density maps of pore features across different groups are shown in Fig. 2. The distribution of the angle (α), shown in Fig. 2a, reveals the same pattern across all groups: four symmetrical peaks appear. This occurs because SEM images are rectangular, with longer diagonals and shorter heights. If pores are randomly and evenly distributed, their occurrence at each angle should be proportional to the length of the line segment passing through the center at that angle. This confirms that pore identification is unbiased and reliable, unaffected by processing conditions, which is the basis for the subsequent discussion of the correlations between pore features.

Fig. 2
Fig. 2
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Distribution density maps of individual pore features extracted from different samples: (a) angle, (b) dispersion, (c) radius, (d) area, (e) circularity, (f) aspect ratio.

Overall, except for Group #4, the remaining three sample groups exhibited similar pore feature distributions. Group #4, which underwent the highest sintering temperature, pressure, and thermal cycles, generated pores that were smaller (smaller radius), less dispersed (lower dispersion), and more irregular in shape (lower circularity). Statistical descriptors from individual image reflect variation of pore features in different positions with the same area, as shown in Fig. 3, which also generally align with the intuitive impression obtained by observing local morphologies from SEM images. Through comparison of statisitic features across groups, trends in how pore features change with sintering conditions can be analyzed, as summarized in Table 2. However, the density maps also reveal significant overlap in feature distributions between groups. For example, a local area with 15% porosity could belong to any sample group, though its properties greatly differ. Therefore, it is unreliable to evaluate or simulate material performance based on a single feature, such as modeling local morphology for FEM simulation, where the sintered structure is treated as a two-phase mixture of air and metal. On the contrary, incorporating all features simultaneously into one model may reduce errors caused by individual features, yet possible correlations among features and their different influence on properties remain unclear.

Fig. 3
Fig. 3
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Density distribution maps of statistical pore features extracted from different images: (a) porosity, (b) average neck length, (c) pore numbers, (d) average pore area, (e) proportion of pores with aspect ratio in (0.5, 2), (f) dispersion, (g) proportion of pores with circularity > 0.69, (h) average circularity, (i) average aspect ratio.

Table 2 Summary of observed trends in pore features under changing processing conditions.
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For that, a Spearman correlation analysis was conducted on the statistical descriptors obtained from individual SEM images, and a heat map of the results is shown in Fig. 4. Spearman correlation coefficients range from − 1 to 1, with values closer to ± 1 indicating stronger correlations between two variables. In total, eight pairs of features exhibited relatively strong correlations (absolute value > 0.7), which can be categorized into two groups. One group, highlighted in yellow boxes, relates to pore counts and distribution characteristics, including porosity (p), average neck length (\(\:\stackrel{-}{L}\)), average dispersion (\(\:\stackrel{-}{d}\)), and number of pores (n). The others, highlighted in brown boxes, relate to pore shape, including average circularity (c), proportion of pores whose δ \(\in\) (0.5, 2) (nδ), proportion of pores whose c > 0.69 (nc). Due to strong intra-group correlations, nine coefficients exhibit higher values (corresponding to the combinations of four and three features within two groups: \(\:{C}_{4}^{2}+{C}_{3}^{2}\)). The pairwise relationships between features within correlated groups are visualized in Fig. 5, with each point representing one image. In the shape group, a clear trend was observed: images with a higher count of nearly circular pores (c > 0.69) also exhibited higher average circularity (Fig. 5a). Assuming pore circularity follows a Gaussian distribution with mean \(\:\stackrel{-}{c}\) and standard deviation σ, the proportion of pores with c > 0.69 can be estimated:

Fig. 4
Fig. 4
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Spearman correlation matrix among all statistical descriptors of images.

Fig. 5
Fig. 5
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The pairwise relationships between features within correlated groups. The shape group: (a) proportion of pores with circularity > 0.69 against average circularity, and (b) proportion of pores with aspect ratio in (0.5, 2) against average circularity in the shape group. The porosity group: (c) pore numbers against porosity, (d) dispersion against neck length, (e) neck length against porosity, and (f) dispersion against porosity.

$$\:\begin{array}{c}P\left(X>0.69\right)=1-\varphi\:\left(\frac{0.69-\stackrel{-}{c}}{\sigma}\right) .\end{array}$$
(1)

where \(\:\varphi\:\) is the cumulative distribution function (CDF) of the standard Gaussian distribution. The regression line fits the data well, validating this relationship. In addition, Fig. 5b reveals that images with higher average circularity tend to have more pores with aspect ratios between 1:2 and 2:1 (a range corresponding to a circularity value of approximately 0.69), suggesting a potential linear relationship.

From the porosity group, a clear linear relationship is observed between porosity (p) and pore number (n), as shown in Fig. 5c. This can be directly explained by their mathematical relationship:

$$\:\begin{array}{c}p=\frac{n\cdot\:\stackrel{-}{A}}{S}.\end{array}$$
(2)

where S is the area of the SEM image. Notice that \(\:\stackrel{-}{A}\) remains nearly constant across all four groups (Fig. 3), the \(\:\stackrel{-}{A}/S\) can also be regarded as constant, hence the linear regression. Similarly, a strong correlation is observed between average dispersion (\(\:\stackrel{-}{d}\)) and neck length (\(\:\stackrel{-}{L}\)). For any two adjacent pores i and j, the neck length can be approximated by:

$$\:\begin{array}{c}{L}_{ij}={d}_{i}-{r}_{i}-{r}_{j}.\end{array}$$
(3)

where r is the radius of the minimum circumscribed circle of each pore. On average, this relationship holds at the group level as well, with a regression slope of 1.157, close to the theoretical value of 1. In the case where particles are in their initial, un-sintered state (i.e., neck length = 0), dispersion (\(\:\stackrel{-}{d}\)) corresponds to the ideal geometric spacing. Assuming a two-dimensional arrangement of four particles, the theoretical distance is \(\:\sqrt{2}R/3\), where R is the particle radius (see Supplementary 2). The particles used in this study range from 50 to 200 nm in radius with the average radius of 120 nm, which aligns well with the observed regression intercept of approximately 0.118. Notice that the regression may be invalid when porosity approach to 0, as the calculated neck length would tend toward infinity. However, in practical observations, pores are always inevitable and generally well-distributed. Another noteworthy observation is the strong correlation between porosity (p) and dispersion (\(\:\stackrel{-}{d}\)). In particular, the scatter points representing Group #4 are distinctly separated from those of the other three groups. To better understand this relationship, we further examined the relationship between porosity and dispersion, as detailed below:

In an SEM image of area S, assume n pores are evenly distributed. The pore density, therefore, is given by:

$$\:\begin{array}{c}\lambda\:=\frac{n}{S}.\end{array}$$
(4)

When n is large, the positional relationships among pores can be approximated using a two-dimensional Poisson process. Under this assumption, for one pore, the probability that its nearest neighboring pore appears within a distance r follows:

$$\:\begin{array}{c}F\left(r\right)=1-{e}^{-\pi\:\lambda\:{r}^{2}} \end{array}$$
(5)

The corresponding probability density function is:

$$\:\begin{array}{c}f\left(r\right)=2\pi\:\lambda\:r{e}^{-\pi\:\lambda\:{r}^{2}} \end{array}$$
(6)

The expected nearest-neighbor distance, which represents the average pore dispersion d̅, can be calculated as:

$$\:\begin{array}{c}\stackrel{-}{d}={\int\:}_{0\varvec{}}^{\varvec{\infty\:}}rf\left(r\right)dr.\end{array}$$
(7)
$$\:\begin{array}{c}=2\pi\:\lambda\:{\int\:}_{0\varvec{}}^{\varvec{\infty\:}}{r}^{2}{e}^{-\pi\:\lambda\:{r}^{2}}dr.\end{array}$$
(8)

To solve the integral, the gamma function is introduced:

$$\:\begin{array}{c}\varGamma\:\left(n+\frac{1}{2}\right)={\int\:}_{0}^{\varvec{\infty\:}}{r}^{z-\frac{1}{2}}{e}^{-r}dr=\frac{\left(2n\right)!\sqrt{\pi\:}}{n!{4}^{n}}.\end{array}$$
(9)

Taking the value when n = 1, we have:

$$\:\begin{array}{c}\varGamma\:\left(\frac{3}{2}\right)={\int\:}_{0}^{\varvec{\infty\:}}{r}^{\frac{1}{2}}{e}^{-r}dr=\frac{\sqrt{\pi\:}}{2}.\end{array}$$
(10)

Substituting the result into Eq. (8) and using the substitution:

$$\:\begin{array}{c}\stackrel{-}{d}=\frac{1}{2}\sqrt{\frac{S}{n}}=\frac{1}{2}\sqrt{\frac{\stackrel{-}{A}}{p}} \end{array}$$
(11)

That is, when calculating from a cross-sectional image, the average distance (e.g. dispersion) between pores is directly correlated to the average pore area \(\:\stackrel{-}{A}\) and porosity \(\:p\). In the circumstance that \(\:\stackrel{-}{A}\) can be assumed constant, such as within the same sample, or across samples with minimal difference, \(\:\stackrel{-}{d}\) becomes inversely proportional to the square root of \(\:p\). Based on this relationship, regression analyses were conducted for each sample group, and the resulting equations are presented in Fig. 5e,f. In groups #1, #2, and #3, the distribution of \(\:\stackrel{-}{A}\) is relatively narrow (see Fig. 4), allowing it to be treated as constant. As a result, their data points align well with the regression curves. Specifically, \(\:\stackrel{-}{A}\) for these groups are similar (0.020, 0.213, 0.020 µm2, respectively), so their corresponding regression coefficients are nearly identical (0.084, 0.080, and 0.080). In contrast, group #4 exhibits a wider and different \(\:\stackrel{-}{A}\) distribution, hence the great deviation from the regression.

In summary, due to the strong correlations among features within each group, it is reasonable to use a single representative feature to describe the microstructure, when detailed numerical analysis is not required. For example, the observation that many studies have reported, that processing conditions significantly affect porosity and pore shape (typically assessed by circularity), can be regarded as a specific case within the principle proposed in this study.

Significance analysis for features identified by machine learning

Given the strong correlations among features, we first applied PCA to reduce dimensionality before training the machine learning model. PCA transforms the original n-dimensional feature set into a new set of uncorrelated variables called principal components, which retain as much of the original data’s variability as possible. Through this approach, the original, highly correlated features are replaced with independent and fewer variables. If PCA is applied directly to the full set of features (see Supplementary 5a), the first four principal components accounted for over 95% of the total variance. However, the four components will lose clear physical meaning, since PCA is essentially a coordinate transformation. To preserve interpretability, we instead applied PCA separately to the porosity group and shape group and obtained two representative descriptors: porosity factor (PF) and shape factor (SF). Both factors were standardized using z-score normalization. See in Supplementary 5b, each factor explained more than 85% of the variance in its respective group.

After dimensionality reduction, each cross-sectional image is characterized by four features: PF, SF, \(\:\stackrel{-}{\delta\:}\), and \(\:\stackrel{-}{A}\). As shown in Fig. 6a, the Spearman correlation coefficients among these four features are significantly lower. The distribution of images based on PF and SF, as shown in Fig. 6b, shows a weak positive correlation, and group #4 clearly separated from the other three groups. Most studies have commonly reported when the sintering temperature or the pressure increases, pores in the samples become larger and more regular with lower porosity14,38. The above relationship between PF and SF suggests that a similar trend occurs even within the same sample: regions with lower porosity tend to have more regularly shaped pores. This is likely due to uneven heat transfer or pressure distribution during processing. Such non-uniformity can affect the reliability of evaluations based on selected cross-sectional areas: depending on which region is analyzed, local observation may lead to opposite conclusions.

Fig. 6
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Features after dimensionality reduction by PCA: (a) Spearman correlation matrix, (b) relationship between the two principal components: SF and PF.

The four features were used as input for classification tasks. Four commonly used machine learning models were trained: SVM, KNN, RF, and ANN, covering both weak and strong learners. The goal of the four models is to judge the group that an image belongs to basing on the four features. All models were trained on the same training dataset and scored on the same testing dataset. The accuracy of each model and corresponding classification results are shown in Fig. 7. A performance metric (precision, recall, and F1-score) summarizes each model’s precision, while the plots on the right display the detailed classification results. In these plots, green dots indicate the group that the models believe an image belongs to, and red dots indicate the correct group of that image; incorrect classification will leave a red dot without covering by green dot. All four models are able to recognize samples’ group and achieve accuracy above 81%. Among them, the SVM model (weak learner) had the lowest accuracy 81%, while the ANN (strong learner) achieved the highest 90%. These results suggest that when all four features are considered together, the differences in microstructures across groups become clear and consistent. Therefore, it is reasonable to consider that the four features represented most pore morphology changes that processes involved in this study may affect. For comparison, the VAE model encoded the SEM images into a 64-dimensional latent space without manual intervention. Although the strong similarity between the original and reconstructed images (Fig. S4a) indicates that the latent space preserves sufficient image information, it is noted that the latent features lack physical interpretability. Another ANN classification model was then trained using these latent features as inputs. The optimal hidden-layer structure identified during tuning was (34, 12, 34), which is more complex than the ANN based on physical features (3, 7, 9). However, the resulting classification accuracy was only 85%, lower than the 90% achieved using physical descriptors. The result is reasonable as the unsupervised VAE extracts features mainly based on image similarity through convolution, without explicitly focusing on underlying physical correlations. Therefore, visually similar SEM images may be mapped to similar latent spaces, even if their pore features differ. This comparison further demonstrated the effectiveness of the proposed physical descriptors for representing characterization.

Fig. 7
Fig. 7
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Classification accuracy of four learning models: (a) SVM, (b) KNN, (c) RF, (d) ANN, and (e) VAE-ANN framework. Left—accuracy scores including precision, recall, and F1-score; Right—comparison between classified and actual groups on the test set.

As expected, the features that change most across groups should carry greater weight in models, namely, a feature’s importance within a model reflects how significantly it is influenced by the processes. To quantify this, we assessed the contribution of each feature to the model’s prediction accuracy. We selectively replace one feature in the test data with random noise while keeping the others unchanged; the resulting drop in prediction accuracy indicates the relative importance of the feature. In the case of the Random Forest model, which consists of multiple decision trees, feature importance was computed by measuring the average reduction in residual sum of squares caused by each feature across all trees. Figure 8 summarizes the importance of features across all four models. Among the four descriptors, the aspect ratio \(\:\stackrel{-}{\delta\:}\) and the average pore area \(\:\stackrel{-}{A}\) showed relatively lower importance in the four models, suggesting that the process parameters involved in this study (temperature, pressure, and TCT) had a slight impact on these two features. In contrast, PF and SF emerged as the dominant contributors across all models: SF held the highest weight in the SVM model, while PF was the most important feature in the other three models.

Fig. 8
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Feature importance analysis for each machine learning model: (a) SVM, (b) KNN, (c) RF, and (d) ANN.

This result is consistent with the underlying mechanisms that have been inferred from experimental observations. In the initial stage, neighboring particles form necks primarily through surface diffusion paths, and their geometry and applied pressure largely determine the neck contact area, in turn deciding the initial pore shape. As sintering progresses, or during subsequent thermal treatments such as aging or TCT, temperature provides sufficient driving force for bulk diffusion paths to become dominant. This intermediate stage is primarily responsible for densification and the reduction of porosity9. The result that PF and SF show the strongest significance across models matches the microstructural evolution well. However, the remaining descriptors, \(\:\stackrel{-}{\delta\:}\) and \(\:\stackrel{-}{A}\), also play a critical role. As discussed earlier, an ideal uniform process condition is rarely achieved in practice. Even under the same processing parameters, pressure or temperature across local regions sometimes differ due to anisotropy or non-uniformity, which can result in differences in local porosity and pore shape. For example, if the actual local pressure is lower than the setting value, PF and SF calculated from that local region may resemble those of samples processed at a lower set pressure. In such cases, \(\:\stackrel{-}{\delta\:}\) becomes informative. Pressure gradients can induce particle flow from high-pressure regions toward lower-pressure regions, generating shear stress. The shear stress affects the \(\:\stackrel{-}{\delta\:}\), and provides additional information that distinguishes local low-pressure regions from low-pressure processing conditions. In this sense, \(\:\stackrel{-}{\delta\:}\) and \(\:\stackrel{-}{A}\) act as complementary descriptors that capture secondary effects beyond porosity and shape alone. The results not only explain the intuitive observations that processes mainly affect porosity and pore shape, but more importantly, provide a more reliable understanding of microstructural behavior by comprehensive analysis of multiple features.

Eventually, the four physical descriptors, which can be extracted from local SEM images, are expected to remain statistically stable within the same processing group, while showing clear differences across different groups. This stability is maintained even when the intuitive difference can be observed from local images within the same group. We also believe that the conclusion can be extended to other solid-state sintering porous structures in other applications, including cases with different particle size distributions or even non-metallic particles, as long as they share similar processes and microstructural characteristics. Therefore, the extracted descriptors are well suited as inputs for predictive models. In the most straightforward extension, measured physical properties (e.g., thermal conductivity or mechanical strength) may serve as labels, with the input data, a regression model between morphology and performance can be trained. Furthermore, these descriptors could be incorporated into the loss functions of generative models, such as GANs or flow-matching models, to constrain the generated microstructures to keep physical consistency. Future work will explore these directions in detail.

Conclusions

Microstructure morphology is highly related to material properties, yet not all features extracted from morphology reliably describe microstructure, especially considering the inhomogeneity of microstructures. In this study, we systematically assess the accuracy and pertinence of commonly used features in characterizing sintered microstructures, based on large-scale analysis of pore data from cross-sectional SEM images.

The results reveal that the correlated features are categorized into two groups related to pore distribution and shape, and each group exhibits internal mathematical relationships that were inferred and validated through regression. The two groups of features were reduced to two principal physical descriptors (PF and SF) to reduce data redundancy and complexity. Based on these, four classification models (SVM, KNN, RF, and ANN) were trained and achieved high accuracy in identifying microstructures. The advantage of the proposed method has been proven by comparing it with an unsupervised VAE-ANN framework. Moreover, the feature significance analysis demonstrated the effectiveness of the simplified features to describe the porous morphology.

As this study focuses on pore-based features visible in cross-sectional SEM images, other microstructural features, such as grain orientation and size, may also influence bonding performance. Analysis and quantifying the influences of other potential factors remain critical for future investigation.