The chemical composition of cement clinker, particularly the percentage of key oxides, is crucial for evaluating cement quality and is vital for engineering-accident liability tracing. However, traditional methods for chemical composition analysis rely heavily on unhydrated original cement samples, which are often missing, improperly stored, or ambiguously labeled in engineering management, resulting in an insufficient basis for subsequent identification. Once cement hydrates, active minerals undergo irreversible reactions, making it challenging to invert the original clinker composition directly from the hardened paste. In contrast, backscattered electron (BSE) images of the hardened cement paste contain sufficient physical information to distinguish the original chemical compositions of different cement clinkers, thus providing a potential pathway for post-event tracing. Although deep learning methods have been explored for microstructural image mapping to infer macroscopic properties, conventional regression models are typically "black-box" point predictors that cannot quantify prediction uncertainty. The conditional probability density estimation method can not only provide the most probable predicted value, but also output the complete estimated probability-density curve along with its confidence assessment, thereby enabling quantitative characterization of prediction uncertainty. Accordingly, this study proposes a conditional probability density estimation method based on a Deconvolutional Density Network (DDN). Utilizing BSE images of hardened paste together with encoded water-to-cement (W/C) ratios and curing ages as conditional inputs, this model directly predicts the full conditional probability density of the main clinker oxide contents, providing a new objective and quantitative pathway for quality liability tracing.

Cement clinker samples from 47 different plants were used to prepare cement paste specimens with W/C ratios ranging from 0.25 to 0.70 and curing ages from 1 d to 28 d. The hydration was terminated by immersion in anhydrous ethanol, and the samples were polished for subsequent scanning. BSE images were acquired via a JSM-7610F field-emission SEM at 200× magnification and an accelerating voltage of 15 kV. For each sample, two scans were performed, and nine non-overlapping regions were captured per scan, resulting in a dataset of 844 independent BSE images. The proposed model consists of a feature extraction layer based on a residual network (ResNet) with self-supervised learning and cross-layer skip connections, a conditional encoder, and deconvolutional layers. The ResNet extracts multiscale morphological features from local textures to global patterns. The conditional encoder embeds the W/C ratio and curing-age conditions into the image features. The DDN is then employed to output the conditional probability density distributions for the percentages of eight oxides (CaO, SiO₂, Al₂O₃, Fe₂O₃, SO₃, Na₂O, MgO, and K₂O). The deconvolutional layers map the fused features to a smooth probability distribution over 256 discretized intervals. To evaluate the model, accuracy (ACC) under different confidence levels (α), mean absolute error (MAE), and mean squared error (MSE) were adopted as evaluation metrics.

The estimation results of the proposed model indicate that, at a confidence level of α = 90%, the model achieves an accuracy of over 90% in estimating the percentage contents of five oxides (SiO₂, Al₂O₃, Fe₂O₃, SO₃, and Na₂O) on the test set. When the confidence level is relaxed to α = 75%, the estimation accuracy for all eight oxides exceeds 90%. This could be attributed to the DDN's capability of modeling complete probability density distributions instead of simple point estimates, effectively resolving the nonunique-solution problem stemming from the intrinsic information ambiguity of BSE imaging. In terms of MAE and MSE metrics, the model exhibits relatively higher absolute errors for components with higher contents, such as CaO. This is generally attributed to the nature of absolute error metrics, which are positively correlated with the oxide content. Conversely, low-content oxides like Na₂O show smaller estimation errors, possibly because their larger fluctuation ranges among different samples allow the model to capture clearer mapping relationships. In addition, ablation experiments further reveal that removing the W/C ratio and age inputs leads to a decrease in estimation accuracy across all confidence levels, demonstrating the effectiveness of the conditional encoder in integrating critical hydration parameters. Compared with the deterministic ResNet regression model, the proposed DDN model exhibits significantly better stability of estimation performance under higher precision requirements. This result demonstrates that the DDN model effectively mitigates the inherent uncertainty in the mapping from BSE images to chemical compositions, a limitation typical of deterministic regression models that lack reliability quantification.

The main conclusions of this study are summarized as follows. The proposed probability density estimation method successfully maps BSE images of hardened cement paste to the oxide contents of clinker. Experimental results demonstrate that the model achieves an accuracy of over 90% for five oxides at a confidence level of 90%. The adoption of the Deconvolutional Density Network enables the model not only to output the most probable value of each oxide content but also to provide a complete probability density curve with corresponding confidence assessment, thereby overcoming the limitation of traditional regression models that lack quantifiable reliability for quality tracing applications. Moreover, the ResNet-based feature extraction with cross-layer skip connections effectively captures multiscale features from BSE images, while the integration of W/C ratio and curing age via the conditional encoder significantly enhances the model predictive capability. This study demonstrates that the probabilistic retrospective estimation based on BSE images and DDN offers a robust and objective means to quantitatively infer the original clinker composition from hardened paste. As such, it holds promise as a transformative technique for cement-quality arbitration in engineering-accident liability investigations.