Ultra-high-performance concrete (UHPC) has been widely used in bridges, tunnels, prefabricated structures, and other demanding engineering applications owing to its high compressive strength, excellent toughness, and superior durability. However, conventional UHPC typically requires a high cement content, high dosages of reactive powders, and steel fiber reinforcement, leading to considerably higher carbon emissions than ordinary concrete. The partial replacement of ordinary Portland cement (OPC) with low-carbon cementitious materials, such as ground granulated blast-furnace slag, fly ash, silica fume, limestone powder, belite cement, and other supplementary cementitious materials, has therefore become an important strategy for developing low-carbon UHPC. The incorporation of low-carbon binders enhances sustainability but also increases the complexity of mixture proportion design. Variations in oxide composition, reaction activity, particle size distribution, and hydration products jointly affect the compressive strength of low-carbon UHPC through hydration reactions, particle packing, fiber reinforcement, and curing conditions. Although machine learning models have shown good prediction performance in concrete strength prediction, most of them remain black-box approaches and provide limited physical interpretability. Post-hoc explanation methods, such as feature importance analysis, SHAP, and LIME, can identify statistical associations, but they may generate physically unrealistic virtual mixture proportions when strict mix proportion constraints and material boundaries are ignored. To address these limitations, this study proposes an interpretable Bayesian network (BN) modelling framework for the intelligent design of low-carbon UHPC. The proposed framework integrates hydration-packing mechanism priors with data-driven structure learning, aiming to describe the conditional dependencies among mixture proportion parameters, intermediate mechanism variables, and 28 d compressive strength. By introducing physically meaningful intermediate variables, the model is expected to provide both reliable strength prediction and mechanistic interpretation for low-carbon UHPC mixture design.

A total of 491 original UHPC mixture proportion records were collected through a systematic literature review following the PRISMA guidelines. After unit standardization, variable transformation, outlier checking, and valid sample screening, 136 valid samples were retained for model construction. The target variable was the 28 d compressive strength of UHPC, denoted as Z₁. To improve the physical interpretability of the model, the original high-dimensional mixture proportion variables were transformed into a hydration-packing feature system. The material/process variables included the cementitious material reaction activity index X₀, water-to-binder ratio X₁, activator-to-binder ratio X₂, binder-to-sand ratio X₃, and steel fiber volume fraction X₄. The intermediate mechanism variables included the modified CaO/SiO₂ molar ratio Y₁, the modified SO₃/Al₂O₃ molar ratio Y₂, and the initial particle packing density Y₃. These variables were used to represent C-S-H formation potential, sulfate-aluminate reaction balance, and the initial physical compactness of the particle system, respectively. Based on the mechanism of low-carbon UHPC strength formation, a three-layer BN structure was constructed, consisting of a material and process parameter layer, an intermediate mechanism layer, and a target performance layer. Mechanism-informed constraints were introduced to prevent physically unreasonable paths, such as reverse links from compressive strength to mix design variables. Structure learning was performed using a combined strategy involving the hill-climbing, MMHC, and Tabu Search algorithms. bootstrap resampling was adopted to evaluate edge stability, and the BGe score was used as the main scoring criterion, with BIC and AIC used for sensitivity comparison. For parameter learning, maximum likelihood estimation was used for auxiliary variables, while Bayesian ridge regression was adopted for continuous compressive strength prediction. To reduce the risk of data leakage during model evaluation, all preprocessing steps, including variable transformation, standardization parameter estimation, outlier threshold determination, BN structure learning, bootstrap-based stable edge selection, and parameter learning, were embedded within the training folds of a 10-fold cross-validation procedure. The validation folds were used only for final performance evaluation.

The proposed BN model achieved good prediction accuracy for the 28 d compressive strength of low-carbon UHPC. Based on 10-fold cross-validation and Bootstrap evaluation, the model obtained an average RMSE of 0.281 3, an average MAE of 0.203 1, and an average R² of 0.910 1. The predicted mean compressive strength was 162.11 MPa, with a standard deviation of 24.53 MPa and an average confidence interval width of 46.04 MPa. Compared with random forest and support vector regression models, the BN model provided comparable prediction performance while also offering explicit conditional dependency paths and uncertainty representation. Correlation analysis indicated that curing temperature, pozzolanic cementitious materials, quartz sand, and steel fiber content generally were positively correlated with compressive strength, whereas the water-to-binder ratio and cement content were weakly negatively correlated. However, Pearson correlation analysis describes only linear statistical relationships and could not reveal hierarchical conditional dependencies among variables. Therefore, BN modeling was necessary to identify indirect effects and mechanism-related pathways. The final BN structure demonstrated that the modified CaO/SiO₂ molar ratio Y₁ acted as an important upstream node associated with compressive strength. This indicates that the active oxide composition of cementitious materials plays a fundamental role in low-carbon UHPC strength formation. The modified SO₃/Al₂O₃ molar ratio Y₂ served as an important intermediate node, reflecting the role of sulfate-aluminate reaction balance in hydration product formation. The initial particle packing density Y₃ was directly related to the physical compactness of the matrix and contributed substantially to strength development. Mechanism-informed intervention sensitivity analysis further showed that Y₁, Y₂, Y₃, and their interaction contributed 28.7%, 19.2%, 35.1%, and 17.0% to compressive strength, respectively. When Y₁ and Y₂ were combined as the hydration effect, the hydration effect, packing effect, and hydration-packing interaction contributed 47.9%, 35.1%, and 17.0%, respectively. These results suggest that the strength development of low-carbon UHPC is jointly controlled by hydration reactions, particle packing, and their nonlinear coupling. Therefore, intelligent mix design should prioritize the optimization of active oxide ratios, initial particle packing density, and their compatibility with the binder-to-sand ratio, water-to-binder ratio, and steel fiber content. A preliminary independent validation using five unseen low-carbon UHPC mixtures showed that the measured 28 d compressive strengths were consistent with the BN-predicted values and fell within the 95% prediction intervals. The standard deviation of prediction bias was 6.46 MPa, and the average relative deviation was approximately 0.45%. This provides preliminary support for the applicability of the proposed framework for candidate mix screening and strength prediction. Nevertheless, because the number of validation mixtures was limited, further experimental data covering broader cementitious systems, curing regimes, and fiber contents are still needed to improve the generalizability of the model.

In this study, an interpretable Bayesian network modelling framework was developed for the intelligent design of low-carbon UHPC. By integrating hydration-packing mechanism priors with data-driven structure learning, the proposed model described a hierarchical probabilistic relationship among material and process parameters, intermediate mechanism variables, and 28 d compressive strength. The mechanism-constrained BN model achieved high prediction accuracy under small-sample and high-dimensional conditions, with an average R² of 0.91 based on 10-fold cross-validation. Compared with conventional black-box machine learning approaches, the proposed BN framework not only maintained competitive predictive performance but also provided interpretable conditional dependency paths and uncertainty estimation. The modified CaO/SiO₂ molar ratio, modified SO₃/Al₂O₃ molar ratio, and initial particle packing density were identified as key mechanism variables governing compressive strength. The intervention sensitivity analysis showed that hydration effects (47.9%), packing effects (35.1%), and hydration-packing interactions (17.0%) jointly controlled the strength formation of low-carbon UHPC. Independent experimental validation provided preliminary support for the applicability of the proposed framework for candidate mix screening and strength prediction. This method provides a useful probabilistic modeling tool for the interpretable and intelligent design of low-carbon UHPC.