Hydraulic normal concrete is required to satisfy multiple performance demands during mixture proportioning, construction, and long-term service. Its fresh-state workability affects transportation, placement, and vibration; early-age compressive strength governs construction sequence and early load-bearing capacity; standard-age compressive strength is directly related to strength grade assessment; and drying shrinkage reflects dimensional stability of hardened concrete. In engineering practice, these properties are not controlled only by deterministic target values, but are usually associated with allowable fluctuations, quality assurance requirements, and strength guarantee rates. Owing to the multiphase and heterogeneous nature of concrete, the measured properties may still fluctuate even under similar mixture proportions, raw material conditions, and curing conditions. Therefore, deterministic point prediction is insufficient for describing the reliability of performance predictions.

Machine learning has provided an effective data-driven approach for modeling the nonlinear relationship between mixture composition and concrete properties. Previous studies have demonstrated its applicability in predicting compressive strength, slump, shrinkage, durability-related indicators, and other engineering properties of cement-based materials. Multi-output and multitask learning methods further enable simultaneous prediction of multiple performance indicators by exploiting shared information among related objectives. However, most existing models for concrete performance prediction still focus mainly on deterministic outputs, while uncertainty quantification, prediction interval construction, and credibility evaluation remain insufficiently explored, especially for hydraulic normal concrete. To address this issue, this study developed a hierarchical multi-task heteroscedastic probabilistic neural network (HMT-HPNN) for probabilistic multitarget performance prediction and credibility evaluation of hydraulic normal concrete.

A multitarget performance database of hydraulic normal concrete was established. The database contained 457 samples, with 45 raw input variables related to cement, fly ash, aggregates, and mixture proportions. Four output targets were selected, including slump, 7 d compressive strength, 28 d compressive strength, and 28 d drying shrinkage. The development database was divided into a training set and a test set at an 8:2 ratio, corresponding to 365 and 92 samples, respectively. In addition, 46 homologous engineering samples were used as an engineering validation set to evaluate the generalization ability of the proposed model in an engineering scenario.

Four neural network models were constructed and compared, including Single-target MLP, Multi-output MLP, MLP + MC Dropout, and HMT-HPNN. The Single-target MLP was used as an independent deterministic modeling baseline, while the Multi-output MLP was adopted as a conventional deterministic multi-output baseline. The MLP + MC Dropout was selected as a classical probabilistic neural-network baseline, in which multiple stochastic forward passes were used to estimate predictive uncertainty. The proposed HMT-HPNN consisted of a shared representation module, a hierarchical task representation module, task-specific branches, and heteroscedastic probabilistic subnetworks. For each target, the model simultaneously output the predictive mean and log-variance, and was trained using a heteroscedastic Gaussian negative log-likelihood (NLL) loss.

To ensure a fair comparison, all models were optimized using the same data partition and preprocessing strategy. The training set was further divided using fixed five-fold stratified cross-validation based on the quantiles of 28 d compressive strength. Randomized grid search was adopted for hyperparameter optimization. Deterministic models were mainly selected according to the five-fold average NRMSE, while probabilistic models were selected by jointly considering prediction accuracy and negative log-likelihood. The final models were retrained on the complete training set and evaluated on the test set. The evaluation metrics included R², MAE, NRMSE, PICP90, NLL, calibration curves, 90% prediction intervals, composite uncertainty, composite error, and credibility classification.

The target distribution and correlation analysis showed that the four properties had different statistical characteristics and material-control mechanisms. The 7 d and 28 d compressive strengths exhibited a strong positive correlation, reflecting the continuity of strength development. In contrast, slump and 28 d drying shrinkage showed weak linear correlations with strength targets, indicating that workability and dimensional stability should be treated as independent prediction targets rather than being inferred from compressive strength alone. For prediction accuracy, the Single-target MLP achieved the lowest average test NRMSE among the four models. However, the overall accuracy differences among Single-target MLP, Multi-output MLP, MLP + MC Dropout, and HMT-HPNN were relatively small. HMT-HPNN retained reliable regression capability after introducing probabilistic outputs. On the test set, its R² values for 7 d compressive strength, 28 d compressive strength, and 28 d drying shrinkage were approximately 0.875, 0.868, and 0.872, respectively. Slump showed relatively lower prediction accuracy, which was associated with its higher sensitivity to aggregate moisture content, water demand, admixture dispersion, and construction-related disturbances.

For uncertainty quantification, HMT-HPNN outperformed MLP+MC Dropout. The PICP90 values of HMT-HPNN for slump, 7 d compressive strength, 28 d compressive strength, and 28 d drying shrinkage were 86.96%, 90.22%, 91.30%, and 93.48%, respectively, which were closer to the nominal confidence level of 90%. In addition, HMT-HPNN achieved lower NLL values for all four targets, indicating a more appropriate balance between prediction error and predicted variance. The calibration curves also showed that HMT-HPNN generally provided better agreement between nominal confidence levels and empirical coverage rates. The 90% prediction intervals generated by HMT-HPNN achieved an overall coverage rate of 90.49% on the test set, demonstrating that the heteroscedastic uncertainty output could be effectively transformed into reliable prediction intervals. To evaluate the risk-indication ability of model uncertainty, composite uncertainty was calculated from the normalized predictive standard deviations of the four targets, while composite error was calculated from the normalized absolute prediction errors. The testing results showed that composite error increased with the composite uncertainty level, confirming that the uncertainty output of HMT-HPNN was informative in identifying sample-level prediction risk.

For the homologous engineering validation set, HMT-HPNN maintained satisfactory generalization performance. The R² values for slump, 7 d compressive strength, 28 d compressive strength, and 28 d drying shrinkage were 0.70, 0.94, 0.90, and 0.87, respectively. The PICP90 values for all four targets reached or exceeded 90%, and an effective correspondence was observed between the level of uncertainty and the prediction error. These results indicate that HMT-HPNN maintained its accuracy, uncertainty quantification capability, and credibility evaluation ability under homologous engineering conditions.

A probabilistic multitarget prediction framework was developed for hydraulic normal concrete. The proposed HMT-HPNN simultaneously predicted slump, 7 d compressive strength, 28 d compressive strength, and 28 d drying shrinkage by integrating shared representation, hierarchical task representation, task-specific branches, and heteroscedastic probabilistic subnetworks. Although the point prediction accuracy of HMT-HPNN was not the highest among all compared models, its testing accuracy was comparable to that of conventional neural network models, confirming its reliable regression capability. Compared with MLP + MC Dropout, HMT-HPNN provided superior uncertainty quantification, with PICP90 values closer to the nominal 90% confidence level and lower NLL values for all four targets. The overall coverage rate of the 90% prediction intervals reached 90.49%, indicating that the model uncertainty could be effectively converted into reliable prediction intervals. The composite uncertainty was consistent with the composite prediction error, which confirmed the ability of HMT-HPNN to identify sample-level prediction risks. Independent homologous engineering validation further demonstrated the generalization capability of HMT-HPNN. The model maintained satisfactory accuracy, reliable prediction interval coverage, and effective credibility classification on 46 engineering validation samples. Therefore, the proposed framework provides a data-driven method for multitarget performance prediction, prediction interval construction, and credibility evaluation of hydraulic normal concrete, and can support mixture proportion design and performance risk assessment.

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.

Float glass is a fundamental material for architecture, automotive applications, electronic displays and photovoltaics, and its quality stability directly affects downstream processing efficiency and end-use performance. In continuous float glass production, defects such as bubbles, knots, stones and point defects reduce product grade, increase cutting losses and rework costs, and disturb stable operation. Existing online defect inspection systems usually identify defects after they have already formed, so they cannot provide sufficient support for feed-forward adjustment of furnace thermal conditions, combustion state, glass level or tin-bath atmosphere. The melting and forming process is a highly nonlinear, strongly coupled, and long-lag industrial thermal process. Process disturbances are transmitted through the temperature field, pressure field, atmosphere field and glass flow field, and their influence may appear at the inspection end only after tens of minutes or longer. Therefore, this study focuses on short-term prediction of four types of defects with 10 levels in float glass and proposes a lag-aware variable-coupled TimeMixer model, named LVC-TimeMixer, to establish a dynamic mapping from historical process states and defect sequences to future defect trends.

Historical records from a float glass production line were used to construct the prediction dataset. The input data consisted of 31 process variables covering furnace thermal state, pressure and liquid level, combustion and atmosphere, tin-bath protective gas, feeding and control settings, together with 40 historical defect labels corresponding to Bubble, Knot, Stone and PDefect at ten levels. The raw 10 s records were sorted by timestamp and resampled to a 1 min interval. Window averaging was applied to process variables to suppress high-frequency noise, while defect labels were aggregated by counts in each window; short missing values were treated using forward filling and linear interpolation. Supervised samples were generated by a sliding-window strategy, with a 360 min historical input window and a 30 min prediction horizon. On the basis of the original TimeMixer framework, LVC-TimeMixer introduces a lag-aware encoder and a variable-coupled mixer before the multiscale mixing backbone. The lag-aware encoder explicitly extracts delayed representations under candidate lag scales of 5, 10, 30, 60, 120 min and 240 min and uses a gating mechanism to adaptively select effective lag information. The variable-coupled mixer groups variables according to process semantics, including temperature, pressure, flow, atmosphere and historical defects, and then learns intra-group representations and inter-group coupling relationships. The encoded representation is subsequently fed into the TimeMixer backbone, where multiscale decomposition, Past-Decomposable-Mixing and Future-Multipredictor-Mixing are used to predict the multi-step defect sequence.

The proposed model was compared with DLinear, PatchTST, iTransformer, the original TimeMixer and WPMixer under the same data split, input window, prediction horizon and evaluation metrics. The results show that LVC-TimeMixer achieved the best overall performance in MAE, RMSE, R² and 30 min cumulative defect MAE, with corresponding values of 10.525, 30.368, 0.616 and 5 194.10, respectively. Although its sMAPE and severe-defect F1 were not the best among all models, they remained close to the leading level, indicating that the proposed lag-aware and variable-coupled structures improve trend fitting and cumulative risk estimation while maintaining competitive severe-defect warning capability. At prediction horizons of 5, 10, 15, 20 min and 30 min, LVC-TimeMixer consistently maintained the lowest MAE and a relatively high R², demonstrating stable short- and medium-term prediction capabilities.

For different defect categories, the optimal model varied because Bubble, Knot, Stone and PDefect have different temporal patterns and process response mechanisms. LVC-TimeMixer did not achieve the best MAE for every single defect type, but its errors for all four defect categories were close to those of the corresponding optimal models, and its overall defect prediction and 30 min cumulative risk estimation were superior. Rolling prediction curves and predicted-vs.-true scatter distributions further indicate that LVC-TimeMixer can better track the rise, fall and local peaks of total defects on chronological test data. Ablation experiments confirm that the lag-aware encoder improves prediction by aligning process disturbances with delayed defect responses, whereas the variable-coupled mixer used alone may be limited when lag alignment is absent. When both modules were introduced simultaneously, the MAE and 30 min cumulative defect MAE were further reduced by 21.22% and 16.60%, respectively, compared with TimeMixer-Lag, proving the synergistic effect between lag-aware encoding and variable-group coupling. Engineering evaluation showed that the average and maximum inference times for a single prediction window were 11.6 ms and 21.3 ms, respectively, on a computing platform equipped with custom 2 × 24 GB VRAM NVIDIA GeForce RTX 4090 GPUs, which is far below the 1 min rolling warning threshold in industrial applications.

In this study, an LVC-TimeMixer model was proposed for short-term multiclass defect prediction in the float glass melting-forming process. By embedding a lag-aware encoder and a variable-coupled mixer into the TimeMixer multiscale trend-fluctuation modeling framework, the model explicitly describes both the delayed relationship between process disturbances and defect responses and the coupling relationship among temperature, pressure, flow, atmosphere and historical defect variables. The comparative experiments, prediction-curve analysis and ablation study demonstrate that the proposed model improves overall prediction accuracy, cumulative risk estimation and prediction stability. The model can provide data-driven support for rolling defect early warning, furnace operating-condition optimization and intelligent quality control in float glass production. However, further research is still required to improve generalization under extreme operating conditions, to verify long-term real-time deployment, and to integrate synchronized stripe-image diagnosis or other multi-source visual information.

Externally applied electric fields are widely used in rapid chloride migration coefficient tests and electrochemical repair or protection of concrete structures. These techniques accelerate the directional migration of ions in the pore solution and are important for evaluating chloride resistance and controlling reinforcement corrosion risks. However, concrete is a heterogeneous porous material composed of mortar, interfacial transition zones, and aggregates. Under strong electric fields, the distributions of electric-potential, ion concentration and migration flux become strongly coupled. The conventional Poisson-Nernst-Planck model usually assumes a constant ionic mobility, implying that the electromigration flux increases approximately linearly with the electric field intensity. This assumption may be insufficient under strong electric-field conditions because local electric-field concentration, concentration polarization and ion crowding can reduce the effective ionic mobility. As a result, the chloride migration front and migration flux may show nonlinear evolution. To address this issue, this work develops a physics-informed neural network framework for identifying a field-dependent ionic-mobility relation and analyzing multi-ion electromigration in concrete under strong electric fields.

A multi-ion electromigration model was established based on the Poisson-Nernst-Planck equations and the Langmuir chloride-binding relation. Four representative ions in RCM tests, i.e., Na⁺, K⁺, Cl^- and OH^-, were considered. A field-dependent mobility-suppression function was introduced into the electromigration term to describe the nonlinear attenuation of effective ionic mobility under strong electric fields. The concrete specimen was represented as a two-dimensional, three-phase mesoscopic domain consisting of mortar, interfacial transition zones, and aggregates. Aggregates were regarded as impermeable, while ion transport occurred in mortar and interfacial transition zones.

To identify the mobility suppression parameters from limited chloride profiles, an importance-adaptive-training physics-informed neural network was adopted. The governing equations, boundary conditions, initial conditions and experimental chloride-concentration profiles were embedded into the loss function. The neural network took dimensionless coordinates, time and voltage as inputs, and output the normalized electric potential and ion concentrations. The shape parameters of the mobility suppression function were optimized together with the neural network parameters. RCM data reported by Spiesz and Brouwers were used for model validation. The chloride profile at 47.5 V was used for parameter identification, while the profiles at 35 V and 60 V were used to evaluate the cross-voltage prediction capability. Prediction accuracy was evaluated using RMSE and R².

The identified mobility suppression parameters obtained by IAT-PINN were (α, γ) = (0.74, 1.37), while the least-squares (LSQ) method gave (α, γ) = (0.03, 1.04). The LSQ result was closer to the boundary of the search domain, indicating possible parameter nonuniqueness when only discrete chloride profiles were fitted. In contrast, the IAT-PINN introduced governing equations and boundary constraints into the inversion process, which improved the stability of parameter identification. When the proportion of training samples decreased, the IAT-PINN maintained lower RMSE and higher R² than LSQ, showing better data efficiency under sparse observations.

The same mobility parameters identified at 47.5 V were then used to predict chloride profiles at 35 V and 60 V without recalibration. The model captured the voltage-dependent advancement of the chloride migration front and agreed well with the measured profiles. The improvement was more evident at high voltage, indicating that the field-dependent mobility relationship reduced the prediction bias caused by the constant-mobility assumption.

The identified suppression function further revealed a transition from a driving force-controlled regime to a mobility-limited regime. At low electric field intensity, the suppression factor was close to 1 and the migration response was nearly linear. As the electric field increased, the effective mobility decreased, causing the chloride migration contribution to reach a peak and then decline. Multi-ion concentration profiles showed localized migration bands for Cl^-, K⁺, Na⁺ and OH^-, whose fronts gradually slowed with time. The data-assimilation correction was mainly concentrated near the chloride migration front, confirming that experimental data selectively constrained the high-gradient transport region rather than shifting the entire concentration field.

A field-dependent mobility-suppression relation was incorporated into a multi-ion Poisson-Nernst-Planck model and identified using an importance-adaptive physics-informed neural network. The proposed framework enabled the joint reconstruction of ionic concentration fields, electric-potential fields and mobility parameters from limited RCM data. Compared with least-squares (LSQ) fitting, the IAT-PINN improved parameter stability under sparse data by introducing physical constraints. The identified mobility relation showed that chloride migration under strong electric fields is not linearly enhanced with increasing voltage. Instead, effective mobility attenuation can lead to saturation or reduction of the migration contribution. Multi-ion concentration profiles and data assimilation corrections further confirmed the localized and migration-limited characteristics of strong electric-field electromigration. The proposed method provides a theoretical basis for interpreting RCM test results and identifying field-dependent mobility parameters in concrete.