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.