Method for predicting multi-age rheological properties of foamed cement-based paste and low-carbon proportioning design

Abstract

The present application belongs to the technical field of cement-based material rheological property prediction and mix proportion design, and discloses a foamed cement-based slurry multi-age rheological property prediction and low-carbon mix proportion design method. The method comprises: obtaining real data through multi-age independent testing; dividing according to records at random or dividing as a whole according to mix proportion numbers; training a physically constrained diffusion model to generate virtual samples and screening; outputting dynamic yield stress and plastic viscosity through Bayesian optimization RBF-ANN; analyzing feature contribution through SHAP; taking minimum cost and carbon emission as the target and rheological parameters as the constraint, outputting a Pareto candidate set through NSGA-II; if the real retest error is less than or equal to 5%, outputting a recommended mix proportion, otherwise backfilling and iterating. The present application realizes accurate prediction of the rheological property of foamed slurry and reverse design of low-carbon mix proportion under a small sample.

Description

Technical Field

[0001] This invention belongs to the field of rheological property prediction and mix design technology of cement-based materials, and specifically discloses a method for predicting the rheological properties of foamed cement-based slurry at multiple ages and for low-carbon mix design. Background Technology

[0002] Foamed cement-based grout is a lightweight, porous material formed by introducing pre-made foam into a cementitious grout. It possesses advantages such as low density, strong flowability and filling capacity, and lightweight insulation, and is widely used in engineering fields such as grouting in goaf areas, backfilling voids, sealing air leakage channels, and lightweight filling. During grouting, backfilling, or sealing processes, the grout must maintain suitable fluidity during the transportation stage to ensure pumpability and diffusion filling effect, while gradually developing structural load-bearing capacity after settling to suppress bleeding, segregation, bubble rise, aggregation, and bubble collapse. Therefore, dynamic yield stress, plastic viscosity, and their evolution over time are key rheological indicators for evaluating the pumpability, diffusion, and foam stability of foamed cement-based grout.

[0003] Foamed cement-based pastes contain compressible and easily aggregated and broken air bubbles. Their rheological properties are influenced by multiple factors, including the dosage of fly ash (FA) or carbonized fly ash (CFA), the dosage of polycarboxylate superplasticizer (PS), the dosage of hydrophobic nano-silica (HNS), the dosage of triethanolamine (TEA), the foam ratio, and the settling age. Significant nonlinear coupling effects exist among these factors: on the one hand, they alter the dispersion state and free water content of the cementitious particles; on the other hand, they act on the bubble interface film and the early hydration process, resulting in complex time-varying nonlinear characteristics in the paste's rheological behavior.

[0004] Currently, the mix design and rheological performance evaluation of foamed cement-based slurries mainly rely on single-factor experiments, orthogonal experiments, response surface methodology, or empirical formulas. However, when the composition of cementitious materials, foam ratio, admixture dosage, and curing age all change simultaneously, the experimental workload of these methods increases exponentially, and it is difficult to reveal the interactive coupling laws between multiple factors. In multi-age rheological tests, if the same sample is repeatedly sheared, the early shear history will damage the internal structure of the slurry, seriously interfering with the test results at subsequent ages. If samples are prepared independently for different ages, batch differences in raw materials and fluctuations in the mixing process are inevitably introduced, resulting in insufficient comparability and repeatability of age evolution data.

[0005] In recent years, methods such as artificial neural networks, support vector regression, and ensemble learning have been gradually applied to the performance prediction and mix proportion optimization of cement-based materials. For example, Chinese patent CN111986737A discloses a method for optimizing the mix proportion of durable concrete based on random forest and NSGA-II. It establishes a predictive model for the frost resistance and impermeability of concrete using random forest, and employs the NSGA-II algorithm to perform multi-objective optimization of the mix proportion with durability indicators as the objective. Another example is Chinese patent CN119379115A, which discloses a modified red mud mix proportion optimization method based on strength, cost, and carbon emissions. It aims to maximize compressive strength, minimize economic cost and carbon emissions, uses NSGA-II to solve for the Pareto optimal solution set, and combines the entropy-weighted TOPSIS method for scheme decision-making. Yet another example is Chinese patent CN121439046A, which discloses an intelligent design method for the mix proportion of high-performance low-carbon concrete. By constructing a basic database of cementitious materials, training a pore structure prediction sub-model, and setting multi-dimensional optimization constraints and objectives, it achieves intelligent design of concrete mix proportions.

[0006] However, the aforementioned existing technologies still have the following shortcomings: First, existing methods primarily target solid mechanical properties such as compressive strength, flexural strength, impermeability, or frost resistance after hardening, lacking the ability to predict and design for the rheological behavior of slurry in its fresh state (dynamic yield stress, plastic viscosity, and its age evolution). Rheological property testing of foamed cement-based slurry requires specific shear conditions, and the coalescence and destruction of air bubbles must be avoided during the testing process. This differs fundamentally from the mechanical property testing of hardened concrete in terms of testing principles, data characteristics, and physical constraints.

[0007] Second, existing data augmentation methods often employ random interpolation, resampling, or unconstrained generative models. The resulting virtual samples are prone to violating fundamental materials science principles such as cement + FA + CFA ≠ 100%, water-cement ratios deviating from experimental design, and rheological parameters exceeding the physically feasible region. This renders the augmented data unusable and reduces model reliability. Furthermore, existing methods rarely consider the correlation between records of the same mix proportion at different ages when partitioning data, leading to data leakage and an overestimation of the model's generalization ability.

[0008] Third, dynamic yield stress and plastic viscosity exhibit significant differences in their numerical scale (Pa and mPa·s) and age-related evolution. Existing models often employ a uniform network structure and manually set hyperparameters based on experience, making it difficult to simultaneously ensure the prediction accuracy and stability of both rheological parameters within a multi-output prediction model. Furthermore, for black-box models, existing methods lack quantitative explanations of the direction and intensity of the influence of factors such as cementitious material composition, foam ratio, admixtures, and age on rheological properties.

[0009] Fourth, existing methods mostly remain at the performance prediction level, failing to couple rheological prediction with unit volume cost and unit volume carbon emissions to form an feasible reverse design scheme; at the same time, they lack a closed-loop update mechanism from prediction to physical verification, making it impossible to continuously optimize the model using verification test data. Summary of the Invention

[0010] To address the aforementioned issues, this invention discloses a method for predicting the rheological properties of foamed cement-based slurry at multiple ages and for designing low-carbon mix proportions.

[0011] This invention includes the following technical solutions: This invention provides a method for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions, comprising the following steps: S1, set the material mixing ratio, complete the ingredient mixing, foaming and low-shear mixing, and take the end of mixing as the starting point of the age; S2, the same batch of slurry is divided into independent age samples and sealed and left to stand under the same environmental conditions; S3. At each target age, a rheological test is performed on the corresponding independent specimens, and the rotational speed and torque signals are collected to calculate the dynamic yield stress. and plastic viscosity ; S4. Establish a real-world experimental database. Select the data partitioning method based on the model's purpose: Mode A involves random partitioning by individual records to complete data for known mix proportions at different ages; Mode B involves partitioning by mix proportion number to ensure that all age records for the same mix proportion only enter the training or test set, used to evaluate the model's predictive ability for new mix proportions. All standardized and normalized parameters are calculated only from the corresponding training set.

[0012] S5, when it is necessary to expand the sample, only the real training set is used to train the physical constraint diffusion model and generate candidate virtual samples, and the real test set does not participate in the diffusion model training and sample generation. S6: Only when the internal validation accuracy of the real training set does not reach a set threshold is a candidate virtual sample generated using the diffusion model. The variable range, cementitious material quality rules, rheological parameter range, and distance between the sample and the real training domain are checked sequentially; samples are discarded if any condition is not met. For qualified samples, the amounts of cement, water, reference slurry, and foam are calculated according to the water-cement ratio and volume relationship, and then combined with the real training set to form an expanded training set. If the baseline model accuracy has already met the threshold, virtual samples are not generated, and the real training set is used directly.

[0013] S7 uses Bayesian optimization within the real training set to determine the radial basis function type, kernel width, smoothing parameter, age distance weight, and target transformation method, and establishes an RBF-ANN multi-objective prediction model. The dynamic yield stress and plastic viscosity two-dimensional prediction results are output once through a unified training and prediction interface. S8, calculate the determination coefficients of the two rheological outputs on the real training set and the real test set, respectively. Mean absolute error and root mean square error SHAP analysis was used to analyze the contribution and direction of each input variable to the two rheological outputs. S9, within the fixed target age and model applicability domain, the new mix design branch only calls the RBF-ANN multi-objective model that has reached the threshold of step S8 after verification by mode B or a new external batch, predicts dynamic yield stress and plastic viscosity, and deterministically calculates unit volume cost. and carbon emissions per unit volume And adopt constrained NSGA-II to and Minimize as the goal, and The Pareto candidate combination ratios are obtained by using the preset range as a constraint; Mode A is only used for screening within the known combination ratio data domain, and its results are not presented as new combination ratio recommendations. S10, representative Pareto candidate mix proportions were selected for physical preparation and rheological retesting at the target age, and the unit volume cost was recalculated. and carbon emissions per unit volume ; S11, based on whether the measured rheological properties are within the engineering good range and whether the prediction-measured relative error is not greater than the set threshold, a pass or backfill determination is made; when the measured... and All are within the preset range and the relative error , If the verification is successful, otherwise the actual test results are backfilled and S4 to S8 are re-executed. S12 outputs the validated recommended mix proportions, rheological properties, economic and environmental indicators, and model traceability information.

[0014] Preferably, in step S1, the cementitious material is composed of cement, fly ash (FA), and carbonized fly ash (CFA), satisfying the requirements of cement... Polycarboxylate superplasticizer (PS), hydrophobic nano silica (HNS) and triethanolamine (TEA) are all measured based on the total mass of the cementitious materials; the water-cement ratio is preferably 0.20; the pre-made foam is added in a low-shear manner, with the end of mixing as the starting point of the curing period, and the batch of raw materials, the set dosage, the actual dosage, the mixing time and the foam state are recorded.

[0015] Preferably, in step S2, the same batch of slurry obtained in step S1 is divided into multiple sample containers in equal amounts, with each sample corresponding to only one target age. The containers are then sealed and left to stand, thereby obtaining independent age sample sequences of 0 min, 20 min, 40 min, 60 min and 80 min, avoiding interference from repeated shearing history and batch differences on the age evolution results.

[0016] Preferably, in step S3, when the target age is reached, the corresponding independent sample is placed at the rheometer measurement station, the six-bladed rotor is slowly lowered to the calibrated immersion depth, and the low-speed start-up and multi-speed steady-state programs are executed in sequence; the speed and torque of each speed level are recorded, and the validity of the data is judged based on the torque range, speed tracking error and sample appearance; records with obvious bleeding, overflow or collapse are marked as abnormal; after the test, the sample is no longer used for subsequent age measurements.

[0017] Preferably, in step S3, the calculation of rheological parameters specifically includes: first, calibrating the six-bladed rotor-sample cup system using a standard fluid or reference slurry, and recording the angular velocities at each rotational speed level. and steady-state torque The values ​​were converted into equivalent shear rate and shear stress, respectively. After truncating and averaging the steady-state speed ranges for each speed level, the dynamic yield stress was obtained by fitting the Bingham model. and plastic viscosity Peak torque is extracted separately during the startup phase. And calculate the static yield stress. Least square fitting is performed using at least three effective steady-state measurement points to output a uniformly calibrated dynamic yield stress. and plastic viscosity .

[0018] Preferably, in step S4, mode A is randomly divided according to individual records to complete the data of known mix proportions at different ages; mode B is divided as a whole according to the mix proportion number, and based on the FA / CFA content, PS content, HNS content, TEA content, and foam ratio. A unique mix designation number is constructed based on the water-cement ratio. In Model B, samples with the same number but different ages must be grouped into the same data subset to ensure that all ages of the same mix designation are only included in the training or test set for evaluating the model's predictive ability for new mix designs. Missing value handling rules, outlier criteria, and variable boundaries are determined before data partitioning. All standardized and normalized parameters are calculated only from the corresponding training set and are fixedly applied to the validation and test sets of this model. Bayesian optimization and model parameter selection are only performed within the real training set; the real test set does not participate in sample addition, preprocessing parameter fitting, or model training.

[0019] Preferably, in step S5, the training and candidate virtual sample generation of the physical constraint diffusion model includes: constructing a nine-dimensional joint sample Z0=(FA,CFA,PS,HNS,n) using the real training set. f ,TEA,t,τ0,μ p and information on material systems and target age conditions, including This refers to the mass fraction of fly ash. This refers to the mass fraction of carbonized fly ash. This refers to the mass fraction of polycarboxylate superplasticizer. The mass fraction of hydrophobic nano-silica. As a multiple of foam, This represents the mass fraction of triethanolamine. The age is set; the nine-dimensional variables are standardized according to the statistics of the real training set, and forward diffusion is performed according to the linear noise schedule; a denoising network is trained to receive noisy samples, diffusion steps and conditional information; the Gaussian noise is gradually inverted under the condition of no classifier; the generated results are de-standardized and candidate virtual samples are output; cement mass ratio, water content, reference slurry volume and foam volume are not used as diffusion generation variables, but are deterministically reconstructed after de-standardization according to FA, CFA and foam ratio.

[0020] Preferably, the denoising network adopts a fully connected structure. Nine-dimensional noisy samples are mapped through two hidden layers with 128 neurons each, outputting a nine-dimensional noise estimate. The hidden layers use SiLU activation and a 0.10 dropout rate. The diffusion step uses 32-dimensional sinusoidal position encoding, and the material system and target age conditions are encoded using 32-dimensional conditional encoding, which is then concatenated with the hidden features. The training batch size is 32, the optimizer is AdamW, and the initial learning rate is... Weight decay The training run is limited to a maximum of 2000 epochs, and the process is stopped early if the internal validation loss fails to decrease for 100 consecutive epochs. During training, conditional guidance is discarded with a probability of 0.10 to achieve classifier-free guidance. The conditional guidance weights are... The search is performed based on minimizing the sum of the denoising error and the physical constraint violation rate verified within the real training set.

[0021] Preferably, in step S6, candidate virtual samples are generated using a diffusion model only when the internal validation accuracy of the real training set does not reach a set threshold. The physical consistency screening sequentially includes: variable range checking (each generated variable should be within the experimental value range); and cementitious material quality rule checking. The requirement for mutual exclusion or compounding of FA / CFA should be included in the quality rules for cementitious materials; the range of rheological parameters should be checked. and The sample should be within a reasonable range; the Mahalanobis distance between the sample and the actual training domain should be checked; samples that do not meet any condition should be rejected. For qualified samples, the water and cement dosages should be determined according to a water-cement ratio of 0.20, and based on the reference slurry volume. and foam volume Determine the volume parameters, where The volume of the finished material is used; the mass, water-cement ratio, and volume relationship of the cementitious material are checked and found to be no greater than the numerical tolerance; qualified samples are combined with the real training set to form an expanded training set; if the accuracy of the baseline model has reached the standard, virtual samples are not generated and the real training set is used directly; the real test set always contains only real experimental records.

[0022] Preferably, the initiation condition for the diffusion augmentation is: any rheological output in the five-fold cross-validation within the real training set. Below 0.95, or dynamic yield stress Above 5 Pa, or plastic viscosity Higher than 500 mPa·s; if the two rheological outputs All are not less than 0.95, dynamic yield stress Not higher than 5 Pa and plastic viscosity Diffusion augmentation will not be enabled if the pressure is not higher than 500 mPa·s.

[0023] Preferably, in step S7, the RBF-ANN multi-objective prediction model uses the real training record as the radial basis center, and the dynamic yield stress adopts a multi-quadratic radial basis kernel with a kernel width of 0.18 and a smoothing parameter. The age distance weight is 0.60, and the target value is logarithmically transformed before an exponential inverse transform is performed on the output. The plastic viscosity adopts a thin plate spline radial basis kernel with a kernel width of 1.0, a smoothing parameter of 0.03, and an age distance weight of 0.05, while the target value is kept at its original scale. The mapping, parameters, and preprocessor of the two outputs are encapsulated in the same model object, and the two-dimensional prediction results of dynamic yield stress and plastic viscosity are returned in one go through a unified fitting and prediction interface.

[0024] Preferably, in step S7, the search range for the Bayesian optimization includes: the radial basis function is selected from multiple quadratic kernels, thin-plate spline kernels, Gaussian kernels, and inverse multiple quadratic kernels; the kernel width is 0.05–5, and the smoothing parameter is... The age distance weight is 0.01 to 1, and the target variable is transformed by either the original value or logarithmic transformation. The cross-validation error of the two rheological outputs is used as the objective function, and the next set of parameters to be evaluated is selected iteratively. After the preset number of iterations or the objective function converges, the optimal parameters are used to refit the final model on the expanded training set.

[0025] Preferably, in step S8, the model accuracy acceptance threshold is set to: the two rheological outputs of the training set and the real test set. All are not less than 0.95, dynamic yield stress All are not higher than 5 Pa, plastic viscosity The values ​​are all no higher than 500 mPa·s; after the two rheological outputs pass the above thresholds, the RBF-ANN multi-objective model is frozen and a permutation-type SHAP interpretation is performed; real test records are used as interpretation samples, and a fixed background set is formed by extracting from real training records; 128 feature permutations are randomly generated for each interpretation record, and a starting record is extracted with replacement from the background records for each permutation, and the features of the record to be interpreted are replaced one by one in the order of permutation and the marginal prediction change is calculated; SHAP only calls the prediction interface of the frozen model and does not use test labels to refit the model; the predicted value of a single sample is decomposed into the sum of the baseline value and the contribution of each input feature.

[0026] Preferably, in step S9, the decision variables of the constrained NSGA-II are: , , , , and The optimization objective is to minimize the cost per unit volume. carbon emissions per unit volume The constraint condition is dynamic yield stress. and plastic viscosity Within the pre-set good engineering range, i.e. and Where X=(FA,CFA,PS,HNS,n) f (TEA) For the target age, and These are model predictions; candidate individuals only enter the final Pareto set when the total normalization constraint violation is zero; after the material unit price or carbon emission factor is updated, all candidates are recalculated using the new version of the factor. , And Pareto Frontier.

[0027] Preferably, in step S9, the unit volume cost by Finished materials are accounted for using functional units with a deterministic approach, summing the unit volume usage of each material with the batch unit price of the project; carbon emissions per unit volume. by Finished materials are accounted for using a functional unit deterministic approach, summing the unit volume usage of each material with the carbon emission factor, and deducting the fixed carbon dioxide amount with actual measurement or reliable evidence. The unit price of the project and the carbon emission factor are managed by version; any change in the version of any factor triggers a recalculation based on the original bill of materials. and And re-execute the non-dominated sort.

[0028] Preferably, in step S10, representative schemes with low cost, low carbon emissions, compromise inflection points, and safety margins from the rheological constraint boundary are selected from the Pareto candidate set; candidate schemes used to demonstrate the reverse design capability of the new mix proportion should be assigned new mix proportion numbers, and their complete mix proportions and all age records have not been involved in model training, internal verification, and real testing; subsequently, independent samples are re-prepared according to the material measurement, foaming, mixing, dispensing, and testing procedures used in the real test database; dynamic yield stress and plastic viscosity are re-measured at the target age, and unit volume cost and unit volume carbon emissions are recalculated using the same functional units.

[0029] Preferably, in step S11, the relative error of dynamic yield stress Relative error of plastic viscosity The calculation is performed by dividing the absolute value of the difference between the measured value and the predicted value by the absolute value of the measured value, with a decimal point included in the denominator to prevent the denominator from being zero. When measured and All are within the good range of this round of projects, and , The system is considered passed upon verification; any measured value exceeding the limit or any relative error exceeding 5% will immediately trigger backfilling; backfilling will not overwrite historical records, and will include candidate mix proportions, target age, and measured values. and Actual measurement and The test batch and the current model version are added as new records to the real database; then S4 to S10 are re-executed; the maximum number of closed-loop updates is fixed at 3 times, and the schemes that still do not meet the conditions after three times are marked as pending verification and are not recommended for engineering.

[0030] This invention also provides a system for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions, used to implement the method described in any of the above-mentioned embodiments, comprising: The data acquisition interface is used to acquire rotational speed, torque, and sample state data for multi-age rheological testing. The data partitioning and database management module is used to select the data partitioning method according to the model's purpose: Mode A is randomly partitioned by individual records, and Mode B is partitioned as a whole by the combination ratio number, and stores the standardization and normalization parameters and indexes of the corresponding training set; The physical constraint diffusion augmentation module is used to generate candidate virtual samples when the verification accuracy within the real training set does not reach a set threshold. It sequentially checks the variable range, cementitious material quality rules, rheological parameter range, and training domain distance, and combines qualified samples with the real training set to form an expanded training set. If the baseline model accuracy has reached the target, the real training set is used directly. The Radial Basis Function Artificial Neural Network Modeling and Interpretation Module is used to construct an RBF-ANN multi-objective prediction model on an expanded training set and determine hyperparameters using Bayesian optimization. The model accuracy is evaluated on a real test set, and the model is frozen to perform SHAP interpretability analysis after the accuracy reaches the target. The constrained NSGA-II optimization module is used to call models that have passed pattern B validation, at unit volume cost. and carbon emissions per unit volume Minimization as the objective, with dynamic yield stress and plastic viscosity Constrained by the preset good engineering range, a non-dominated sorting genetic algorithm is executed to output a Pareto candidate combination ratio set; The physical verification and closed-loop feedback module is used to prepare physical samples and re-test rheological properties of Pareto candidate mix proportions. When the measured value exceeds the limit or the relative prediction error is greater than 5%, backfilling is triggered, the measured record is added to the real experimental database and iterative updates are triggered.

[0031] Compared with the prior art, the present invention has the following beneficial effects: (1) By preparing samples in the same batch, allowing them to stand independently at different ages, and conducting single tests, the repetition of shear history and differences between different batches of samples can be avoided, thereby improving the comparability and repeatability of rheological data at multiple ages.

[0032] (2) Two non-interchangeable data partitioning methods are set according to the purpose of the model: Mode A is randomly partitioned by single record to complete the data of known combination ratios at different ages; Mode B is partitioned by combination ratio number to ensure that all age records of the same combination ratio are only included in the training set or test set to evaluate the model’s predictive ability for new combination ratios; all standardization, normalization, physical constraint sampling and model optimization use only the corresponding training set to reduce the risk of data leakage and capability boundary confusion.

[0033] (3) Only when the internal validation accuracy of the real training set does not reach the set threshold, the diffusion model is used to generate candidate virtual samples; the variable range, cementitious material quality rules, rheological parameter range and the distance between the sample and the real training domain are checked in turn; the amount of cement, water, reference slurry and foam is deterministically calculated by the generated variables to avoid the loss of setting uncalculated variables; if the baseline model accuracy has reached the standard, virtual samples are not generated and the real training set is used directly; at the same time, a denoising network fully connected structure, SiLU activation, dropout rate of 0.10, positional encoding and conditional encoding, AdamW optimizer and conditional guided weight search are adopted to improve the quality and physical feasibility of virtual samples under small sample conditions.

[0034] (4) Bayesian optimization is used to determine the kernel function, kernel width, smoothing parameter, age distance weight and target transformation method of RBF-ANN, and the dynamic yield stress and plastic viscosity are output by one RBF-ANN multi-objective model at one time, which improves the efficiency of parameter selection and the stability of dual output prediction. RBF-ANN takes the real training record as the radial basis center, the dynamic yield stress adopts multiple quadratic radial basis kernel and logarithmic target transformation, and the plastic viscosity adopts thin plate spline radial basis kernel and original value target. The two outputs are encapsulated in the same model object.

[0035] (5) SHAP is used to quantitatively identify the contribution and direction of material composition, foam ratio, admixture and age to the two rheological outputs, thereby improving the interpretability of the model prediction results.

[0036] (6) The RBF-ANN rheological prediction is combined with the deterministic accounting of unit volume cost and unit volume carbon emission and the constraint NSGA-II. A verifiable and updatable closed-loop mechanism is formed by physical retesting of candidate mix proportions, error threshold determination, data backfilling and a maximum of 3 closed-loop iterations, which can reduce the number of repeated experiments required for screening low-carbon mix proportions. Attached Figure Description

[0037] Figure 1 Overall flowchart for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions; Figure 2 A partial unfolded diagram for data modeling and optimization; Figure 3 The correlation and marginal distribution of measured and predicted values ​​in the training and test sets of the RBF-ANN multi-objective model are shown in the figure, where (a) represents the dynamic yield stress and (b) represents the plastic viscosity. Figure 4 The image shows a comparison of measured and predicted values ​​of the training and test sets of the RBF-ANN multi-objective model, where (a) represents the dynamic yield stress and (b) represents the plastic viscosity. Figure 5The image shows the global feature importance of the RBF-ANN multi-objective model SHAP, where (a) represents the dynamic yield stress and (b) represents the plastic viscosity. Figure 6 The diagram shows the SHAP dual output summary of the RBF-ANN multi-objective model, where (a) represents the dynamic yield stress and (b) represents the plastic viscosity. Detailed Implementation

[0038] like Figure 1 As shown, the method for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions according to the present invention includes the following steps: I. Variable System and Evaluation Indicators

[0039] The cementitious material consists of cement, FA, and CFA, satisfying the condition that cement + FA + CFA = 100%; the cement mass fraction is obtained by subtracting the FA and CFA mass fractions from 100%. PS, HNS, and TEA are all measured based on the total mass of the cementitious material, and the water-cement ratio is preferably 0.20. The original input vector X = (FA, CFA, PS, HNS, nf, TEA, t) is defined, the rheological output vector Yr = (τ0, μp) is defined, and the economic and environmental index Ye = (C, E). The RBF-ANN takes X as input and outputs only two rheological prediction values, forming the vector Ŷr; Yr represents the measured rheological label used for training and testing. For the same candidate mix proportion, first, under the given finished product volume Vp and the measured or volumetrically calibrated cementitious material mass concentration ρb, determine the amount of cementitious material, foam, and each material according to Equation (1) and the material mass ratio. Then, combined with the project unit price, carbon emission factor, and the amount of fixed carbon dioxide with evidence, obtain Ye=(C, E) according to Equations (6) and (7) respectively. The rheological prediction value and the calculated value are associated with the same candidate mix proportion, and then, according to Equation (17), C and E are used as optimization objectives, and the predicted τ0 and μp are used as constraints; C and E are not the predicted outputs of RBF-ANN.

[0040] When the volume of the finished material Vp is taken as 1 m 3 When the foam ratio nf is defined as Vf / Vb, where Vb is the reference slurry volume before adding foam, Vf is the foam volume, and Vp satisfies Vp=(1+nf)Vb. The reference slurry volume Vb and the cementitious material mass Mb are calculated according to formula (1), where ρb is defined as the cementitious material mass concentration of the reference slurry, in kg / m³. 3 ρb is the mass of cementitious material corresponding to a unit reference slurry volume, not the total mass density of the reference slurry; ρb is determined by the measured ratio of the mass of cementitious material in the actual batch to the volume of the slurry before foaming, or by the volume model calibrated by experiment, and Vf=nfVb.

[0041] (1); After obtaining Mb, let rFA, rCFA, rPS, rHNS, and rTEA be the mass ratios of the corresponding materials relative to the total mass of the cementitious materials. Then, the masses of cement, FA, CFA, PS, HNS, TEA, and water are Mb(1-rFA-rCFA), MbrFA, MbrCFA, MbrPS, MbrHNS, MbrTEA, and 0.20Mb, respectively, and the foam volume is Vf=nfVb. When FA and CFA are used mutually exclusively, the input value of the unused material is 0; when multiple admixtures are allowed, rFA≥0, rCFA≥0, and rFA+rCFA≤1 should be satisfied. The dosage of each material, foam ratio, and age are all limited within the experimental boundaries that can be prepared in practice.

[0042] II. Acquisition of real experimental data and multi-age rheological testing.

[0043] In step S1, cement, FA or CFA, PS, HNS, TEA and water are weighed in sequence according to the set mix ratio to prepare a uniform reference slurry. Then, the pre-made foam is added in a low-shear manner. The time of end of mixing is taken as the starting point of the curing period, and the batch of raw materials, the set dosage, the actual dosage, the mixing time and the foam state are recorded.

[0044] In step S2, the same batch of slurry obtained in step S1 is divided into multiple sample containers in equal quantities. After recording the initial state, each container is sealed and left to stand. Each sample corresponds to only one target age, thus obtaining independent age sample sequences of 0 min, 20 min, 40 min, 60 min, and 80 min, avoiding interference from repeated shearing history and batch differences on the age evolution results.

[0045] In step S3, when the target age is reached, the corresponding independent sample is placed at the rheometer measurement station, the sample state is recorded, the six-bladed rotor is slowly lowered to the calibrated immersion depth, and the low-speed start-up and multi-speed steady-state programs are executed in sequence; after the test, the sample is no longer used for subsequent age measurements.

[0046] Record the rotational speed and torque at each speed level, and determine the validity of the data based on the torque range, speed tracking error, and sample appearance. Records with obvious bleeding, overflow, or bubble collapse are marked as abnormal. Output the valid raw signals, abnormal markers, and sample state records for each age period for rheological parameter calculation and the establishment of a real experimental database.

[0047] III. Calculation of rheological parameters over multiple ages.

[0048] Test results with different mix proportions and ages need to be compared on the same scale. To this end, the six-bladed rotor-sample cup system is first calibrated using a standard fluid or reference slurry, and then the rheological parameters are calculated in the order of "original signal conversion - rheological model fitting - parameter output"; when absolute calibration is not completed, the results are clearly marked as equivalent values.

[0049] At the i-th steady-state speed stage, the rheometer records the angular velocity Ωi and the steady-state torque Ti; these are converted into equivalent shear rate and shear stress according to equation (2), respectively. Where Kγ is the shear rate calibration coefficient and KT is the torque-shear stress calibration coefficient.

[0050] (2); After truncating and averaging the steady-state speed ranges for each speed level, the Bingham model was used to establish a quantitative relationship between shear stress and shear rate. In Equation (3), the intercept τ0 represents the dynamic yield stress that the slurry needs to overcome to begin sustained flow, and the slope μp represents the viscous resistance corresponding to the unit shear rate increment after yielding. In Equation (3), the units of shear stress τi and τ0 are Pa, and the unit of shear rate is s. -1 The plastic viscosity μp in the database is recorded in mPa·s, therefore multiply by 10 -3 Convert to Pa·s.

[0051] (3); The peak torque Tmax is extracted separately during the start-up phase, and the static yield stress τs is calculated according to Equation (4) based on the outer envelope diameter Dv and effective height Hv of the six-bladed rotor; the blade thickness, cup diameter, and end effect are calibrated using reference materials. τs is used to characterize the starting resistance when the static structure is first damaged and is not used in conjunction with the dynamic yield stress.

[0052] (4); At least three effective steady-state measurement points are used to perform least-squares fitting according to Equation (5), where the underlined quantity is the average value of the steady-state measurement points; at the same time, the goodness of fit, the number of measurement points, and the record of outlier removal are saved. This step outputs the uniformly calibrated dynamic yield stress τ0 and plastic viscosity μp as two rheological property labels for the subsequent database. In Equation (5), i is the serial number of the effective steady-state measurement point, and the underlined quantity represents the arithmetic mean of the corresponding measurement point; the unit of the least-squares slope is Pa·s, multiplied by 10 3 The output is μp in mPa·s.

[0053] (5); IV. Unit volume cost and carbon emission accounting.

[0054] Cost accounting follows the sequence of "unifying functional units—determining material usage—matching project unit prices—summing". Using 1 m... 3The finished material is a functional unit. First, Vb and Mb are determined according to formula (1), and the foam volume is determined by Vf=nfVb. Then, the actual usage of cement, FA, CFA, PS, HNS, TEA and water is converted according to the above mass ratio relationship. Finally, the unit volume cost C is calculated according to formula (6). Where mi and pi are the usage of the i-th material and the unit price of the engineering batch, Vf and pf are the foam volume and unit price, respectively, and Ce is the energy consumption cost within the accounting boundary. The total mass density of the reference slurry shall not be directly multiplied by Vb as the mass of the cementitious material. N in formula (6) and formula (7) is the total number of material categories in the material list. The summation is only for materials measured by mass. When Vf is in m 3 Timing, pf must be expressed in yuan / m 3 When quoting prices in yuan / L, first convert the unit.

[0055] (6); Carbon emission accounting and cost accounting use the same 1 m 3 The functional units and material list are used to calculate the carbon emission per unit volume E according to formula (7). Where ei is the carbon emission factor of the i-th material, Ef and Ee are the energy consumption emissions from foam making, production and transportation, respectively, and mCO2,fix is ​​the fixed amount of carbon dioxide included when there is actual measurement or reliable evidence. FA and CFA are calculated according to actual mass and corresponding emission factors, respectively, and are not calculated by replacing the material list with a fixed reduction ratio; this step outputs two economic and environmental indicators, C and E.

[0056] The unit price and carbon emission factor are managed according to version v. When the unit price or carbon emission factor of material i is updated, the formulas pi(v+1)=αp,i·pi(v) and ei(v+1)=αe,i·ei(v) are used respectively, where αp,i and αe,i are the ratios of the new and old values, respectively. The same version update method is used for foam, energy consumption, and transportation factors. Any change in the version of a factor triggers a recalculation of C(v+1) and E(v+1) based on the original bill of materials, and a re-execution of the non-dominated sorting, without directly proportionally converting to the old Pareto results. Each calculation saves the factor source, unit, applicable region, and effective date.

[0057] (7); V. Real data segmentation and verifiable preprocessing.

[0058] In step S4, the effective original signal, dynamic yield stress τ0, plastic viscosity μp, unit volume cost C, unit volume carbon emission E, mix proportion number, and age are written into the real test database. The data partitioning method is selected according to the model's purpose: Mode A randomly partitions data based on individual records to complete data for known mix proportions at different ages; Mode B partitions data based on the mix proportion number to ensure that all age records for the same mix proportion only enter the training or test set, used to evaluate the model's predictive ability for new mix proportions. Both modes can partition the real training set and the real test set at an 8:2 ratio, and save the partitioning mode, mix proportion number, and index to ensure the results are verifiable. The local unfolding process for data partitioning and model optimization is as follows: Figure 2 As shown.

[0059] All standardized and normalized parameters are calculated only from the corresponding training set and are fixedly applied to the validation data and real test set of this mode. The missing value handling rules, outlier criteria and variable boundaries are determined before data partitioning; Bayesian optimization and model parameter selection are only completed within the real training set, and the real test set does not participate in sampling, preprocessing parameter fitting and model training. The evaluation results of mode A are only used for data completion and data domain screening of known combination ratios at different ages; only when the real test results of mode B or the completely independent external new batch validation results reach the accuracy threshold specified in step S8 will the new combination ratio reverse design branch of step S9 be activated, otherwise only the data domain screening results will be output and marked as data to be expanded. In equation (8), i is the record number and j is the input feature number; the superscript train indicates that the corresponding maximum and minimum values ​​are determined only by the real training set of this mode.

[0060] (8); VI. Physical constraints for diffusion augmentation and sample screening.

[0061] In step S5, the physical constraint diffusion model is trained only using the real training set of the corresponding partitioning mode, provided that the internal validation accuracy of the real training set does not reach the set threshold. The diffusion model learns the joint distribution of input variable X and rheological output Yr under conditions of mix proportion, target age, and process boundary, and generates candidate virtual samples. When used for reverse design of new mix proportions, the diffusion model is trained only on the real training set of mode B; the real test set of mode B and external new batch data are not allowed to enter the diffusion model training, condition construction, or candidate sample screening process.

[0062] Step S5 further includes: S501, constructing a nine-dimensional joint sample Z0=(FA, CFA, PS, HNS, foam ratio, TEA, age, τ0, μp) and material system and target age conditions using the real training set; S502, standardizing the nine-dimensional variables according to the statistics of the real training set, and performing 1000 forward diffusion steps according to the linear noise schedule, β1=1×10 -4 , β1000=0.02; S503, train the denoising network that receives noisy samples, diffusion steps and conditional information; S504, gradually invert from Gaussian noise under the condition of no classifier; S505, denormalize the generated results and output candidate virtual samples; cement mass ratio, water content, reference slurry volume and foam volume are not used as diffusion generation variables, but are deterministically reconstructed after denormalization according to FA, CFA, foam ratio and equation (1).

[0063] In one preferred implementation, the denoising network employs a fully connected structure. Nine-dimensional noisy samples are mapped through two hidden layers with 128 neurons each, outputting a nine-dimensional noise estimate. The hidden layers use SiLU activation and a 0.10 dropout rate. The diffusion step uses 32-dimensional sinusoidal positional encoding, and the material system and target age conditions are encoded using 32-dimensional conditional encoding, which is then concatenated with the hidden features. The training batch size is 32, the optimizer is AdamW, and the initial learning rate is 1×10⁻⁶. -3 Weight decay 1×10 -5 The training run is limited to a maximum of 2000 epochs, and is stopped early if the internal validation loss does not decrease for 100 consecutive epochs. During training, conditions are discarded with a probability of 0.10 to achieve classifier-free guidance. The conditional guidance weight w is searched in {0.5, 1.0, 1.5, 2.0, 2.5, 3.0}, with the minimum sum of the denoising error and the physical constraint violation rate in the internal validation of the real training set as the criterion. In this embodiment, w=1.5 is preferred.

[0064] The seven inputs X and two rheological labels Yr=(τ0, μp) in Dtrain are combined into a joint sample Z0 as shown in Equation (9). Cost C and carbon emissions E are calculated by Equations (6) and (7) and do not participate in diffusion generation. Conditional information c includes at least the material system category, target age, upper and lower bounds of each generation variable, FA / CFA mutual exclusion or compounding rules, and preparation process. For each inverse standardized sample, let the cement mass ratio Water volume mw=0.20Mb, reference slurry volume Vb=Vp / (1+nf) and foam volume Vf=nfVb; all of the above variables are deterministically reconstructed from the generated quantities and are no longer independent constraint variables that the denoising network cannot directly calculate.

[0065] (9); The forward diffusion process progressively adds Gaussian noise ε to the normalized joint sample z0 according to equation (10), resulting in a noisy sample zt at step td. The conditional denoising network takes zt, the diffusion step, and condition c as inputs, learns the noise or reverse recovery quantity, and progressively inverts from the Gaussian noise to obtain candidate virtual samples. This process learns the conditional joint distribution of Dtrain, rather than simply interpolating the original samples. Equation (10) uses t_d to mark the diffusion step number, distinguishing it from the slurry age t; the noisy samples mentioned above refer to the samples corresponding to the diffusion step.

[0066] (10); To ensure the feasibility of candidate samples, the composite loss function shown in equation (11) is adopted. Each physical loss is first made dimensionless by the exponential moving average of the corresponding loss in the initial training phase. Lnoise controls the denoising accuracy; Lbound uses the variable-wise ReLU over-limit penalty function; let... Lmass takes ReLU (-rFA). 2 +ReLU(-rCFA) 2 +ReLU(-rcem) 2 When FA and CFA are used mutually exclusively, add min(rFA, rCFA). 2 The mutual exclusion term is not added when multiple doping is allowed; Lrheo is the normalized distance of τ0 and μp exceeding the rheological envelope of the training set; Ldist is the overlimit of the sample exceeding the Mahalanobis distance threshold of the real training set. The initial weights of Lnoise, Lbound, Lmass, Lrheo and Ldist are 1, 1, 10, 1 and 0.5 respectively. The water-gel ratio and foam volume closure are no longer set as diffusion loss, but are deterministically guaranteed by mw=0.20Mb, Vb=Vp / (1+nf) and Vf=nfVb. When the violation rate of any hard constraint in the internal validation is higher than 5%, the corresponding weight is multiplied by 1.5, and the maximum is no more than 10 times its initial value; the remaining weights remain unchanged. All weight adjustments only use the internal training fold and validation fold of the real training set. The cumulative coefficient in Equation (10) is obtained by multiplying (1−β_s) of each term from step 1 to step t_d, where β_s is the noise schedule parameter of the s-th diffusion step.

[0067] (11); In step S6, candidate virtual samples are generated using a diffusion model only when the internal validation accuracy of the real training set does not reach the set threshold. The variable range, cementitious material quality rules, rheological parameter range, and distance between the sample and the real training domain are checked in sequence, and the sample is removed if any condition is not met. Among them, the variable range check includes the upper and lower limits of the generated variables, non-negativity, and age fit; the cementitious material quality rules include rcem≥0 and the requirement of mutual exclusion or compounding of FA / CFA; the rheological parameter range check uses the rheological envelope; the training domain distance uses the standardized Euclidean distance or Mahalanobis distance. For qualified samples, the cement, water, reference slurry Vb and foam volume Vf are calculated according to formula (1) and the deterministic quality relationship, and the values ​​of |cement+FA+CFA-100%|, |mw / Mb-0.20| and |Vp-Vb-Vf| are checked to be no greater than the numerical tolerance of 10. -6 Since these quantities are deterministically calculated from the same set of generating variables, this check is used to detect numerical or unit conversion errors and is not considered as loss in the denoising network. Qualified samples are combined with the real training set to form the expanded training set Daug, and the sample source, confidence level, and training weights are preserved. If the baseline model accuracy has already met the requirements, virtual samples are not generated, and the real training set is used directly, i.e., Daug = Dtrain. The real test set always contains only real experimental records.

[0068] Distribution consistency is determined using standardized Euclidean or Mahalanobis distance; the distance threshold is calculated solely by Dtrain, preferably the sum of the upper quartile of the nearest neighbor distance distribution of the training samples and 1.5 times the interquartile range. The number of virtual samples after screening is preferably 0.5 to 3 times the number of real training samples to maintain the dominant role of real experimental information in training.

[0069] The activation condition for diffusion augmentation is: any rheological output R in the five-fold validation within the real training set. 2 Below 0.95, or dynamic yield stress RMSE above 5 Pa, or plastic viscosity RMSE above 500 mPa·s. If the two rheological outputs R 2 If the RMSE of dynamic yield stress is not lower than 0.95, the RMSE of plastic viscosity is not higher than 5 Pa, and the RMSE of plastic viscosity is not higher than 500 mPa·s, then diffusion augmentation is not enabled and Daug = Dtrain is set. The RBF-ANN baseline model in this embodiment has met the above accuracy requirements, so Tables 1 to 3 and the SHAP results are all obtained from training with real samples, and virtual samples that have not been actually generated are not included in the model evaluation. In Equation (12), h is the candidate virtual sample number, which is different from the rheological output marker q in Equation (13).

[0070] (12); VII. Bayesian optimization of RBF-ANN multi-objective prediction, model evaluation and SHAP interpretation.

[0071] In step S7, an RBF-ANN multi-objective rheological prediction model is constructed. The model receives mix proportion composition, foam ratio, TEA content, and age characteristics, and constructs an adaptive radial basis mapping for the numerical scale and evolution law of dynamic yield stress and plastic viscosity. The mappings, parameters, and preprocessor of the two outputs are encapsulated in the same model object, and the two-dimensional prediction results of dynamic yield stress and plastic viscosity are returned at once through a unified fit and predict interface. The output of this model is only dynamic yield stress and plastic viscosity; cost C and carbon emission E are calculated independently according to the bill of materials and equations (6) and (7), and are associated with the rheological prediction results according to the candidate mix proportion number.

[0072] In one embodiment, the RBF-ANN multi-objective model uses real training records as the radial basis function centers. The dynamic yield stress employs a multi-quadratic radial basis kernel with a kernel width of epsilon=0.18 and a smoothing parameter of smoothing=1×10⁻⁶. -7 The age distance weight is 0.60, and the target value is logarithmically transformed before the exponential inverse transformation is performed at the output. The plastic viscosity adopts the standardized original input and the radial basis kernel of the thin plate spline, with kernel width epsilon=1.0, smoothing parameter smoothing=0.03, age distance weight 0.05, and the target value is kept at the original scale. In Equation (13), q∈{τ,μ} identifies the corresponding rheological output, Zq, ψq and epsilonq are the characteristic transformation, radial basis kernel and kernel width of the output, respectively, and cqj, wqj and bq are the center, weight and bias, respectively. In Equation (13), j is the radial basis center index, and all centers are used for summation and traversal; fτ is the logarithmic scale output and fμ is the original scale output.

[0073] (13); In step S7, Bayesian optimization is used to search for the kernel function combination, radial basis center number, kernel width, smoothing or regularization parameters, age distance weights, and objective transformation method of the RBF-ANN multi-objective model. Bayesian optimization performs cross-validation within the real training set, using the validation error of the two rheological outputs as the objective function. A surrogate model and acquisition function are constructed based on the evaluated parameter combinations, and the next set of parameters to be evaluated is iteratively selected. After reaching a preset number of iterations or when the objective function converges, the optimal parameters are used to refit the final model on the expanded training set.

[0074] The search range for Bayesian optimization can be set as follows: the radial basis functions can be selected from multiple quadratic kernels, thin-plate spline kernels, Gaussian kernels, and inverse multiple quadratic kernels; the kernel width epsilon can be 0.05–5, and the smoothing parameter smoothing can be 10. -8~1, the age distance weight is 0.01~1, and the target variable transformation method is original value or logarithmic transformation. The entire search process only uses the training fold and validation fold within the real training set or the expanded training set. The real test set must not be used for parameter search. In equation (14), K is the number of internal cross-validation folds, and Lval ( ᵏ ) (ξ) is the verification loss of the hyperparameter combination ξ at the k-th fold.

[0075] (14); In step S8, the model accuracy acceptance threshold is set to: the two rheological outputs R of the training set and the real test set. 2 All values ​​are no less than 0.95, the dynamic yield stress RMSE is no more than 5 Pa, and the ductile viscosity RMSE is no more than 500 mPa·s. The 160 data points in the dataset were divided in an 8:2 ratio to form the dynamic yield stress training set R. 2 =0.982006, MAE=1.307720 Pa, RMSE=3.022247 Pa, test set R 2 =0.969406, MAE=3.060267 Pa, RMSE=4.211307 Pa; Plastic viscosity training set R 2 =0.961620, MAE=311.711273 mPa·s, RMSE=445.825051 mPa·s, test set R 2 =0.952746, MAE=333.402343 mPa·s, RMSE=399.421098 mPa·s, all four results passed the threshold. Figure 3 The correlation and marginal distribution between measured and predicted values ​​are given. Figure 4 Provide a sample-by-sample comparison of training and testing records; Figure 4 The horizontal axis represents the display sequence number and does not indicate continuous changes in age. In equation (15), n is the number of records in the corresponding dataset, and the summation iterates from i=1 to n; ȳ is the arithmetic mean of the measured values, and ŷi is the predicted value of the i-th record.

[0076] (15); Once the two rheological outputs pass the aforementioned thresholds, the RBF-ANN multi-objective model is frozen and a permutational SHAP interpretation is performed. Thirty-two real test records are used as interpretation samples, and 48 records are drawn without replacement from 128 real training records to form a fixed background set. For each interpretation record, 128 feature permutations are randomly generated. For each permutation, one starting record is drawn with replacement from the 48 background records, and these are replaced item by item in the permutation order with the features of the record to be interpreted, with marginal prediction changes calculated. SHAP only calls the frozen model's predict interface and does not refit the model using test labels.

[0077] Equation (16) decomposes the predicted value of the i-th record and the q-th output into the sum of the baseline value φ0,iq and the contributions φijq of the seven input features. Here, φ0,iq is the arithmetic mean of the predicted values ​​of the background starting records extracted from 128 permutations of that record; φijq is the average marginal contribution of feature j in the 128 replacement paths. The global average absolute SHAP value Ijq = (1 / 32)Σi|φijq| is calculated for the 32 test records, and the normalized relative importance Pjq = Ijq / ΣjIjq × 100%. The maximum summation error in this embodiment is 1.819 × 10⁻⁶. -12 . Figure 5 According to the order given by Pjq, CFA, FA and TEA contribute significantly to dynamic yield stress, while age, foam ratio and CFA contribute significantly to plastic viscosity.

[0078] Figure 6 The summative SHAP distributions of the two rheological outputs are given. Positive and negative values ​​on the x-axis represent increases or decreases in predicted values, respectively, and colors from blue to red indicate increasing feature values. The SHAP dependency plot is generated as follows: for each output, the feature with the largest average absolute SHAP value is selected as the x-axis, and its SHAP value is used as the y-axis, colored with the actual value of the second most important feature; dynamic yield stress uses CFA as the x-axis and is colored with FA, and plastic viscosity uses age as the x-axis and is colored with foam ratio. Scatter points are taken from 32 real test records without smoothing or extrapolation.

[0079] (16); Table 1: Global Importance of SAP in RBF-ANN Multi-Objective Models 8. Constraints on the low-carbon mix design and closed-loop verification of NSGA-II.

[0080] In step S9, constrained NSGA-II is used for multi-objective optimization of mix proportions. For the reverse design of new mix proportions, it is first confirmed that the actual test results of mode B or the external new batch verification results reach the accuracy threshold specified in step S8; under the condition of fixed target age, the optimizer calls the above-verified RBF-ANN multi-objective model to predict the dynamic yield stress τ0 and plastic viscosity μp, and calculates the unit volume cost C and unit volume carbon emission E according to equations (6) and (7) based on the candidate material usage, engineering unit price and carbon emission factor; with the goal of minimizing C and E, and with the constraints that τ0 and μp are in the engineering good range, the material composition satisfies mass conservation and the candidate points are within the applicable domain of the model, the Pareto candidate mix proportion set is output. When only the results of mode A are available, the optimization range is limited to screening within the known mix proportion data domain, and no new mix proportion engineering recommendation is output.

[0081] The rheologically favorable range is determined based on pumping, diffusion distance, stacking, and static stability requirements. When engineering limits are lacking, the 5%–95th percentile of 128 real training records is used as the initial range: dynamic yield stress 3.022–64.078 Pa, plastic viscosity 1368.375–8051.770 mPa·s; the example can also set a narrower construction window for specific target ages. The decision variables for NSGA-II are CFA, FA, PS, HNS, foam ratio, and TEA, totaling nvar=6; the population size is max(100, 10nvar)=100, the maximum number of iterations is 200, the simulated binary crossover probability is 0.9, the distribution index is 15, and the variable-wise polynomial mutation probability is 1 / nvar=0.1667 with a distribution index of 20. If the relative increment of the Pareto front hypervolume is less than 1×10 for 20 consecutive generations... -4 If the process fails, the process will terminate early; otherwise, it will run until generation 200.

[0082] Constraint handling employs the Deb feasibility rule: feasible individuals are prioritized over infeasible individuals; two feasible individuals are compared based on non-dominance level and congestion distance; two infeasible individuals are compared based on the total normalized constraint violation (CV). The CV is the sum of the positive exceedances of the lower bound of τ0, the upper bound of τ0, the lower bound of μp, the upper bound of μp, the mass closure of cementitious materials, the mutual exclusion or compounding rule of FA / CFA, the variable boundary, and the distance to the model's applicable domain. Candidate individuals only enter the final Pareto set when the CV = 0. After updating the material unit price or carbon emission factor, the C, E, and Pareto fronts of all candidates are recalculated using the new version of the factor.

[0083] (17); In step S10, representative schemes with low cost, low carbon content, compromise inflection points, and safety margins from the rheological constraint boundary are selected from the Pareto candidate set obtained in step S9. Candidate schemes used to demonstrate the reverse design capability of the new mix proportion should be assigned a new Mix_ID, and their complete mix proportions and all age records have not been involved in model training, internal validation, and real testing; subsequently, independent samples are re-prepared according to the material measurement, foaming, mixing, dispensing, and testing procedures used in the real test database.

[0084] The dynamic yield stress and plastic viscosity were re-measured at the target age, and the unit volume cost and unit volume carbon emission were recalculated using the same functional units. The measured rheological values ​​were compared with the predicted values ​​of the RBF-ANN multi-objective model item by item, and the relative prediction error δk of the kth rheological index was calculated according to formula (18) to form a candidate mix ratio prediction-measured comparison record.

[0085] (18); In step S11, the relative error of dynamic yield stress is calculated using equation (18): Relative error to plastic viscosity: Only when the measured τ0, μ p All are within the good range of this round of engineering, and δ τ ≤5%, δ μ A pass is only determined when the percentage is ≤5%; any measured value exceeding the limit or any relative error exceeding 5% will immediately trigger backfilling. k corresponds only to τ0 and μ. p C and E are recalculated but not included in the 5% criterion.

[0086] Backfilling does not overwrite historical records. Instead, candidate mixing ratios, target ages, measured τ0 and μp, C, E, experimental batches, Loop_ID, data source (physical retesting), and the current model version are added as new records to the real database. Duplicate imports of identical batches are rejected based on Sample_ID, while independent parallel samples are retained as different records. The original 32 real test records and their indexes are frozen in the evaluation version of this application. Backfilled records only enter the next round of real training sets. Subsequently, S4 to S10 are re-executed: the training set version is updated and the preprocessor is refitted; whether to execute S5 to S6 is determined based on the internal validation threshold; RBF-ANN parameter optimization and training, fixed test set evaluation, and SHAP calculation are re-executed; and NSGA-II search and retesting of new candidates are performed again. The maximum number of closed-loop updates, Nmax, is fixed at 3 times. Schemes that still do not meet the conditions after three updates are marked as pending validation and are not recommended for engineering.

[0087] In step S12, for the candidate schemes that passed the judgment in step S11, the FA or CFA, PS, HNS, foam ratio, TEA, water-cement ratio and target age are output. At the same time, the predicted value, measured value and relative error of dynamic yield stress and plastic viscosity are given, as well as unit volume cost C, unit volume carbon emission E, model version and test batch, forming a verifiable and traceable recommendation record.

[0088] In one embodiment, five test ages are set: 0 min, 20 min, 40 min, 60 min, and 80 min. Each mix proportion is divided into multiple independent samples from the same batch of slurry. Only the corresponding sample is tested when the target age is reached, and the tested sample is not used for subsequent age measurements.

[0089] In one embodiment, the real-world experimental database contains 160 mix design-age records. Each record includes seven raw inputs, two rheological outputs, and unit volume cost and unit volume carbon emissions determined according to the bill of materials. Using Mode A, each record is randomly divided into 128 real training records and 32 real test records in an 8:2 ratio. Records of the same mix design at different ages are allowed to be included in the training or test sets respectively. Therefore, this embodiment is used to supplement data on known mix designs at different ages.

[0090] In this embodiment, the water-cement ratio is set to 0.20 according to the experimental scheme and is not used as a model input variable. An RBF-ANN multi-objective model outputs dynamic yield stress and plastic viscosity through a unified interface: the dynamic yield stress adopts multiple quadratic radial basis mapping and logarithmic objective transformation, and the plastic viscosity adopts thin-plate spline radial basis mapping and original value objective. Model parameters, data partitioning index, preprocessor, predicted values, and residuals are all saved with the model version.

[0091] The following examples use the same RBF-ANN multi-objective model from the uploaded dataset and Notebook, a random 8:2 partitioning of a single record in the same pattern A, and saved prediction results. They provide an evaluation of the pattern A model, Pareto screening candidates within the two data domains, and their five-year-old measured-predicted data. Each candidate target age record belongs to one of 32 real test records and was not used for model fitting; the measured values ​​of these records are used to verify data completion and screening results for different ages of known combination ratios. Other ages are also listed to provide complete multi-age data. Since other ages with the same Mix_ID can be included in the training set, the following examples are not used to evaluate the model's predictive ability for new combination ratios; the reverse design of new combination ratios is performed separately according to the pattern B or external new batch validation branch and steps S9–S12.

[0092] IX. Evaluation of the RBF-ANN model of pattern A and candidate instances within the data domain.

[0093] Table 2 summarizes the evaluation results of real training records and real test records. 2 MAE and RMSE are calculated directly from the saved measured and predicted values; the test results obtained using Mode A are used to evaluate the multi-age completion ability of known mix proportions.

[0094] Table 2: Evaluation Results of RBF-ANN Multi-Objective Model on Training and Testing Sets In this embodiment, a unified RBF-ANN multi-objective estimator uses 128 real training records as the radial basis function centers, and manages the adaptive radial basis mapping of two targets through a unified fit and predict interface. The dynamic yield stress is set using a multiquadric kernel, epsilon=0.18, and smoothing=1×10⁻⁶. -7 The age option weight is 0.60, and a logarithmic objective transformation is used; the plastic viscosity is calculated using a thin-plate-spline kernel, epsilon=1.0, smoothing=0.03, and the age option weight is 0.05; the predict interface returns two predicted values ​​at once: dynamic yield stress and plastic viscosity.

[0095] Example 1: (Low-cost candidate screening within the Pattern A data domain, Mix_ID=19): Target age is 80 min; example construction window is set to τ0=15~25 Pa, μp=5000~6000 mPa·s. Mix proportions are: cement 80%, CFA 0%, FA 20%, PS 1.2%, HNS 0.9%, foam ratio 1.7, TEA 0.10%, water-cement ratio 0.20; C=695.922621 yuan / m³ 3 E=403.750562kg CO2e / m 3 Within the candidate set of the data domain, which consists of known combination ratios and target age combinations, there is no other feasible candidate that is non-inferior in both C and E and has at least one superior objective. Therefore, this scheme belongs to the non-dominated solution of this candidate set. At 80 min, the predicted τ0 = 18.6611 Pa, the measured τ0 = 18.2090 Pa, and δτ = 2.48%; the predicted μp = 5279.5 mPa·s, the measured μp = 5219.6 mPa·s, and δμ = 1.15%. Both values ​​do not exceed 5%, thus the scheme passes the data domain screening verification.

[0096] Table 3: Measured-Predicted Comparison of Low-Cost Candidate Five-Year Ages in the Data Domain of Pattern A, Example 1 Example 2: (Low-carbon candidate screening within the Mode A data domain, Mix_ID=20): Target age is 40 min; example construction window is set to τ0=14~20 Pa, μp=3000~3500 mPa·s. Mix proportions are: cement 80%, CFA 20%, FA 0%, PS 1.2%, HNS 0.9%, foam ratio 1.7, TEA 0.10%, water-cement ratio 0.20; C=695.922621 yuan / m³ 3 E = 323.004483 kg CO2e / m 3 The proposed scheme is a non-dominated candidate within the known mix proportion data domain that meets the corresponding construction window requirements. At 40 min, the predicted τ0 = 15.0640 Pa, the measured τ0 = 14.8110 Pa, and δτ = 1.71%; the predicted μp = 3279.0 mPa·s, the measured μp = 3165.3 mPa·s, and δμ = 3.59%. Both values ​​are less than 5%, thus the scheme passes the data domain screening verification.

[0097] Table 4: Measured-Predicted Comparison of Low-Carbon Candidate Five-Year Ages in the Data Domain of Model A, Example 2 Tables 3 and 4 disclose the complete mix proportions, measured and predicted values ​​at five ages, costs, carbon emissions, model parameters, and relative errors at target ages for candidates within the data domain of two sets of Model A. The 80-minute test records from Example 1 and the 40-minute test records from Example 2 were not used for model fitting, and the measured values ​​of τ0 and μp were both within the corresponding example construction window, with both δk values ​​not exceeding 5%. Therefore, verification was achieved through target age completion and data domain screening for known mix proportions. The relative errors at other ages are only used to demonstrate the model's performance on the complete age sequence. The above results are not considered evidence of the generalization ability of unknown mix proportions; when used for reverse design of new low-carbon mix proportions, verification must be performed separately using Model B or a completely independent external new batch. After NSGA-II outputs candidates with new Mix_IDs, the mix proportions should be re-prepared, retested, and closed-loop determination completed according to steps S10–S12.

[0098] The data flow for each step is as follows: Steps S1 to S3 form real multi-age rheological records, and equations (6) and (7) form deterministic economic environment indicators; Step S4 selects mode A by random division of single records or mode B by overall division of mix proportion number according to the model's purpose, and outputs the corresponding real training set and real test set; Steps S5 to S6 generate and screen candidate virtual samples only when the verification accuracy within the real training set does not reach the set threshold, and form an expanded training set with the real training set. If the baseline model accuracy has reached the standard, the real training set is used directly; Step S7 outputs the Bayesian-optimized RBF-ANN multi-objective rheological prediction results; Step S8 outputs the evaluation indicators of the training set and test set and the SHAP interpretation results. The results of mode A are only used for data completion and data domain screening for different ages of known mix proportions; only after the verification of mode B or a new batch from outside reaches the threshold will the model enter the new mix proportion reverse design branch in step S9 and output Pareto candidates to be verified; Steps S10 to S12 complete the physical retesting of new candidates, error judgment, data backfilling and recommendation result output.

[0099] In summary, this invention provides a detailed explanation of the entire process, including real-world testing, data partitioning, physical constraint diffusion sampling, RBF-ANN prediction, SHAP interpretation, NSGA-II optimization, and physical closed-loop verification. It clarifies the definitions of the cementitious material system and variables, and provides a complete method for calculating rheological parameters, costs, and carbon emissions. Two data partitioning methods, Mode A and Mode B, are designed for completing known mix proportions and evaluating the generalization of new mix proportions, respectively. A physical constraint diffusion model is constructed for small-sample sampling, and an RBF-ANN multi-objective prediction model is established and its hyperparameters are determined through Bayesian optimization. After achieving the target, the model is frozen for SHAP interpretation. In the optimization phase, the goal is to minimize cost and carbon emissions, constrained by rheological parameters. The Pareto candidate set is output via NSGA-II, and physical retesting and closed-loop updates are completed. The examples verify the effectiveness of the method, with prediction errors not exceeding 5%. The steps are clearly connected, forming a complete and traceable closed loop from data acquisition to mix proportion recommendation.

[0100] The above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions, or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A method for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions, characterized in that, Includes the following steps: S1, mix ingredients according to the set ratio, foam and mix at low shear, with the end of mixing as the starting point of the age; S2, the same batch of slurry is divided into multiple independent samples and left to stand, with each sample corresponding to a target age; S3. Perform a rheological test on an independent specimen at each target age, collect rotational speed and torque data, and calculate the dynamic yield stress. and plastic viscosity ; S4. Establish a real experimental database and select the data partitioning method according to the model's purpose: Mode A is randomly partitioned by single record to complete the data of known combination ratios at different ages; Mode B is partitioned as a whole by combination ratio number to ensure that all age records of the same combination ratio only enter the training set or test set, and is used to evaluate the model's predictive ability for new combination ratios; all standardized and normalized parameters are calculated only from the corresponding training set. S5, when the training set verification accuracy is lower than the threshold, use the training set to train the diffusion model to generate candidate virtual samples; S6. Only when the internal validation accuracy of the real training set does not reach the set threshold, a diffusion model is used to generate candidate virtual samples. The variable range, cementitious material quality rules, rheological parameter range, and distance between the sample and the real training domain are checked in turn. If any condition is not met, the sample is removed. For qualified samples, the amount of cement, water, reference slurry, and foam is calculated according to the water-cement ratio and volume relationship. Then, the samples are merged with the real training set to form an expanded training set. If the baseline model accuracy has reached the standard, no virtual samples are generated, and the real training set is used directly. S7. Construct a radial basis function neural network prediction model on the extended set. Determine the kernel function, kernel width, smoothing parameter, aging weight, and target transformation through Bayesian optimization, and output the first time. , ; S8 evaluates accuracy on the test set, freezes the model after it meets the target, and performs SHAP analysis on the contribution of each input feature. S9, the new mix design uses the model frozen in S8 to predict... , Calculate the unit volume cost and carbon emissions ,by , Minimum as the target , With a preset range as a constraint, the Pareto candidate set is output via NSGA-II; S10, Select Pareto candidate solutions for physical preparation and target age retesting, recalculate. , ; S11, Actual Test , If the result is within the preset range and the relative error is ≤5%, it passes; otherwise, it is backfilled and S4-S8 are executed again. S12 outputs the validated recommended combination ratio, including predicted and measured values. , And traceability information.

2. The method according to claim 1, characterized in that, The training and virtual sample generation of the diffusion model in step S5 includes: constructing a nine-dimensional joint sample Z0=(FA,CFA,PS,HNS,n) using the real training set. f ,TEA,t,τ0,μ p ) and condition information, among which This refers to the mass fraction of fly ash. This refers to the mass fraction of carbonized fly ash. This refers to the mass fraction of polycarboxylate superplasticizer. The mass fraction of hydrophobic nano-silica. As a multiple of foam, This represents the mass fraction of triethanolamine. The age is determined by the standardization of the nine-dimensional variables. Forward diffusion is then performed to train the denoising network, and candidate virtual samples are obtained by stepwise inversion using Gaussian noise.

3. The method according to claim 1, characterized in that, In step S6, candidate virtual samples are generated using a diffusion model only when the internal validation accuracy of the real training set does not reach a set threshold. These samples are then filtered sequentially based on variable range, cementitious material quality rules, rheological parameter range, and Mahalanobis distance of the training domain; samples failing any of these conditions are discarded. For qualified samples, the water consumption is determined based on a water-cement ratio of 0.20, and then... ; ;and ; Determine the reference slurry volume With foam volume ,in The volume of the finished material is used; the amount of cement, water, reference slurry and foam used is calculated for qualified samples, and then combined with the real training set to form an expanded training set; if the accuracy of the baseline model has reached the standard, virtual samples are not generated and the real training set is used directly.

4. The method according to claim 1, characterized in that, The radial basis function artificial neural network multi-objective prediction model described in step S7 uses the real training record as the radial basis center. The dynamic yield stress adopts a quadratic radial basis kernel and logarithmic objective transformation, and the plastic viscosity adopts a thin plate spline radial basis kernel and original objective. The two outputs are encapsulated in the same model and returned once. and The accuracy threshold mentioned in step S8 is the ratio of the training set to the real test set. and coefficient of determination All are not lower than 0.95, and The root mean square error is no higher than 5 Pa. The root mean square error is no higher than 500 mPa·s.

5. The method according to claim 1, characterized in that, In step S9, the decision variables constraining NSGA-II are: , , , , , The optimization objective is to minimize and , constraint is ;and ; Where X = (FA, CFA, PS, HNS, n) f (TEA) For the target age, and The model predicts the cost per unit volume. Carbon emissions per unit volume Based on material usage and unit price of the project, and carbon emission factor, respectively, 1 m 3 Finished materials are accounted for using functional units with a definite accounting approach.

6. The method according to claim 1, characterized in that, Relative error of dynamic yield stress in step S11 according to Calculate the relative error of plastic viscosity. according to Calculation, where This represents the measured value of dynamic yield stress. This is the predicted value of dynamic yield stress. This is the measured value of plastic viscosity. This is the predicted value for plastic viscosity. To prevent small quantities with a denominator of zero; when measured , All are within the preset range and , If successful, the candidate combination ratio, target age, and measured value will be backfilled into the database and S4 to S8 will be re-executed. The maximum number of closed-loop updates is 3.

7. A computer-readable storage medium, characterized in that, The device contains a computer program that, when executed by a processor, implements the steps of the method as described in any one of claims 1 to 6.

8. A system for predicting the rheological properties of foamed cement-based slurry at multiple ages and designing low-carbon mix proportions, used to perform the method as described in any one of claims 1 to 6, characterized in that, include: The data acquisition interface is used to acquire rotational speed, torque, and sample state data for multi-age rheological testing. The data partitioning and database management module is used to execute step S4, selecting mode A or mode B according to the model's purpose; mode A partitions randomly according to individual records, while mode B partitions the entire dataset according to the matching ratio number, and stores the preprocessing parameters and indexes of the corresponding training set. The physical constraint diffusion augmentation module is used to execute steps S5 to S6. Only when the internal validation accuracy of the real training set does not reach the set threshold, the diffusion model is trained to generate candidate virtual samples, and the variable range, cementitious material quality rules, rheological parameter range and training domain distance are checked in sequence. The qualified samples are combined with the real training set to form an expanded training set; if the baseline model accuracy has reached the standard, the real training set is used directly. The radial basis function artificial neural network modeling and interpretation module is used to execute steps S7 to S8, build and train a multi-objective prediction model, evaluate accuracy and perform SHAP analysis; The constrained NSGA-II optimization module is used to perform step S9, in order to and Minimum as the goal, with and To constrain the output Pareto candidate set; The physical verification and closed-loop feedback module is used to execute steps S10 to S11, perform physical retesting, and trigger backfilling and iterative updates when the deviation exceeds the standard.

9. The system according to claim 8, characterized in that, The constraint NSGA-II optimization module uses the Deb feasibility rule to handle constraint violations, and sequentially performs non-dominated sorting, crowding calculation, crossover and mutation, and outputs Pareto candidates only when the total number of constraint violations is zero.

10. The system according to claim 8, characterized in that, The physical verification and closed-loop feedback modules respectively calculate and ,when , If the measured value exceeds the limit, backfilling will be triggered, and S4 to S8 will be re-executed. The maximum number of closed-loop updates is 3.

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