Phosphogypsum solidification agent reverse design method based on integrated machine learning and pareto optimization

Abstract

The present application relates to the technical field of road engineering solid waste resource utilization, solves the technical problem that traditional phosphogypsum solidification is prone to post-stage strength collapse, especially relates to a reverse design method of phosphogypsum solidification agent based on integrated machine learning and Pareto optimization, including global test point distribution and original data collection; core evaluation index and comprehensive satisfaction degree construction; global hyperparameter optimization based on heterogeneous machine learning model library driven by global data; high-throughput virtual deduction and safe formula feasible region demarcation; comprehensive performance index and multi-objective iterative optimization of Pareto algorithm, etc. The phosphogypsum solidification agent formula obtained by reverse design in the present application takes into account the anti-erosion performance, low-carbon economy and mechanical strength, completely solves the pain point of traditional phosphogypsum solidification prone to post-stage strength collapse, and realizes low-carbon low-cost efficient intelligent reverse design.

Description

Technical Field

[0001] This invention relates to the field of solid waste resource utilization technology in road engineering, and in particular to a reverse design method for phosphogypsum curing agents based on integrated machine learning and Pareto optimization, which is particularly suitable for the reverse design of multi-element solid waste-based materials that take into account erosion resistance, low carbon economy and mechanical strength. Background Technology

[0002] Phosphogypsum, as a major industrial solid waste, has broad prospects for resource utilization in roadbed engineering. However, due to the presence of soluble phosphorus and fluorine impurities and acidic substances within it, phosphogypsum commonly suffers from bottlenecks such as low early strength, subsequent strength reduction, and poor water stability when used as roadbed filler. Conventional single alkali sources provide OH... - Under continuous acidic erosion, phosphogypsum will be rapidly consumed, leading to a decrease in the pH value of the system, interruption of the hydration reaction, and even decomposition of the hydration products.

[0003] A multi-component solid waste solidification system is constructed by introducing cement, steel slag powder, and blast furnace slag powder. Utilizing a dual-alkali source synergistic activation mechanism, it effectively generates a relay release of alkalinity, providing long-term resistance to acid erosion. However, existing mix design methods largely rely on trial and error based on human experience, often focusing solely on maximizing compressive strength as the sole optimization objective. This neglects the long-term evolution of the material's internal activity and fails to consider macro-level constraints such as economic costs and carbon emissions in actual engineering projects. There is a lack of a scientific, intelligent reverse-engineering system that can achieve optimal microscopic performance and macroscopic benefits across the entire product chain. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a reverse design method for phosphogypsum curing agents based on integrated machine learning and Pareto optimization. By using objective dimensionality reduction based on the Derringer satisfaction function, data-driven integrated machine learning mapping, and Pareto frontier iterative optimization, efficient formulation design of curing agents is achieved.

[0005] To address the aforementioned technical problems, this invention provides a reverse design method for phosphogypsum curing agents based on integrated machine learning and Pareto optimization. This method includes the following steps: Based on the global simplex lattice design, point tests were conducted on curing agents with different ratios within the mixing constraint space, and data on various core mechanical and durability properties were collected to establish an original database for characterizing the relationship between curing agent formulation and performance. Based on the original database, the core evaluation index is calculated to eliminate the interference of the cementitious material's own activity and quantify the degree of erosion of the curing agent by the acidic substances inside the phosphogypsum. The Derringer satisfaction function is introduced to transform the multi-dimensional target response into a single prediction target of high confidence comprehensive satisfaction. Using overall satisfaction as the single prediction target, hyperparameter global optimization is performed based on a heterogeneous machine learning model library driven by global data. The global optimal fusion weighting coefficient of each heterogeneous base learner is then solved iteratively under the weight boundary constraints using a sequence minimum optimization algorithm. In the mixing constraint space, a gridded virtual formula set covering the continuous phase diagram of the full ratio is generated. The heterogeneous machine learning model library optimized and weighted by hyperparameters is called to perform forward extrapolation of the global performance. Forced physical boundary cutting is performed by the comprehensive satisfaction watershed threshold to determine the safe formula feasible domain of the curing agent. Economic and environmental factors are introduced within the feasible domain of safe formulations to construct a comprehensive performance index. The Pareto algorithm is used to output the globally optimal formulation of the curing agent through non-dominated sorting with the goal of maximizing the comprehensive performance index.

[0006] Furthermore, establishing the original database includes: The curing agent is composed of various solid waste materials, and the mass fraction of its components is: ,satisfy A point-based experiment was conducted using a global simplex lattice design within a mixed-material constrained space. Different proportions of curing agents are mixed into phosphogypsum at a fixed external dosage, and then mixed, compacted and cured at the optimal moisture content to form a multi-element solid waste system. Experiments were conducted to test the performance of a multi-component solid waste system during the curing period. The core mechanical and durability performance data of the underlying solid waste system were obtained, including the compressive strength of the mortar. The compressive strength of mortar for PO 42.5 cement at the same curing age And the unconfined compressive strength of the cured body after the curing agent is incorporated into phosphogypsum. This led to the establishment of a discrete original database of curing agent formulations and properties.

[0007] Furthermore, the fixed amount of the curing agent is 8% of the dry basis mass of the phosphogypsum.

[0008] Furthermore, this invention proposes and calculates an activity index. With strength retention rate Two core evaluation indicators were used to eliminate the influence of differences in the activity of the cementing materials themselves and to directly quantify the inhibitory effect of the acidic environment of phosphogypsum on the strength development of the curing agent. Among these, the core evaluation indicator includes an activity index, which characterizes the cementing potential of a multi-component solid waste system under ideal conditions. And the strength retention rate, used to intuitively quantify the resistance of curing agents to degradation and strength shrinkage under acidic phosphogypsum erosion environments. .

[0009] Activity index For the same maintenance period Below, the compressive strength of mortar in a multi-element solid waste system Compressive strength of mortar compared to PO 42.5 cement The ratio, expressed as: Strength retention rate For the same curing agent formulation at the same curing age Below, the unconfined compressive strength of the cured body With compressive strength of mortar The ratio, expressed as: This invention addresses the problem of interdependent and inconsistent dimensions among multiple performance indicators within a material by introducing a Derringer satisfaction function to construct a comprehensive objective variable. The response to each individual objective is then calculated. Set an acceptable lower limit and upper limit And calculate the individual satisfaction of the core evaluation indicators. The calculation formula is: in, The target importance weight index.

[0010] Using the geometric mean method based on individual satisfaction levels Building overall satisfaction The expression is: in, This represents the total number of target responses. The geometric mean calculation incorporates an implicit mathematical penalty mechanism at the underlying level: failure to meet any single indicator will result in a decrease in overall satisfaction. Approaching 0.

[0011] Furthermore, solving for the globally optimal fusion weighting coefficients of each heterogeneous base learner includes: A heterogeneous machine learning model library, including support vector regression machine, random forest and extreme gradient boosting tree, is used, with a full-domain test dataset including a multi-dimensional solid waste system and a large number of failure samples as input features. Based on overall satisfaction As a single prediction target, hyperparameter global Bayesian optimization is performed on each heterogeneous base learner through cross-validation. Leave-one-out cross-validation is used to train and evaluate each heterogeneous base learner after hyperparameter optimization, and the global prediction matrix of each heterogeneous base learner is output. An ensemble loss function is defined with the goal of minimizing the overall mean squared error. The sequence minimum optimization algorithm is used to perform nonlinear programming iterative solution under the constraint that the sum of the weighting coefficients of each heterogeneous base learner must be 1.0 and all of them must be in the interval [0, 1.0] to obtain the globally optimal fusion weighting coefficients of each heterogeneous base learner.

[0012] Furthermore, determining the safe formulation feasibility domain of the curing agent includes: Within the mixing constraint space, a gridded virtual formula set covering the full proportion continuous phase diagram is generated, wherein the component scanning step size is precisely set to 1%, and a total of tens of thousands of high-resolution virtual mix proportion samples are generated. By calling a heterogeneous machine learning model library optimized and weighted by hyperparameters, high-throughput virtual prediction results are generated by performing a full-domain positive batch extrapolation of the overall satisfaction of virtual combination ratio samples. After the full-domain forward batch simulation is completed, a watershed threshold is set and a forced post-physical boundary cut is performed on the high-throughput virtual prediction results to eliminate invalid virtual formulas with a comprehensive satisfaction prediction value lower than the watershed threshold. This enables the accurate delineation of the feasible domain of safe formulas that ensure long-term resistance to acid erosion and activation of activity within a continuous space.

[0013] Furthermore, outputting the globally optimal formulation of the curing agent includes: Calculate the combined cost of each set of gridded virtual recipes within the feasible region of the safe recipe. C With carbon emissions E ; Based on comprehensive cost C With carbon emissions E Calculate the relative economic index or C and relative carbon emission index or E The calculation formula is: in, C The cost of phosphogypsum roadbed filler; C ref The cost of cement-cured phosphogypsum roadbed filler; E Carbon emissions from phosphogypsum roadbed filler; E ref Carbon emissions of cement-cured phosphogypsum roadbed filler; The overall satisfaction level predicted by using a heterogeneous machine learning model library to perform a global positive batch extrapolation of the current gridded virtual recipe is obtained. D And calculate the relative satisfaction index. or D The calculation formula is: in, D Overall satisfaction with the curing agent-cured phosphogypsum roadbed filler; D ref The overall satisfaction level of pure cement-cured phosphogypsum roadbed filler; According to the relative economic index or C Relative carbon emission index or E Relative satisfaction index or D To construct a comprehensive performance index for achieving a balance between mechanical properties, carbon emissions, and economic benefits of solidified materials. The expression is: Using the Pareto algorithm to maximize the overall performance index The algorithm iteratively optimizes within the feasible domain of safe formulations to ultimately output the globally optimal formulation of the curing agent.

[0014] Furthermore, the Pareto algorithm performs visualization-based multi-objective screening based on dimensionality reduction mapping, including: Constructing a three-dimensional Pareto frontier space: using relative economic indices or C Relative carbon emission index or E Relative satisfaction index or D Plot the comprehensive efficiency index using independent three-dimensional coordinate axes. Color mapping dynamically displays the original three-dimensional spatial game relationship between cost, carbon emissions, and mechanical durability, and eliminates globally disadvantageous solutions. Constructing a two-dimensional Pareto frontier dimensionality reduction space: using the merged relative economic index or C Relative carbon emission index or E The obtained composite index The horizontal axis represents the relative satisfaction index. or D Using the vertical axis, perform a two-dimensional spatial topological filter to obtain a non-dominated solution set; Based on the non-dominated solution set, the Pareto optimal frontier curve representing the extreme value of multi-objective performance is obtained, and the global optimal solution is finally output as the global optimal formulation for the reverse design of the curing agent.

[0015] By employing the above technical solution, the present invention provides a reverse design method for phosphogypsum curing agents based on integrated machine learning and Pareto optimization, which has at least the following beneficial effects: 1. The present invention adopts a phosphogypsum curing agent formula obtained by reverse design, which takes into account the anti-corrosion performance, low carbon economy and mechanical strength, and completely solves the pain point of traditional phosphogypsum curing being prone to later strength shrinkage, and realizes a high-efficiency intelligent reverse design with low carbon and low cost.

[0016] 2. This invention proposes a mathematical model of activity index and strength retention rate, quantifies the degree of erosion of the curing agent activity by acidic substances inside phosphogypsum, and proposes a high-quality formula that is not prone to strength shrinkage, thus ensuring the long-term durability of road phosphogypsum mixtures.

[0017] 3. This invention replaces the traditional massive physical trial-and-error experiments with nonlinear regression and bi-level optimization models, which greatly shortens the research and development and adaptation cycle of solidifying agents for multi-component solid waste systems. Attached Figure Description

[0018] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of the reverse design method for phosphogypsum curing agent in this invention; Figure 2 This is a distribution diagram showing the overall satisfaction level of the ternary solid waste-based solidifier components in this invention; Figure 3 This is a diagram of the optimal Pareto front curve in the two-dimensional Pareto front reduction space of this invention. Figure 4 This is a diagram of the optimal Pareto front curve in the three-dimensional Pareto front space of this invention; Figure 5 This is a SHAP swarm analysis diagram of the curing agent formulation in this invention; Figure 6 This is a SHAP dependency graph of the interaction between the curing agent formulation and cement in this invention; Figure 7 This is the SHAP dependency graph of the steel slag interaction in the curing agent formulation of this invention. Detailed Implementation

[0019] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0020] Example 1: This embodiment proposes a reverse design method for phosphogypsum curing agents based on ensemble machine learning and Pareto optimization. Through objective dimensionality reduction based on the Derringer satisfaction function, data-driven ensemble machine learning mapping, and Pareto front iterative optimization, efficient formulation design of the curing agent is achieved. Figure 1 As shown, the method includes the following steps: S1. Based on a global simplex lattice design, point-based experiments were conducted on curing agents with different formulations within a mixture constraint space, and data on various core mechanical and durability properties were collected. A raw database was established to characterize the relationship between curing agent formulations and performance. This embodiment uses a global simplex lattice design to conduct point-based experiments and collect multiple core performance data of the curing agent, providing fundamental data support for subsequent index calculations and model construction. These include: Specifically, in this embodiment, the ternary solid waste-based solidifier is configured to consist of P.O42.5 cement, steel slag powder, and S95 slag powder, with the component mass fractions meeting the following requirements. Furthermore, point-based tests were conducted within a confined space for the mixture. This step employed a fixed component step size of 10% to perform a high-throughput point-based test of the mixture design throughout the entire space, molding and preparing specimens of the multi-component solid waste mortar system and the unconfined compressive strength mixture specimens of the cured body after incorporating the curing agent into phosphogypsum. During the preparation of the unconfined compressive strength mixture specimens, the fixed external dosage of the curing agent was set to 8% of the dry weight of the phosphogypsum. The prepared ternary solid waste-based curing agent was uniformly mixed with the undisturbed phosphogypsum, and an appropriate amount of water was added to achieve the optimal moisture content of the mixture. After thorough mixing, the mixture was compacted in a standard mold using a static pressing method to prepare unconfined compressive strength cylindrical specimens. After demolding, the molded specimens were transferred to a standard curing room for curing under the following conditions: temperature 20±2℃, relative humidity ≥95%.

[0021] After standard curing, the molded specimens were tested to obtain the compressive strength of the mortar in the multi-component solid waste system at a specific curing age of 28 days. The compressive strength of mortar for PO 42.5 cement at the same curing age And the unconfined compressive strength of the cured body after the curing agent is incorporated into phosphogypsum This led to the establishment of a discrete original database of curing agent formulations and properties.

[0022] S2. Based on the original database, core evaluation indicators are calculated to eliminate interference from the activity of the cementitious material itself and quantify the degree of erosion of the curing agent by acidic substances inside phosphogypsum. The Derringer satisfaction function is introduced to transform the multidimensional target response into a high-confidence comprehensive satisfaction. Single prediction target.

[0023] Specifically, this embodiment, based on the collected original database, first proposes and calculates the activity index. With strength retention rate Two core evaluation indicators. Due to the complex interrelationships and nonlinear interactions among the three indicators—compressive strength, activity index, and strength retention rate—this embodiment introduces the Derringer satisfaction function to reduce the dimensionality and compress the features of the core evaluation indicators, such as... Figure 2 As shown. The acceptable lower limit for the 28-day compressive strength was set at 0.7 MPa, and the upper limit at 3.0 MPa; regarding the activity index... The lower limit is set at 20%, and the upper limit at 50%; regarding the strength retention rate Set the lower limit to 8% and the upper limit to 20%.

[0024] By calculating the individual satisfaction level of the core evaluation indicators Then, the geometric mean method was used to construct the overall satisfaction level. The expression is: in, The target total number of responses. This geometric mean calculation introduces an implicit mathematical penalty mechanism at the underlying level, where failure to meet any single indicator will cause the overall satisfaction level to approach 0.

[0025] S3, based on overall satisfaction As a single prediction target, hyperparameters are globally optimized based on a heterogeneous machine learning model library driven by global data. The global optimal fusion weighted coefficients of each heterogeneous base learner are then iteratively optimized under weight boundary constraints using a sequence minimum optimization algorithm.

[0026] This invention constructs a high-precision integrated prediction model based on full-domain data-driven approach, which is used to input a full-domain test dataset containing penalty features into a heterogeneous machine learning model library to establish a precise nonlinear mapping relationship between curing agent formulation and overall satisfaction.

[0027] Specifically, in this embodiment, to enable artificial intelligence to transcend local optima and fully identify the nonlinear evolution of the mix proportion in the extreme value boundary region, 66 sets of discrete data points across the entire domain are extracted for training a heterogeneous machine learning model library. The ternary component mix proportion matrix corresponding to the multi-component solid waste system is used as the input feature. X Overall satisfaction D As a prediction target y The input features are processed using the Standard Scaler algorithm. X Standardization and dimensionless processing are performed. The heterogeneous machine learning model library includes various heterogeneous base learners, such as Support Vector Regression (SVR), Random Forest (RF), and Extreme Gradient Boosting Tree (XGBoost).

[0028] First, using 5-fold cross-validation (KFold) combined with grid search CV, we perform global Bayesian optimization of hyperparameters for each heterogeneous base learner to minimize the negative mean squared error, including the core search penalty coefficient for the support vector regression (SVR) machine. C and e Spaces for the number of search trees and maximum depth in Random Forest (RF), and for the search learning rate and subsampling rate in Extreme Gradient Boosting Tree (XGBoost).

[0029] Secondly, leave-one-out cross-validation (LOOCV) is applied to rigorously train and evaluate each heterogeneous base learner after hyperparameter optimization, and output the global prediction matrix of each heterogeneous base learner.

[0030] Finally, an ensemble loss function is constructed with the goal of minimizing the overall mean squared error (MSE). The Sequence Minimum Optimization (SLSQP) algorithm is used to perform nonlinear programming iterative solution under the strict constraint that the sum of the weighting coefficients of each heterogeneous base learner must be 1.0 and all of them must be in the interval [0, 1.0], so as to obtain the globally optimal fusion weighting coefficients of each heterogeneous base learner, thereby constructing a highly generalizable weighted ensemble prediction model.

[0031] S4. Generate a gridded virtual formula set covering the continuous phase diagram of the full proportion in the mixing constraint space, call the heterogeneous machine learning model library optimized and weighted by hyperparameters to perform positive extrapolation of global performance, and use the comprehensive satisfaction watershed threshold to perform forced physical boundary cutting to determine the safe formula feasible domain of the curing agent.

[0032] This embodiment employs high-throughput virtual simulation and safe formulation feasible region delineation to achieve comprehensive satisfaction assessment of high-throughput gridded virtual formulations within a mixing constraint space. Predictions are made, and slicing is performed based on underlying physical thresholds to determine the viable domain for safe formulations.

[0033] Specifically, in this embodiment, a gridded virtual formula set covering the full proportion of continuous phase diagrams is generated within the mixing constraint space, wherein the component scanning step size is precisely set to 1%, generating tens of thousands of high-resolution virtual mix proportion samples.

[0034] By calling upon a heterogeneous machine learning model library optimized and weighted by hyperparameters, high-throughput virtual prediction results are generated through full-domain positive batch extrapolation of the overall satisfaction of virtual combination ratio samples.

[0035] After the full-domain forward batch simulation is completed, a watershed threshold for overall satisfaction is set, and a forced post-physical boundary cut is performed on the high-throughput virtual prediction results. Invalid virtual formulas with overall satisfaction prediction values ​​lower than the watershed threshold are eliminated. This enables the accurate delineation of the feasible domain of safe formulas that ensure long-term resistance to acid erosion and activation of activity in a continuous space, thereby completing the delineation of the feasible domain of formula reverse design and the underlying physical boundary constraints.

[0036] S5. Introduce economic and environmental factors within the feasible domain of safe formulations to construct a comprehensive performance index. The Pareto algorithm is used to maximize the overall performance index. Output the globally optimal formulation of the curing agent for the target non-dominated sorting.

[0037] This embodiment introduces the economic and carbon emission factors of ternary materials into the feasible domain of safe formulations, and uses a multi-objective non-dominated sorting algorithm to construct a three-dimensional and two-dimensional Pareto objective mapping space, dynamically searching for non-dominated solutions to output a globally optimal formulation that takes into account mechanical performance, environmental performance and economy.

[0038] Specifically, this embodiment extracts updated baseline parameters for each component within the feasible domain of the safe formulation, including: cement cost determined to be 250 yuan / t and carbon emissions of 820 kgCO2e / t; steel slag cost determined to be 90 yuan / t and carbon emissions of 88 kgCO2e / t; and slag powder cost determined to be 130 yuan / t and carbon emissions of 78 kgCO2e / t. The overall satisfaction of the 100% pure cement baseline formulation within the feasible domain of the safe formulation is calculated using a heterogeneous machine learning model library. D ref As a baseline for comparison.

[0039] For each set of high-fidelity gridded virtual recipes remaining within the feasible domain of safe recipes, calculate their comprehensive cost. C With carbon emissions E Then, the relative economic index is obtained. or C =C / C ref Relative carbon emission index or E =E / E ref The overall satisfaction level of the current gridded virtual recipe is predicted by global positive batch extrapolation using a heterogeneous machine learning model library. D Compare it with the baseline. D ref The ratio is defined as the relative satisfaction index. or D =D / D ref This allows for the construction of a multi-dimensional comprehensive performance index within the feasible domain of safe formulations. The expression is: Finally, a multi-objective non-dominated sorting optimization algorithm is used to perform a visualization-based multi-objective screening of curing agents to obtain the globally optimal formulation, including the following steps: Constructing a three-dimensional Pareto frontier space: using relative economic indices or C Relative carbon emission index or E Relative satisfaction index or D Plot the comprehensive efficiency index using independent three-dimensional coordinate axes. Color mapping dynamically displays the original three-dimensional spatial game relationship between cost, carbon emissions, and mechanical durability, and eliminates globally disadvantageous solutions to obtain a non-dominated solution set, such as... Figure 4 As shown.

[0040] Constructing a two-dimensional Pareto frontier dimensionality reduction space: To facilitate intuitive engineering decision-making, the two macroeconomic cost indicators of economics and carbon emissions are merged. The merged relative economic index is used as the basis for this approach. or C Relative carbon emission index or E The obtained composite index The horizontal axis represents the relative satisfaction index. or D Using the vertical axis, perform a two-dimensional spatial topological filter to obtain a non-dominated solution set, such as... Figure 3 As shown.

[0041] Finally, based on the non-dominated solution set, the Pareto optimal front curve representing the extreme values ​​of multi-objective performance is output, and the global optimal solution is finally output as the global optimal formulation for the hardener reverse design. Engineering decision-makers can accurately extract the global optimal solution on the boundary of the 2D or 3D Pareto optimal front curve, based on the cost budget or carbon emission limit of a specific road section, as the final mix design result for the hardener reverse design.

[0042] S6. The game theory-based SHAP (Shapley Additive exPlanations) attribution algorithm is introduced to perform global visualization and perspective deconstruction of the decision chain of the heterogeneous machine learning model library, and to quantify the marginal contribution rate of each solid waste raw material to the comprehensive satisfaction prediction result in order to achieve rigorous cross-validation of the physicochemical mechanism of the intelligent decision model.

[0043] Specifically, after the intelligent reverse mix design is output as the optimal formula, a game theory-based SHAP attribution algorithm is introduced to perform global deconstruction of the core tree-like regression surrogate model in the integrated framework. By calculating the Shapley values ​​of each ternary component (cement, steel slag powder, blast furnace slag powder, etc.) for the final comprehensive satisfaction prediction result, a long-tailed distribution of feature importance is established on the full-domain experimental dataset, and the SHAP swarm analysis graph and the feature interaction dependency graph of each component are output, such as... Figure 5-Figure 7 As shown.

[0044] In-depth attribution analysis shows that when the cement content is in the range of 10% to 20% or the steel slag powder content is at a specific level, the corresponding SHAP value exhibits significant positive or polarized extreme value characteristics. This quantitatively verifies the underlying logic of the synergistic stimulation of multiple solid wastes from a data-driven perspective. That is, the rapid exothermic reaction of cement hydration provides an initial high alkalinity start-up environment, while the continuous release of calcium hydroxide by steel slag powder in the middle and later stages provides a long-term alkalinity buffer, thereby effectively stimulating the potential pozzolanic activity of large-volume slag powder. This completely breaks the black-box nature of traditional machine learning models and establishes a closed-loop logic across the entire chain between pure data deduction and physicochemical mechanism verification.

[0045] The present invention will be further described below with reference to specific embodiments, but the scope of protection of the present invention is not limited thereto. The ternary solid waste raw materials used in this embodiment and their economic and environmental benchmark parameters are as follows: Example 2: Based on the heterogeneous machine learning model library and the feasible region of safe recipes constructed in Example 1, for conventional renovation and expansion projects with limited costs and requirements for stable overall performance, the non-dominated solution set corresponding to the Pareto optimal frontier curve is first output. Based on the project's need for a balance between performance and cost, a comprehensive performance index is extracted from the Pareto optimal frontier curve. The non-dominated solution corresponding to the peak value is taken as the optimal equilibrium point.

[0046] The optimal mix proportions output from this equilibrium point are: 12% cement, 35% steel slag powder, and 53% blast furnace slag powder. The total cost of preparing the curing agent according to this proportion is 130.4 yuan / t, with carbon emissions of 170.5 kg CO2e / t. Testing verified that this formula is within the safe and feasible range. When added to phosphogypsum at an 8% mass ratio and compacted, the 28-day unconfined compressive strength of the cured specimen reached 2.32 MPa, demonstrating sufficient long-term alkalinity buffering. With continued curing, the unconfined compressive strengths at 90 days and 180 days steadily increased to 2.78 MPa and 3.12 MPa, respectively, with a CBR value reaching 43.3%, completely eliminating the risk of deterioration in acidic environments. This successfully achieved optimal overall performance while meeting basic corrosion resistance and durability requirements.

[0047] Example 3: The heterogeneous machine learning model library constructed in Example 1, targeting road engineering projects in ecologically sensitive areas with strict red lines for environmental protection and carbon emissions, also first generates the non-dominated solution set corresponding to the Pareto optimal frontier curve. The region with extremely low carbon emissions is directly identified from the non-dominated solution set, and the relative carbon emission index is extracted. or E The minimum nondominated solution. Under the premise of satisfying the overall satisfaction watershed threshold, the lower limit of the limit for the use of high carbon emission cement is explored.

[0048] The optimal low-carbon mix ratio was finally determined to be: 10% cement, 40% steel slag powder, and 50% blast furnace slag powder. The total cost of this formula was 126 yuan / t, with carbon emissions drastically reduced to 156.2 kg CO2e / t. Actual measurements showed that after being added to phosphogypsum at an 8% mass ratio and compacted, the 28-day unconfined compressive strength of this ultra-low-carbon formula was 1.95 MPa. Its 90-day and 180-day strengths increased to 2.41 MPa and 2.86 MPa, respectively, with a CBR value reaching 37.8%. Compared to traditional high-cement hardeners, this formula demonstrates significantly significant carbon reduction benefits while fully meeting the standards for compressive strength and long-term durability.

[0049] Example 4: For heavy-load traffic roadbeds, the curing agent is required to have stronger activity activation and long-term resistance to degradation in acidic environments. A relative satisfaction index is selected from the non-dominated solution set corresponding to the Pareto optimal frontier curve. or D The highest undominated solution was obtained. The final high-strength mix proportion was: 22% cement, 18% steel slag powder, and 60% slag powder. This formula maximizes the potential activity of the slag while providing sufficient initial alkalinity. Actual measurements showed that when this formula was added to phosphogypsum at an 8% mass ratio and compacted, its 28-day unconfined compressive strength reached 3.48 MPa, and its 90-day and 180-day strengths were 4.23 MPa and 4.71 MPa, respectively, with a CBR value of 81.9%, exhibiting excellent mechanical properties.

[0050] Comparative Example 1: Based on the same 66 discrete data points across the entire domain as in Example 1, a nonlinear polynomial was used to establish the response surface equation for each component and its individual 28-day unconfined compressive strength index. The objective function was set to maximize the 28-day compressive strength, and a single-objective optimization programming was run, resulting in a final mix proportion of 30% cement, 10% steel slag powder, and 60% blast furnace slag powder. The unit cost of the curing agent was 162 yuan / t, and the carbon emission was 301.6 kgCO2e / t. Although the 28-day unconfined compressive strength reached 3.52 MPa and the CBR value reached 105.8%, due to the lack of sufficient steel slag powder to provide a long-term alkalinity buffer, the strength experienced a precipitous decline at 90 days, significantly degrading to 2.18 MPa; by 180 days, it further collapsed to 1.89 MPa, indicating severe microstructural deterioration within the roadbed filler. This data definitively confirms that traditional single-objective optimization is prone to falling into the empirical trap of high early intensity, while the present invention can completely eliminate the hidden danger of late-stage intensity reduction through satisfaction baseline constraints and Pareto front post-screening.

[0051] Comparative Example 2: Based on experience with traditional road-use phosphogypsum curing agent formulations and following a high cement content design approach, the synergistic alkali-activation effect between solid wastes was ignored. The industry-standard high-strength formula was selected: 60% cement, 0% steel slag powder, and 40% blast furnace slag powder.

[0052] The unit cost of the molded specimens according to this formula reached as high as 202 yuan / t, and the carbon emissions surged to 523.2 kgCO2e / t. Because the long-term resistance to degradation was not assessed using the Derringer satisfaction function, although the 28-day unconfined compressive strength reached 2.40 MPa and the CBR value reached 49.1%, after further curing to 90 days, the compressive strength significantly decreased to 1.89 MPa, and further to 1.65 MPa at 180 days, resulting in severe structural failure. This comparative data comprehensively demonstrates the superior advancement of the method of this invention in engineering applications.

[0053] To more intuitively demonstrate the significant technological advancements achieved by this invention, the curing agent formulations, macroeconomic and environmental indicators, as well as the core mechanical properties, durability, and environmental safety indicators at various ages of the above embodiments and comparative examples are summarized and compared, as shown in Table 1.

[0054] Table 1. Performance Comparison of Examples and Comparative Examples As shown in Table 1, the intelligent optimized formulations output in Examples 2 to 4 of this invention achieve a 28-day CBR value of 37.8%–81.9%, far exceeding the specification requirement of ≥8% for high-grade highway subgrade fillers. This indicates that the multi-element solid waste formulation can directly meet the engineering needs of heavy-load traffic at all levels. Furthermore, relying on the synergistic regulation of bottom-level satisfaction and top-level Pareto optimization, the overall cost of the curing agent in each embodiment is significantly reduced by 26.1%–37.6% compared to traditional formulations, and carbon emissions are reduced by 53.5%–70.1%, achieving a high degree of unity between high performance and low-carbon economy.

[0055] In contrast, the formulation output by the method of this invention, through the implicit satisfaction penalty mechanism at the bottom layer and Pareto optimization at the top layer, not only reduces the cost of the curing agent by 26.1% to 37.6% and sharply reduces carbon emissions by 53.5% to 70.1%, but also completely overcomes the hidden danger of strength reduction from the underlying material mechanism. The 28-day strength retention rate of each embodiment reaches more than 16%, and its 90-day and 180-day compressive strengths show a good long-term growth trend, achieving a perfect multi-objective synergy of mechanical properties, durability, economy, and low carbon environmental protection.

[0056] Although Comparative Examples 1 and 2 achieved high CBR values ​​and 28-day unconfined compressive strength by piling up conventional activating cement with high carbon emissions, they still suffered severe erosion at 90 and 180 days due to the lack of sufficient steel slag powder to provide a long-term buffer alkali source under the continuous erosion of the highly acidic environment of phosphogypsum. This resulted in irreversible strength reduction. In the field of phosphogypsum curing, evaluating formulations based solely on short-term CBR or early strength indicators is deceptive.

[0057] This invention innovatively introduces the Derringer satisfaction function as a low-level implicit penalty mechanism to determine the high strength retention rate of the formulation. The 90-day and 180-day compressive strengths of each embodiment not only did not decrease, but also showed a good long-term growth trend, ensuring the long-term service performance of the roadbed engineering from both the data foundation and physicochemical mechanism perspectives.

[0058] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0059] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. Since the above embodiments are substantially similar to the method embodiments, their descriptions are relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0060] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A reverse design method for phosphogypsum curing agents based on ensemble machine learning and Pareto optimization, characterized in that, The method includes the following steps: Based on the global simplex lattice design, point tests were conducted on curing agents with different ratios within the mixing constraint space, and data on various core mechanical and durability properties were collected to establish an original database for characterizing the relationship between curing agent formulation and performance. Based on the original database, the core evaluation index is calculated to eliminate the interference of the cementitious material's own activity and quantify the degree of erosion of the curing agent by the acidic substances inside the phosphogypsum. The Derringer satisfaction function is introduced to transform the multi-dimensional target response into a single prediction target of high confidence comprehensive satisfaction. Using overall satisfaction as the single prediction target, hyperparameter global optimization is performed based on a heterogeneous machine learning model library driven by global data. The global optimal fusion weighting coefficient of each heterogeneous base learner is then solved iteratively under the weight boundary constraints using a sequence minimum optimization algorithm. In the mixing constraint space, a gridded virtual formula set covering the continuous phase diagram of the full ratio is generated. The heterogeneous machine learning model library optimized and weighted by hyperparameters is called to perform forward extrapolation of the global performance. Forced physical boundary cutting is performed by the comprehensive satisfaction watershed threshold to determine the safe formula feasible domain of the curing agent. Economic and environmental factors are introduced within the feasible domain of safe formulations to construct a comprehensive performance index. The Pareto algorithm is used to output the globally optimal formulation of the curing agent through non-dominated sorting with the goal of maximizing the comprehensive performance index.

2. The reverse design method for phosphogypsum curing agent according to claim 1, characterized in that, Establishing the original database includes: The curing agent is composed of various solid waste materials, and the mass fraction of its components is: ,satisfy A point-based experiment was conducted using a global simplex lattice design within a mixed-material constrained space. Different proportions of curing agents are mixed into phosphogypsum at a fixed external dosage, and then mixed, compacted and cured at the optimal moisture content to form a multi-element solid waste system. Experiments were conducted to test the performance of a multi-component solid waste system during the curing period. The core mechanical and durability performance data of the underlying solid waste system were obtained, including the compressive strength of the mortar. The compressive strength of mortar for PO 42.5 cement at the same curing age And the unconfined compressive strength of the cured body after the curing agent is incorporated into phosphogypsum. This led to the establishment of a discrete original database of curing agent formulations and properties.

3. The reverse design method for phosphogypsum curing agent according to claim 1, characterized in that, The fixed amount of the curing agent is 8% of the dry weight of the phosphogypsum.

4. The reverse design method for phosphogypsum curing agent according to claim 1, characterized in that, Calculate the core evaluation indicators and convert them into overall satisfaction. include: The calculation, including the activity index, is based on the established original database. and strength retention rate Core evaluation indicators; Response to each individual objective Set an acceptable lower limit and upper limit And calculate the individual satisfaction of the core evaluation indicators. The calculation formula is: ； in, The target importance weight index; Using the geometric mean method based on individual satisfaction levels Building overall satisfaction The expression is: ； in, The total number of target responses.

5. The reverse design method for phosphogypsum curing agent according to claim 4, characterized in that, The activity index The formula used to characterize the cementation potential of a multi-component solid waste system under ideal conditions is as follows: ； The strength retention rate The formula for intuitively quantifying the resistance to degradation and strength reduction of the curing agent in acidic environments of phosphogypsum is as follows: ； in, To maintain the age The compressive strength of mortar in a multi-element solid waste system; To maintain the age The compressive strength of the mortar in P.O42.5 cement; The unconfined compressive strength of the cured body after incorporating phosphogypsum into the curing agent.

6. The reverse design method for phosphogypsum curing agent according to claim 1, characterized in that, Solving for the globally optimal fusion weighting coefficients for each heterogeneous base learner includes: A heterogeneous machine learning model library, including support vector regression machine, random forest and extreme gradient boosting tree, is used, with a full-domain test dataset including a multi-dimensional solid waste system and a large number of failure samples as input features. Based on overall satisfaction As a single prediction target, hyperparameter global Bayesian optimization is performed on each heterogeneous base learner through cross-validation. Leave-one-out cross-validation is used to train and evaluate each heterogeneous base learner after hyperparameter optimization, and the global prediction matrix of each heterogeneous base learner is output. An ensemble loss function is defined with the goal of minimizing the overall mean squared error. The sequence minimum optimization algorithm is used to perform nonlinear programming iterative solution under the constraint that the sum of the weighting coefficients of each heterogeneous base learner must be 1.0 and all of them must be in the interval [0, 1.0] to obtain the globally optimal fusion weighting coefficients of each heterogeneous base learner.

7. The reverse design method for phosphogypsum curing agent according to claim 1, characterized in that, Determining the safe formulation feasibility range of the curing agent includes: Within the mixing constraint space, a gridded virtual formula set covering the full proportion continuous phase diagram is generated, wherein the component scanning step size is precisely set to 1%, and a total of tens of thousands of high-resolution virtual mix proportion samples are generated. By calling a heterogeneous machine learning model library optimized and weighted by hyperparameters, high-throughput virtual prediction results are generated by performing a full-domain positive batch extrapolation of the overall satisfaction of virtual combination ratio samples. After the full-domain forward batch simulation is completed, a watershed threshold is set and a forced post-physical boundary cut is performed on the high-throughput virtual prediction results to eliminate invalid virtual formulas with a comprehensive satisfaction prediction value lower than the watershed threshold. This enables the accurate delineation of the feasible domain of safe formulas that ensure long-term resistance to acid erosion and activation of activity within a continuous space.

8. The reverse design method for phosphogypsum curing agent according to claim 7, characterized in that, The globally optimal formulation of the curing agent is output as follows: Calculate the combined cost of each set of gridded virtual recipes within the feasible region of the safe recipe. C With carbon emissions E ; Based on comprehensive cost C With carbon emissions E Calculate the relative economic index η C and relative carbon emission index η E The calculation formula is: ； ； in, C The cost of phosphogypsum roadbed filler; C ref The cost of cement-cured phosphogypsum roadbed filler; E Carbon emissions from phosphogypsum roadbed filler; E ref Carbon emissions of cement-cured phosphogypsum roadbed filler; The overall satisfaction level predicted by using a heterogeneous machine learning model library to perform a global positive batch extrapolation of the current gridded virtual recipe is obtained. D And calculate the relative satisfaction index. η D The calculation formula is: ； in, D Overall satisfaction with the curing agent-cured phosphogypsum roadbed filler; D ref The overall satisfaction level of pure cement-cured phosphogypsum roadbed filler; According to the relative economic index η C Relative carbon emission index η E Relative satisfaction index η D To construct a comprehensive performance index for achieving a balance between mechanical properties, carbon emissions, and economic benefits of solidified materials. The expression is: ； Using the Pareto algorithm to maximize the overall performance index The algorithm iteratively optimizes within the feasible domain of safe formulations to ultimately output the globally optimal formulation of the curing agent.

9. The reverse design method for phosphogypsum curing agent according to claim 8, characterized in that, The Pareto algorithm performs visualization-based multi-objective screening based on dimensionality reduction mapping, including: Constructing a three-dimensional Pareto frontier space: using relative economic indices η C Relative carbon emission index η E Relative satisfaction index η D Plot the comprehensive efficiency index using independent three-dimensional coordinate axes. Color mapping dynamically displays the original three-dimensional spatial game relationship between cost, carbon emissions, and mechanical durability, and eliminates globally disadvantageous solutions. Constructing a two-dimensional Pareto frontier dimensionality reduction space: using the merged relative economic index η C Relative carbon emission index η E The obtained composite index The horizontal axis represents the relative satisfaction index. η D Using the vertical axis, perform a two-dimensional spatial topological filter to obtain a non-dominated solution set; Based on the non-dominated solution set, the Pareto optimal frontier curve representing the extreme value of multi-objective performance is obtained, and the global optimal solution is finally output as the global optimal formulation for the reverse design of the curing agent.

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