Small sample bayesian optimization sampling method and system based on cloud drop data enhancement

Abstract

This invention discloses a small-sample Bayesian optimization sampling method and system based on cloud droplet data augmentation, belonging to the field of transportation engineering material design. Addressing the problems of limited initial samples and poor fit of the Bayesian optimization surrogate model in modified asphalt formulation design, this invention acquires initial small-sample data; constructs a clustering cloud model to obtain expectation, entropy estimates, and hyperentropy estimates; generates cloud droplet virtual samples using a forward cloud generator and merges them with the original samples to expand the dataset; trains a Gaussian process regression surrogate model based on the expanded data; constructs an expectation-improved acquisition function to optimize and solve candidate formulations; updates the data after physical testing verification; and iterates repeatedly until the termination condition is met to output the optimal formulation. This invention mines the distribution information of small-sample data through cloud droplet data augmentation, improves the accuracy and sampling efficiency of the surrogate model, significantly reduces the number of expensive physical tests, and lowers the cost of formulation design. It can be widely applied to the formulation optimization of modified asphalt and similar high-cost experimental materials.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of transportation engineering material design and artificial intelligence optimization, specifically involving a high-cost, small-sample black-box function optimization method and system for modified asphalt concrete formulation design. This invention is particularly suitable for complex engineering material formulation optimization problems where initial data is extremely scarce due to long indoor testing cycles and high raw material costs, and where target performance indicators exhibit significant physical uncertainties. It can be widely applied to formulation design and optimization scenarios for modified asphalt and similar high-cost experimental materials. Background Technology

[0002] In the formulation design of modified asphalt concrete, to obtain the optimal mix proportion that meets road performance requirements, a large number of laboratory tests (such as rutting tests and flexural tests) are usually required to verify the formulation performance. However, such tests are not only time-consuming and labor-intensive, but also involve expensive modifiers (such as SBS and PE) and complex preparation processes, making the cost of a single test high. Therefore, in practical engineering applications, engineers often can only conduct a very small number of trial mixes within a limited budget, resulting in a very small initial sample data set (usually less than 10 sets), forming a typical small sample optimization scenario.

[0003] Bayesian optimization, as a highly efficient global optimization algorithm, predicts material properties by constructing a surrogate model (such as a Gaussian process) and uses a data acquisition function to balance "developing" the current optimal formulation with "exploring" unknown formulation regions. Theoretically, it can find the global optimum with the fewest trials, making it a preferred solution for optimization in small-sample scenarios. For example, Chinese invention patent CN121034488A discloses a data-driven method for predicting and optimizing the rutting resistance of asphalt materials, which uses Bayesian optimization to design and optimize asphalt material formulations. However, the performance of Bayesian optimization is highly dependent on the accuracy of the surrogate model. In small-sample scenarios of modified asphalt formulation design, the initial data often cannot cover complex nonlinear response surfaces, leading to significant cognitive uncertainty in the surrogate model, distorted prediction variance, and consequently, difficulty in providing effective experimental suggestions through the data acquisition function. This ultimately results in low optimization efficiency and even getting trapped in local optima.

[0004] To address the data scarcity problem in small-sample optimization, existing technologies mainly employ two types of data augmentation schemes: one is the Bootstrap resampling method, which essentially involves simple repetition or sampling with replacement of existing formulation data. The resulting data distribution tends to be uniform, making it difficult to reflect the complex nonlinear coupling relationship between asphalt mixture components and performance. Furthermore, it fails to increase the potential information content of the data and cannot fundamentally improve the fitting accuracy of the surrogate model. The second is data augmentation methods based on neural networks such as Generative Adversarial Networks (GANs). Although these methods can generate new virtual samples, the training process itself requires massive amounts of historical data, which contradicts the realistic premise of "small samples" in formulation design and has poor practicality.

[0005] Existing technologies propose cloud models, using three numerical features—expectation (Ex), entropy (En), and hyperentropy (He)—as cloud model parameters. This enables the transformation of uncertainty between qualitative concepts and quantitative values, exhibiting good handling capabilities for randomness and fuzziness, and effectively uncovering potential distribution patterns in small sample data. For example, Chinese invention patent CN110046427A discloses a method for mix proportioning of manufactured sand T-beam concrete based on orthogonal design and a normal cloud model, which uses the cloud model to evaluate concrete mix proportion schemes. However, existing literature primarily applies cloud models to data evaluation and classification scenarios, failing to deeply integrate them with Bayesian optimization frameworks. It also fails to utilize their generation mechanism to provide high-quality virtual augmented samples for asphalt formulation proxy models, thus unable to effectively solve the small-sample optimization problem in expensive material experiments. Therefore, this invention proposes a small-sample Bayesian optimization sampling method and system based on cloud droplet data augmentation to fill the gap in existing technologies. Summary of the Invention

[0006] The purpose of this invention is to provide a small-sample Bayesian optimization sampling method and system based on cloud droplet data augmentation, in order to solve the technical problems in the design of existing modified asphalt concrete formulations, where the high cost of testing leads to very few sample data, resulting in poor fitting of the Bayesian optimization surrogate model, low sampling efficiency, and easy getting trapped in local optima.

[0007] In a first aspect, to achieve the above-mentioned objective, the present invention provides a small-sample Bayesian optimization sampling method based on cloud droplet data augmentation, comprising the following steps: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators; A clustering cloud model is constructed based on the initial sample set. The formulation component parameters and the target performance index are used together as a joint vector to construct the clustering cloud model, and the cloud model parameters are obtained. The cloud model parameters include the expected value, the entropy estimate, and the hyperentropy estimate. The dataset is expanded by generating a preset number of cloud droplet recipe data based on the cloud model parameters using a forward cloud generator, and then merging the cloud droplet recipe data with the initial sample set to form an expanded training dataset. The Bayesian optimized surrogate model is trained based on the expanded training dataset, and outputs the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition optimization involves constructing an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and then optimizing the solution to obtain the next candidate sampling formula. Experimental verification: Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated. Based on the updated initial sample set, repeat the above steps of building a clustering cloud model, expanding the dataset, training the Bayesian optimized surrogate model, collecting and optimizing data, and conducting experimental verification until the preset termination condition is met, and output the optimal formula in the initial sample set.

[0008] Furthermore, the specific steps for constructing the clustering cloud model include: Clustering algorithms are used to calculate the cluster centers of the initial sample set, which are then used as the expectation of the cloud model. For each dimension parameter of the joint vector, calculate the class-first absolute central moment and class-second central moment of the initial sample set respectively; Based on the characteristics of Gaussian distribution, the entropy estimate of the cloud model is obtained by using the first-order absolute central moment. The hyperentropy estimate of the cloud model is calculated based on the second-order central moments and the entropy estimate, and the expectation, entropy estimate, and hyperentropy estimate are used as cloud model parameters.

[0009] Furthermore, in the process of generating the cloud droplet formulation data, for each cloud droplet to be generated, an instantaneous entropy following a normal distribution is first generated for it. The instantaneous entropy follows a normal distribution with the entropy estimate as the expectation and the hyperentropy estimate as the standard deviation. Then, based on the instantaneous entropy and the expectation of the cloud model, the formulation component parameters of the cloud droplet and their corresponding certainty are calculated through the probability density function of the normal distribution.

[0010] Furthermore, the Bayesian optimized surrogate model is a Gaussian process regression model, and its kernel function includes the squared exponential kernel function or the Matern kernel function.

[0011] Furthermore, the acquisition function is the desired improvement function, and its expression is:

[0012] in, The mean of the predictions output by the Bayesian optimized surrogate model. The standard deviation of the prediction output by the Bayesian optimized surrogate model is given. This represents the current optimal performance value. To explore parameters, As an intermediate variable, and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

[0013] Furthermore, the process of optimizing and solving to obtain the next candidate sampling formula adopts the L-BFGS-B quasi-Newton method or genetic algorithm, and is carried out under the condition of satisfying the boundary constraints of the formula components and the physical process constraints.

[0014] Furthermore, in the initial sample set, the formulation component parameters include the SBS content, asphalt content, and powder-to-oil ratio of the modified asphalt concrete; the target performance index is the high-temperature rutting resistance performance index, including dynamic stability or rutting factor.

[0015] Furthermore, the preset termination condition is that the target performance index reaches a preset threshold, or the number of iterations reaches a preset maximum value.

[0016] Secondly, the present invention provides a small-sample Bayesian optimized sampling system based on cloud droplet data augmentation, comprising: The data acquisition module is used to acquire an initial sample set, which includes formulation component parameters and corresponding target performance indicators; The cloud model construction module is used to receive the initial sample set, construct a clustered cloud model based on the initial sample set, and obtain cloud model parameters, including expectation, entropy estimate and hyperentropy estimate. The dataset expansion module has a built-in positive cloud generator. Based on the cloud model parameters, it generates a preset number of cloud droplet recipe data through the positive cloud generator. The cloud droplet recipe data is then merged with the initial sample set to form an expanded training dataset. The Bayesian optimization engine module includes a surrogate model training unit and a data acquisition function optimization unit, which are used to train a Bayesian optimization surrogate model based on the expanded training dataset and output the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition and optimization module constructs an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and optimizes the solution to obtain the next candidate sampling formula. The experimental verification module is used to connect to the experimental equipment, perform physical experimental verification on the candidate sampling formula, obtain actual performance indicators, add the candidate sampling formula and the corresponding actual performance indicators to the initial sample set, and update the initial sample set. The loop iteration control module is used to coordinate the operation of each module, update the sample set, determine whether the preset termination condition is met, and output the optimal formula.

[0017] Furthermore, the modules are connected via a data bus to form a closed-loop control automated recipe design system, supporting the automated execution of iterative optimization processes.

[0018] Compared with the prior art, the present invention has the following beneficial effects: 1. High data distribution fidelity and more physical meaning of virtual samples: Based on the cloud droplets generated by the clustering cloud model, this invention can accurately simulate the randomness and fuzziness of asphalt formula data through the dual constraints of entropy and hyperentropy, retain the dense distribution characteristics of the original samples in the component space, avoid the problem of artificial homogenization of data distribution caused by the traditional Bootstrap method, and make the expanded data more in line with the physical laws of actual engineering materials.

[0019] 2. Significantly improved efficiency and reduced experimental costs: Clouddrop data augmentation effectively compensates for the lack of information in small sample data, significantly improves the prediction accuracy and uncertainty estimation ability of the Bayesian optimization surrogate model, and enables the acquisition function to more accurately identify potential high-performance formulations. This greatly reduces the number of expensive experiments required to find the optimal mixing ratio, reduces raw material and experimental costs, and shortens the formulation design cycle.

[0020] 3. Precise quantification of uncertainty and more scientific sampling strategy: This invention establishes an intrinsic mathematical connection between data uncertainty and cognitive uncertainty (quantified by entropy and hyperentropy of cloud models) and Bayesian optimization sampling strategy, enabling the optimization process to achieve a balance that is more in line with the understanding of materials science between "exploring" unknown formulation regions and "developing" existing high-performance formulations, effectively avoiding the optimization process from getting trapped in local optima.

[0021] 4. Highly applicable and easy to implement: The method of this invention does not rely on large-scale historical data and only requires a very small amount (usually 5 to 10 sets) of initial experimental data to start the optimization process. It is particularly suitable for scenarios without prior experience, such as the research and development of new modifiers and the design of special asphalt concrete. At the same time, the algorithm logic is clear, the module design is simple, and it is easy to implement in engineering and connect with existing laboratory equipment, making it highly practical. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0023] Figure 1 This is an overall flowchart of the small-sample Bayesian optimization sampling method based on cloud droplet data augmentation provided in this embodiment of the invention.

[0024] Figure 2 This is a schematic diagram comparing cloud droplet generation before and after in an embodiment of the present invention, wherein, Figure 2 In diagram A, the determination curve is mapped from the original sample before amplification. Figure 2 Image B shows a comparison between the original sample and the virtual amplified sample after amplification.

[0025] Figure 3 This is a schematic diagram showing the comparison between the original sample distribution, the Bootstrap amplified distribution, and the cloud droplet amplified distribution in an embodiment of the present invention.

[0026] Figure 4 This is a comparison chart of the Bayesian optimization iterative convergence curve based on the cloud droplet enhanced agent model in this embodiment of the invention and the traditional method.

[0027] Figure 5 This is a schematic diagram of the module structure of the small sample Bayesian optimized sampling system based on cloud droplet data augmentation provided in an embodiment of the present invention.

[0028] Figure 6 This is a schematic diagram of the physical structure of an electronic device provided in an embodiment of the present invention. Detailed Implementation

[0029] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments of this invention are only for explaining the invention and are not intended to limit the scope of protection of this invention.

[0030] Example 1 like Figure 1 As shown, this embodiment provides a small-sample Bayesian optimization sampling method based on cloud droplet data augmentation, including the following steps: Step 1: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators.

[0031] Step 2: Construct a clustering cloud model. Based on the initial sample set, construct a clustering cloud model by using the formulation component parameters and the target performance index together as a joint vector to obtain cloud model parameters, which include expectation, entropy estimate and hyperentropy estimate.

[0032] Step 3: Dataset expansion. Based on the cloud model parameters, a preset number of cloud droplet recipe data are generated using a forward cloud generator as virtual samples. The cloud droplet recipe data is then merged with the initial sample set to form an expanded training dataset.

[0033] Step 4: Bayesian optimization surrogate model training. Based on the expanded training dataset, train the Bayesian optimization surrogate model and output the predicted mean and predicted variance of the target performance index for each candidate formulation.

[0034] Step 5: Data Acquisition and Optimization. Based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, a data acquisition function is constructed and optimized to obtain the next candidate sampling formula.

[0035] Step 6: Experimental verification. Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated.

[0036] Step 7: Iterative optimization and termination. Based on the updated initial sample set, repeat steps 2 to 6 above until the preset termination condition is met, and output the optimal formula.

[0037] The core inventive concept of this invention lies in deeply integrating a clustering cloud model, which can accurately quantify the uncertainty of small sample data, with a Bayesian optimization framework that excels at efficient global optimization, to form a closed-loop iterative optimization system of "data augmentation - proxy modeling - intelligent sampling - experimental verification".

[0038] The technical principle is as follows: First, for the initial small amount of real experimental data, a clustering cloud model is constructed to mine and formally express the inherent randomness (quantified by entropy En) and fuzziness (quantified by hyperentropy He) of the data itself. Then, using this cloud model containing uncertainty information, a large number of "cloud droplets" conforming to the physical distribution law of the original data are generated as virtual samples through a forward cloud generator. These virtual samples are not simple noise, but a reasonable extrapolation and filling of the potential distribution space of the original data, effectively expanding the training dataset. Next, this expanded high-quality dataset is used to train a Bayesian optimized surrogate model (such as Gaussian process regression), which significantly improves the model's fitting accuracy to the real response surface and the reliability of uncertainty estimation. Finally, based on the more accurate surrogate model, the acquisition function can more scientifically make a trade-off between "developing" known high-potential regions and "exploring" unknown uncertain regions, thereby guiding subsequent expensive physical experiments directly to the global optimum and significantly reducing the number of experiments. Throughout the process, the cloud model parameters are updated and the Bayesian optimization is iterated simultaneously, which deepens the understanding of data uncertainty as experimental data accumulates, ultimately achieving efficient and accurate formulation optimization.

[0039] The technical effects of this solution include: 1. High data distribution fidelity: The virtual samples generated by the clustering cloud model accurately simulate the randomness and ambiguity of the original data, avoiding the data distribution distortion caused by traditional resampling methods.

[0040] 2. Significantly improved efficiency: It greatly reduces the number of expensive physical tests, lowers test costs, and shortens the design cycle.

[0041] 3. Scientific sampling strategy: It quantifies data uncertainty and cognitive uncertainty, making the exploration and development balance of Bayesian optimization more physically meaningful and avoiding getting trapped in local optima.

[0042] 4. High applicability: It does not rely on large-scale historical data and is applicable to various small-sample, expensive optimization problems.

[0043] Each step in the above scheme will be explained in detail.

[0044] Detailed explanation of step 1 (obtaining the initial sample set): In this invention, initial small-sample test data for modified asphalt concrete formulation design are obtained. The initial small-sample test data includes formulation component parameters (such as SBS modifier dosage, asphalt dosage, powder-to-oil ratio, etc.) and corresponding target performance indicators (such as dynamic stability DS). The initial small-sample test data are preprocessed to remove outliers and fill in missing values ​​to obtain a standardized initial sample set D.

[0045] This invention provides an exemplary method that first clarifies the design variables (i.e., formulation component parameters) and optimization objectives (i.e., target performance indicators) of the formulation. Taking the optimization of the high-temperature rutting resistance of SBS modified asphalt concrete as an example, the formulation component parameters typically include: SBS modifier dosage (mass fraction, %), asphalt dosage (mass fraction, %), powder-to-oil ratio (mass ratio of mineral filler to asphalt, unitless), aggregate gradation parameters, etc. The target performance indicator is dynamic stability DS (cycles / mm), which is determined by standard rutting tests (such as T 0719-2011) and used to evaluate the high-temperature deformation resistance of asphalt mixtures. Then, the Design of Experiments (DoE) method is used to obtain initial small sample test data, such as orthogonal design, uniform design, or Latin hypercube design, and a set of representative initial test points are selected within the design space. The purpose of this is to cover the design space to the greatest extent possible with the fewest sample points, providing a good foundation for subsequent cloud model construction and surrogate model training. The number of initial samples is usually 5 to 20 sets, depending on the dimension of the design space and the experimental cost.

[0046] Finally, data preprocessing was performed on the initial small sample experimental data to ensure data quality. Preprocessing operations included: ① Outlier removal: Using the Grubbs criterion or the Laida criterion, outlier data points caused by experimental errors or recording mistakes were identified and removed; ② Missing value imputation: For incomplete data caused by experimental failures or missing records, K-nearest neighbor imputation or mean imputation could be used; ③ Data standardization: To eliminate the influence of different component parameters' dimensions, each formulation parameter was standardized or normalized to the [0, 1] interval using Z-score.

[0047] Detailed explanation of step 2 (building the clustering cloud model): This step is crucial for achieving high-quality data augmentation, and its core lies in accurately estimating the three theoretical numerical features (expectations) of the cloud model from a small number of measured samples. ,entropy hyperentropy ), to obtain sample estimates for each feature ( , , This serves as a parameter for the cloud model, providing a parameter basis that conforms to the original data distribution pattern for subsequent cloud droplet generation. The specific operation is as follows: Step 2.1: Calculate the cluster centers of the initial sample set using a clustering algorithm, and use them as the expectation of the cloud model.

[0048] Cluster analysis is performed on the preprocessed initial sample set. Since the initial sample size is very small and the goal is to find the optimal formulation, it is typically assumed that these samples are distributed around a potential optimal center. Therefore, this embodiment uses the K-means clustering algorithm, setting the number of clusters K=1. The unique cluster center C output by the algorithm is a multi-dimensional coordinate point. For example, the joint vector includes formulation component parameters (such as SBS content, asphalt content, and powder-to-oil ratio) and target performance indicators (such as dynamic stability DS). In the space of the joint vector, the cluster center is denoted as... ,in, These are the cluster centers for SBS content, asphalt content, powder-to-oil ratio, and dynamic stability DS, respectively. The values ​​of each dimension of this cluster center C are used as the expected values ​​of the cloud model for the corresponding formulation parameters. .expect It represents the central value of the qualitative concept (i.e., the optimal formulation region) in the universe of discourse (i.e., the formulation component space), and is a specific multidimensional coordinate point.

[0049] For example, a sample of the original formulation of a modified asphalt concrete formula. ,in, SBS modifier dosage (unit: %) Asphalt usage (unit: %) The ratio of powder to oil (unit: dimensionless) Dynamic stability DS (unit: cycles / mm).

[0050] After calculation using K-means clustering (K=1), the cluster centers C are obtained as three-dimensional coordinates. For example, the calculated cluster centers are... .

[0051] The corresponding specific physical meaning is: The expected content of SBS modifier This means that, based on the existing data from the nine experimental formulations, the SBS content in the better-performing formulations is roughly concentrated around 4.2%.

[0052] Expected amount of asphalt used The initial sample set showed that the asphalt content tended to be 4.8%.

[0053] The expected value of the powder-to-oil ratio .

[0054] Cluster center C is directly used as the expectation of the cloud model. This represents the center point of the high-quality formula area. When generating virtual samples (cloud droplets), the system will revolve around this center point. Random fluctuations are made to generate more physically meaningful virtual recipes, such as (4.15%, 4.75%, 1.21) or (4.25%, 4.85%, 1.19).

[0055] Step 2.2: Calculate the class-first absolute central moments of the initial sample set. and class second central moments .

[0056] Based on the established expectations Two core statistical moments are calculated for the initial sample set to provide a data foundation for subsequent calculations of entropy and hyperentropy estimates. Each formulation parameter dimension is calculated independently, as shown in the following formula: 1. Quasi-first-order absolute central moments Reflects the expected value of the sample points The mean absolute distance, which characterizes the dispersion trend of the data, is expressed by the formula:

[0057] Where n is the number of samples in the initial sample set. For the first iThe formulation parameter values ​​for each sample; the first-order absolute central moments of SBS content, asphalt content, and powder-to-oil ratio are calculated using the above formulas.

[0058] 2. Second-order central moment S: also known as sample variance, reflects the distance from the sample point to the expected value. The overall dispersion of data is a quantitative indicator of the total uncertainty of the data, and the formula is:

[0059] Step 2.3: Based on the Gaussian distribution characteristics, the entropy estimate of the cloud model is calculated using the first-order absolute central moments. .

[0060] The theoretical entropy (En) of the cloud model is an inherent numerical characteristic characterizing the randomness of the recipe data, while the theoretical hyperentropy (He) is an inherent numerical characteristic characterizing the stability of entropy and reflecting the fuzziness of the data. Neither of these can be directly obtained from small sample data. They need to be estimated based on Gaussian distribution characteristics and the aforementioned statistical moments to obtain the entropy estimate. and hyperentropy estimate The derivation logic and formulas are as follows: For those following a Gaussian distribution The dataset whose theoretical first-order absolute central moments satisfy .

[0061] Using the class-first absolute central moment F of the sample as the theoretical first absolute central moment The approximate estimate is used to inversely deduce the sample estimate of entropy, which is the entropy estimate. As theoretical entropy Approximate value, entropy estimate The calculation formula is: Formula (1) Theoretical Entropy It reflects the degree of random dispersion of the initial small sample data itself, that is, the coverage of the optimal formulation region.

[0062] Step 2.4: Calculate the hyperentropy estimate of the cloud model based on the second-order central moments and the entropy estimate. .

[0063] In the cloud model, the total dispersion of the data (characterized by the second-order central moment S) is contributed by both the randomness of entropy and the fuzziness of hyperentropy. Given the entropy estimate... Based on this, the variance introduced by entropy is subtracted from the total variance S. The remaining part is the variance of the hyperentropy, from which the sample estimate of the hyperentropy can be derived as the hyperentropy estimate. As an approximation of the theoretical hyperentropy He, the hyperentropy estimate is... The calculation formula is: Formula (2) Hyperentropy He reflects the stability of entropy, that is, the uncertainty of entropy itself, which corresponds to the fuzziness of concepts. In this invention, it characterizes the cognitive uncertainty caused by insufficient sample size.

[0064] At this point, the construction of the clustered cloud model is complete, ultimately yielding a set of cloud model parameters (i.e., cloud model parameters) that can be directly used for cloud droplet generation: Expected +Entropy estimate + Hyperentropy estimate .

[0065] Cloud model features: The cloud model achieves the uncertainty transformation between qualitative concepts and quantitative values ​​through three numerical characteristics: expectation Ex (reflecting average level), entropy En (reflecting randomness), and hyperentropy He (reflecting fuzziness). Compared with traditional probability distributions (which only describe randomness) or fuzzy sets (which only describe fuzziness), the cloud model can simultaneously characterize two types of uncertainty, making it particularly suitable for material systems such as modified asphalt that are significantly affected by fluctuations in raw materials and changes in process conditions. Figure 2 This is a schematic diagram illustrating the principle of cloud droplet generation based on clustering cloud model parameter construction in this invention. The left side shows the distribution of the initial sample set, the middle shows the mathematical calculation process, and the right side shows the distribution of the generated cloud droplet formula data.

[0066] Detailed explanation of step 3 (Clouddrop Data Augmentation and Dataset Expansion): The core of this step is to move from specific points to a broader perspective, based on the cloud model parameters extracted from a small number of real samples in step 2. , , A large number of virtual samples are generated through a forward cloud generator. The generated virtual samples are not created out of thin air, but strictly follow the distribution patterns (expected location, degree of dispersion, degree of uncertainty) revealed by real samples.

[0067] Enhancement function selection and comparison: The positive cloud generator uses a random number generator based on a normal distribution, with the specific mechanism as follows: 1. Generate an instantaneous entropy value. It obeys the entropy estimate For expected and hyperentropy estimates Let be a normal distribution with standard deviation, denoted as . , The distribution follows a normal pattern. This step introduces fuzziness through hyperentropy, making the entropy value upon which each generated cloud droplet is based uncertain.

[0068] 2. Generate a cloud droplet value for this dimension. It conforms to the cloud model expectation in this dimension. For expectation, in terms of the absolute value of instantaneous entropy The standard deviation is a normal distribution, i.e. . That is, the quantitative value in this dimension.

[0069] 3. Calculate the degree of certainty of the cloud droplet in this dimension. This reflects that the value represents the central concept. The degree of subordination.

[0070] Repeat the above steps for all dimensions to obtain a set of dimension values. This constitutes a virtual recipe. d represents the total number of virtual formula dimensions (the total number of formula component parameters). correspond , correspond Repeat this process. Second (usually) ), and you can get the contents A set of virtual cloud droplets containing a virtual recipe.

[0071] Comparison with other enhancement functions: Uniform distribution enhancement: Uniform sampling within the parameter range results in a flat data distribution that cannot reflect the clustering trend of the original samples.

[0072] Bootstrap resampling: sampling with replacement on the original sample is essentially a repetition of the original data, which cannot add new information and the resulting distribution tends to be discretized.

[0073] Gaussian noise enhancement: Gaussian noise with a fixed variance is added near the original sample points, resulting in a multi-peaked data distribution that cannot form a continuous distribution that reflects the cluster centers.

[0074] The normal cloud generator used in this invention introduces hyperentropy (He) to generate cloud droplets with a bell-shaped distribution that is dense in the middle and sparse at both ends, with the peak appearing at the desired location. At a certain point, the distribution gradually decreases towards both sides, and the width of the distribution is adaptively adjusted by the entropy En. This distribution pattern best matches the physical laws of modified asphalt formulation data; the optimal formulation is often concentrated in a specific area, and the greater the deviation, the lower the probability of its occurrence. Figure 3 The differences between the original sample distribution, the Bootstrap expanded distribution, and the cloud droplet expanded distribution were compared. The cloud droplet expanded distribution accurately reflects the clustering trend of the original data and has significantly better fidelity than the Bootstrap method.

[0075] The number of cloud droplets generated is typically over 1000 to ensure sufficient coverage of the original distribution space. These virtual samples are then merged with the original samples to form the expanded training dataset D.exp .

[0076] Detailed explanation of step 4 (training the Bayesian optimized agent model): Detailed operation steps: 1. Model Selection: Gaussian Process Regression (GPR) was used as the surrogate model for Bayesian optimization. GPR can output not only the predicted mean... It can also output the prediction variance. This provides an uncertainty estimate for the subsequent acquisition function, where the prediction standard deviation is... To predict variance The arithmetic square root, that is .

[0077] 2. Kernel Function Settings: Choose either the quadratic exponential kernel or the Matern kernel (Matern 5 / 2 is commonly used). Taking the quadratic exponential kernel as an example, its expression is: ,in, and This represents any two distinct input formulations (i.e., formulation component parameter vectors). For signal variance, This is a length-scale hyperparameter.

[0078] 3. Hyperparameter Optimization: Determine the hyperparameters of the kernel function using maximum likelihood estimation or cross-validation (e.g., 5-fold CV). The training input is the augmented dataset. The output is the target performance value y.

[0079] 4. Model Evaluation: Introducing the coefficient of determination R0 2 The root mean square error (RMSE) is used to evaluate model accuracy. With the cloud-drop-enhanced surrogate model, the prediction error can be controlled within 5%, significantly better than the unenhanced model.

[0080] Advantages of Choosing the Expected Improvement (EI) Function: The expression for the Expected Improvement (EI) function used in this invention is as follows: in, The mean of the predictions output by the Bayesian optimized surrogate model. The predicted standard deviation is calculated using the predicted variance output by the Bayesian optimized surrogate model. This represents the current optimal performance value. To explore parameters, , and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

[0081] The advantage of EI functions is that they naturally balance development (utilization) High areas) and exploration (utilization) Large regions). In modified asphalt optimization scenarios, the initial sample size is small and the uncertainty is high. The EI function tends to explore regions with high uncertainty (due to...). (Dominant); as iterations proceed, the sample size increases, uncertainty decreases, and the EI function will gradually shift towards developing high-potential regions (by...). (Dominant). This adaptive mechanism perfectly complements the cloud model's quantification of uncertainty—the cloud model improves the accuracy of the surrogate model, enabling... The estimates are more accurate, which makes the trade-offs of the EI function more scientific.

[0082] Detailed explanation of step 5 (acquisition function optimization and candidate sampling point determination): Detailed operation steps: 1. Construct the acquisition function: Based on the GPR model trained in step 4, calculate the EI(x) value for any candidate formulation x in the design space.

[0083] 2. Set constraints: Formulation optimization must meet physical and process constraints, such as: ① Component boundary constraints: , ② Correlation constraint: The amount of asphalt used is negatively correlated with the powder-to-oil ratio (too high a powder-to-oil ratio will cause the mixture to be dry, while too low a ratio will easily cause segregation).

[0084] 3. Optimization Solution: Employ the L-BFGS-B quasi-Newton method (suitable for gradient optimization with boundary constraints) or a genetic algorithm (suitable for non-convex and discontinuous problems) to maximize the acquisition function EI(x), and obtain the next candidate sampling formula, denoted as... .

[0085] 4. Multi-start point optimization: To avoid getting trapped in local optima, a multi-start point strategy is adopted - multiple starting points are randomly initialized within the feasible region, and the point with the largest EI value is selected as the candidate point after optimization.

[0086] The principle and selection criteria of the data acquisition function: The data acquisition function is the core of Bayesian optimization decision-making. Besides EI, commonly used functions include the probability of improvement (PI) and the upper confidence limit (UCB). The PI function only considers the probability of improvement, ignoring the magnitude of improvement, and is prone to getting trapped in local optima; the UCB function requires manual adjustment of the exploration coefficients. k However, it lacks an adaptive mechanism. The EI function, on the other hand, inherently includes the expected magnitude of improvement and explores the parameters... ξ (Usually set to 0.01) The exploration intensity can be finely adjusted to better suit the application scenarios of this invention.

[0087] Detailed explanation of step 6 (physical experimental verification and sample update): Verification method: This step brings theoretical calculations back to reality by conducting physical experiments to verify the candidate sampled formulations and obtain the actual target performance indicators, denoted as... ;Will Add the initial sample set and update the initial sample set. Taking SBS modified asphalt as an example, the verification method is a standard rutting test (test temperature 60℃, loading rate 42 times / min), performed according to JTG E20-2011 "Test Procedures for Asphalt and Asphalt Mixtures in Highway Engineering". The test requires preparing rutting specimens of specified dimensions, keeping them in a constant temperature chamber, and then loading them back and forth using a rutting apparatus, recording the deformation curves, and calculating the dynamic stability DS.

[0088] Detailed Explanation: In practice, asphalt mixtures must be prepared strictly according to the candidate formulation, controlling process parameters such as mixing temperature and compaction frequency to ensure the repeatability of test results. Each formulation typically undergoes three parallel tests, and the average value is taken as the measured performance value. .Will Add to the initial sample set D to complete the sample set update.

[0089] Common termination parameters: Termination conditions typically fall into two categories: ① Performance threshold: The target performance reaches a preset requirement, such as DS ≥ 5000 times / mm; ② Resource constraints: The preset maximum number of trials is reached (e.g., total iterations ≤ 20) or the budget limit is reached. If a performance threshold is used as the termination condition, it is important to ensure that the threshold is engineering-reasonable—too high a threshold may lead to non-convergence, while too low a threshold renders the optimization meaningless. The innovation of this invention lies in using cloud droplet enhancement to make the surrogate model's prediction of the performance threshold more accurate, reducing over-iteration or under-iteration caused by model bias.

[0090] Detailed explanation of step 7 (iterative optimization and termination): This step constitutes a complete closed loop. Repeat steps 2 through 6 to form an automated optimization cycle of "learning-reinforcement-decision-validation." In each cycle, the cloud model is rebuilt based on updated real data, resulting in a more accurate understanding of uncertainty; the surrogate model is retrained based on more realistic and richer data (including virtual samples), achieving higher fitting accuracy; and the data collection function makes more informed decisions. This cycle continues until a satisfactory formula meeting engineering requirements is found, or the preset limit for the number of trials is reached. Figure 1 The loop in the diagram clearly illustrates this iterative process. A satisfactory formula that meets engineering requirements refers to whether the optimal target performance index in the updated initial sample set meets the preset termination condition (such as dynamic stability DS>5000 times / mm).

[0091] Example 2: Application Case This embodiment aims to optimize the high-temperature rutting resistance of SBS modified asphalt concrete. Based on the method described in Embodiment 1, it demonstrates the entire optimization process using specific data.

[0092] 1. Initial Sample Set Acquisition (corresponding to step 1) The optimization objective is to improve the dynamic stability DS (unit: cycles / mm) to above 5000 cycles / mm. The design variables are three key formulation components: SBS modifier dosage 3.0% ~ 6.0% (mass fraction); Asphalt dosage : 4.0% ~ 5.5% (mass fraction); Powder to oil ratio : 1.0 ~ 1.4 (unitless).

[0093] Due to cost constraints, only 10 sets of randomized formulation indoor rutting tests were initially conducted to obtain an initial small sample test dataset. (i=1, 2, ..., 10), where The measured value of the dynamic stability of the i-th formulation ranges from 2200 to 3000 cycles / mm.

[0094] Preprocessing: One set of outlier samples (dynamic stability below 2000 cycles / mm) was removed using the Grubbs criterion, leaving 9 valid samples. Z-score normalization was used to normalize the formulation component parameters, resulting in an initial sample set consisting of 9 original valid samples. (i=1, 2, ..., 9), as shown in Table 1.

[0095] Table 1 Initial Sample Set

[0096] 2. Construct a clustering cloud model and use the K-means clustering algorithm (K=1, actually calculating the average value) to calculate the cluster centers C for each formula parameter, thus obtaining... The cluster centers were 4.2%. It was 4.8%. It is 1.2. The cluster centers are (Rounded down to the nearest integer), these values ​​are used as the expected values ​​of the cloud models for each component. , , , Calculate the first-order absolute central moment F and the second-order central moment S of the nine valid samples. For example, , Calculate the entropy and hyperentropy of each component according to formulas (1) and (2):

[0097]

[0098] Similarly, the amount of asphalt used can be calculated. Powder to oil ratio cloud model parameters The results for each dimension are shown in the table below: Table 2 Calculation results of the initial sample set

[0099] Note: A hyperentropy of 0 indicates that the dispersion of the current sample is entirely explained by entropy (randomness) and no additional ambiguity has been shown, which is a reasonable phenomenon under the condition of small initial samples.

[0100] At this point, the clustered cloud model is complete, and a set of cloud model parameters that can be directly used for cloud droplet generation is obtained.

[0101] 3. Dataset Expansion (corresponding to step 3): Start the forward cloud generator and set the number of cloud droplets generated to 1000. Based on the cloud model parameters mentioned above, independently generate 1000 cloud droplet data for each dimension component and combine them into 1000 complete virtual recipes. These virtual recipes are merged with 9 sets of original valid samples, and after removing duplicate samples, the expanded training dataset is obtained. (A total of 1007 samples).

[0102] 4. Train a Bayesian optimized surrogate model (corresponding to step 4), to Using cloud droplet augmentation as input, a Gaussian process regression (GPR) model was trained with a squared exponential kernel function, and the kernel function parameters were determined through 5-fold cross-validation. After training, the surrogate model's prediction error for the dynamic stability of any candidate formulation was controlled within 5% (based on validation set evaluation), significantly outperforming the surrogate model without cloud droplet augmentation (prediction error approximately 15%).

[0103] 5. Acquisition function optimization and candidate sampling point determination (corresponding to step 5), using the expected improvement (EI) function as the acquisition criterion, and setting the exploration parameter ξ=0.01. Current optimal dynamic stability. To satisfy the formulation boundary constraints Under the physical constraint that "asphalt content is negatively correlated with the powder-to-asphalt ratio", the L-BFGS-B algorithm is used to maximize the EI function to obtain the first candidate sampling formula. .

[0104] 6. Physical testing verification and sample update (corresponding to step 6): Prepare SBS modified asphalt mixture specimens according to the candidate formulation, conduct standard rutting tests (test temperature 60℃, loading rate 42 cycles / min), and measure the actual dynamic stability. This is significantly higher than the optimal value for the initial sample (3000 times / mm). Add the original sample set D, and the initial sample set is now updated to 10 sets.

[0105] 7. Iteration optimization and termination (corresponding to step 7): The current optimal dynamic stability of 4200 times / mm has not reached the preset target of 5000 times / mm. Therefore, return to step 2, rebuild the cloud model, generate cloud droplets, train the surrogate model, optimize the acquisition function based on the updated 10 sets of samples, and enter the next round of iteration.

[0106] Repeating the above process, the measured dynamic stability of the candidate formulation obtained in the second iteration was 4550 cycles / mm, in the third iteration it was 4780 cycles / mm, and so on until the candidate formulation obtained in the ninth iteration. The measured dynamic stability reached 5120 cycles / mm, meeting the termination condition. Optimization is complete, and the optimal formula is output: SBS content 5.2%, asphalt content 4.3%, powder-to-oil ratio 1.35, dynamic stability 5120 cycles / mm.

[0107] In a comparison of implementation results, the method of this invention only requires 9 iterations (excluding the initial 10 sets of samples) to obtain the optimal formulation that meets the requirements. If traditional Bayesian optimization (without cloud droplet enhancement) is used, more than 22 iterations (excluding the initial 10 sets of samples) are required to achieve similar performance; if Bootstrap data augmentation combined with Bayesian optimization is used, more than 18 iterations are required. Figure 4 The convergence curve of Bayesian optimization based on the cloud droplet enhanced agent model is shown in comparison with that of traditional methods. The blue line (the method of this invention) shows faster convergence speed, higher final performance, and a reduction of about 60% in the number of trials, which verifies the superiority of the method.

[0108] Example 3: Small Sample Bayesian Optimized Sampling System Based on Cloud Drop Data Augmentation This embodiment provides a system for implementing the method described in Embodiment 1. The system includes the following five core modules, which are connected through a data bus to form a closed-loop controlled automated recipe design system.

[0109] The data acquisition module is used to acquire an initial sample set, which includes formulation component parameters and corresponding target performance indicators; The cloud model construction module is used to receive the initial sample set, construct a clustered cloud model based on the initial sample set, and obtain cloud model parameters, including expectation, entropy estimate and hyperentropy estimate. The dataset expansion module has a built-in positive cloud generator. Based on the cloud model parameters, it generates a preset number of cloud droplet recipe data through the positive cloud generator. The cloud droplet recipe data is then merged with the initial sample set to form an expanded training dataset. The Bayesian optimization engine module includes a surrogate model training unit and a data acquisition function optimization unit, which are used to train a Bayesian optimization surrogate model based on the expanded training dataset and output the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition and optimization module constructs an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and optimizes the solution to obtain the next candidate sampling formula. The experimental verification module is used to connect to the experimental equipment, perform physical experimental verification on the candidate sampling formula, obtain actual performance indicators, add the candidate sampling formula and the corresponding actual performance indicators to the initial sample set, and update the initial sample set. The loop iteration control module is used to coordinate the operation of each module, update the sample set, determine whether the preset termination condition is met, and output the optimal formula.

[0110] Figure 5 The diagram shows the modular structure of the system. This system enables full automation from initial data input to final optimal formula output, significantly improving the efficiency and intelligence of formula design.

[0111] Figure 6 An example is a schematic diagram of the physical structure of an electronic device, such as... Figure 6 As shown, the electronic device may include: a processor 810, a communication interface 820, a memory 830, and a communication bus 840, wherein the processor 810, the communication interface 820, and the memory 830 communicate with each other through the communication bus 840. The processor 810 can call logical instructions in the memory 830 to execute a small-sample Bayesian optimization sampling method based on cloud droplet data augmentation, the method including: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators; Construct a clustered cloud model. Based on the initial sample set, construct a clustered cloud model to obtain cloud model parameters, which include expectation, entropy estimate, and hyperentropy estimate. The dataset is expanded by generating a preset number of cloud droplet recipe data based on the cloud model parameters using a forward cloud generator, and then merging the cloud droplet recipe data with the initial sample set to form an expanded training dataset. The Bayesian optimized surrogate model is trained based on the expanded training dataset, and outputs the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition optimization involves constructing an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and then optimizing the solution to obtain the next candidate sampling formula. Experimental verification: Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated. Based on the updated initial sample set, repeat the above steps of building a clustering cloud model, expanding the dataset, training the Bayesian optimized surrogate model, collecting and optimizing data, and conducting experimental verification until the preset termination condition is met, and output the optimal formula in the initial sample set.

[0112] Furthermore, the logical instructions in the aforementioned memory 830 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0113] On the other hand, the present invention also provides a computer program product, the computer program product comprising a computer program that can be stored on a non-transitory computer-readable storage medium, wherein when the computer program is executed by a processor, the computer is able to execute the small-sample Bayesian optimization sampling method based on cloud droplet data augmentation provided by the above methods, the method comprising: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators; Construct a clustered cloud model. Based on the initial sample set, construct a clustered cloud model to obtain cloud model parameters, which include expectation, entropy estimate, and hyperentropy estimate. The dataset is expanded by generating a preset number of cloud droplet recipe data based on the cloud model parameters using a forward cloud generator, and then merging the cloud droplet recipe data with the initial sample set to form an expanded training dataset. The Bayesian optimized surrogate model is trained based on the expanded training dataset, and outputs the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition optimization involves constructing an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and then optimizing the solution to obtain the next candidate sampling formula. Experimental verification: Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated. Based on the updated initial sample set, repeat the above steps of building a clustering cloud model, expanding the dataset, training the Bayesian optimized surrogate model, collecting and optimizing data, and conducting experimental verification until the preset termination condition is met, and output the optimal formula in the initial sample set.

[0114] In another aspect, the present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the few-sample Bayesian optimization sampling method based on cloud droplet data augmentation provided by the methods described above, the method comprising: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators; Construct a clustered cloud model. Based on the initial sample set, construct a clustered cloud model to obtain cloud model parameters, which include expectation, entropy estimate, and hyperentropy estimate. The dataset is expanded by generating a preset number of cloud droplet recipe data based on the cloud model parameters using a forward cloud generator, and then merging the cloud droplet recipe data with the initial sample set to form an expanded training dataset. The Bayesian optimized surrogate model is trained based on the expanded training dataset, and outputs the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition optimization involves constructing an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and then optimizing the solution to obtain the next candidate sampling formula. Experimental verification: Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated. Based on the updated initial sample set, repeat the above steps of building a clustering cloud model, expanding the dataset, training the Bayesian optimized surrogate model, collecting and optimizing data, and conducting experimental verification until the preset termination condition is met, and output the optimal formula in the initial sample set.

[0115] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0116] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0117] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A small-sample Bayesian optimization sampling method based on cloud droplet data augmentation, characterized in that, Includes the following steps: Obtain an initial sample set, which includes formulation component parameters and corresponding target performance indicators; A clustering cloud model is constructed based on the initial sample set. The formulation component parameters and the target performance index are used together as a joint vector to construct the clustering cloud model, and the cloud model parameters are obtained. The cloud model parameters include the expected value, the entropy estimate, and the hyperentropy estimate. The dataset is expanded by generating a preset number of cloud droplet recipe data based on the cloud model parameters using a forward cloud generator, and then merging the cloud droplet recipe data with the initial sample set to form an expanded training dataset. The Bayesian optimized surrogate model is trained based on the expanded training dataset, and outputs the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition optimization involves constructing an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and then optimizing the solution to obtain the next candidate sampling formula. Experimental verification: Physical experiments are conducted on the candidate sampling formulas to verify their actual performance indicators. The candidate sampling formulas and their corresponding actual performance indicators are then added to the initial sample set, and the initial sample set is updated. Based on the updated initial sample set, repeat the above steps of building a cluster cloud model, expanding the dataset, training the Bayesian optimized surrogate model, collecting and optimizing data, and conducting experimental verification until the preset termination condition is met, and output the optimal formula in the initial sample set.

2. The small-sample Bayesian optimized sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, The specific steps for constructing the clustering cloud model include: Clustering algorithms are used to calculate the cluster centers of the initial sample set, which are then used as the expectation of the cloud model. For each dimension parameter of the joint vector, calculate the class-first absolute central moment and class-second central moment of the initial sample set respectively; Based on the characteristics of Gaussian distribution, the entropy estimate of the cloud model is obtained by using the first-order absolute central moment. The hyperentropy estimate of the cloud model is calculated based on the second-order central moments and the entropy estimate, and the expectation, entropy estimate, and hyperentropy estimate are used as cloud model parameters.

3. The small-sample Bayesian optimized sampling method based on cloud droplet data augmentation according to claim 2, characterized in that, In the process of generating the cloud droplet formulation data, for each cloud droplet to be generated, an instantaneous entropy following a normal distribution is first generated. The instantaneous entropy follows a normal distribution with the entropy estimate as the expectation and the hyperentropy estimate as the standard deviation. Then, based on the instantaneous entropy and the expectation of the cloud model, the formulation component parameters of the cloud droplet and their corresponding certainty are calculated through the probability density function of the normal distribution.

4. The small-sample Bayesian optimization sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, The Bayesian optimization surrogate model is a Gaussian process regression model, and its kernel function includes the squared exponential kernel function or the Matern kernel function.

5. The small-sample Bayesian optimization sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, The acquisition function is the desired improvement function, and its expression is: in, The mean of the predictions output by the Bayesian optimized surrogate model. The standard deviation of the prediction output by the Bayesian optimized surrogate model is given. This represents the current optimal performance value. To explore parameters, As an intermediate variable, and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

6. The small-sample Bayesian optimized sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, The process of optimizing and solving to obtain the next candidate sampling formula adopts the L-BFGS-B quasi-Newton method or genetic algorithm, and is carried out under the condition of satisfying the boundary constraints of the formula components and the physical process constraints.

7. The small-sample Bayesian optimization sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, In the initial sample set, the formulation component parameters include the SBS content, asphalt content, and powder-to-oil ratio of the modified asphalt concrete; the target performance index is the high-temperature rutting resistance performance index, including dynamic stability or rutting factor.

8. The small-sample Bayesian optimization sampling method based on cloud droplet data augmentation according to claim 1, characterized in that, The preset termination condition is when the target performance index reaches a preset threshold, or when the number of iterations reaches a preset maximum value.

9. A small-sample Bayesian optimized sampling system based on cloud droplet data augmentation, used to implement the method according to any one of claims 1-8, characterized in that, include: The data acquisition module is used to acquire an initial sample set, which includes formulation component parameters and corresponding target performance indicators; The cloud model construction module is used to receive the initial sample set, construct a clustered cloud model based on the initial sample set, and obtain cloud model parameters, including expectation, entropy estimate and hyperentropy estimate. The dataset expansion module has a built-in positive cloud generator. Based on the cloud model parameters, it generates a preset number of cloud droplet recipe data through the positive cloud generator. The cloud droplet recipe data is then merged with the initial sample set to form an expanded training dataset. The Bayesian optimization engine module includes a surrogate model training unit and a data acquisition function optimization unit, which are used to train a Bayesian optimization surrogate model based on the expanded training dataset and output the predicted mean and predicted variance of the target performance index for each candidate formulation. The acquisition and optimization module constructs an acquisition function based on the predicted mean and predicted variance output by the Bayesian optimized surrogate model, and optimizes the solution to obtain the next candidate sampling formula. The experimental verification module is used to connect to the experimental equipment, perform physical experimental verification on the candidate sampling formula, obtain actual performance indicators, add the candidate sampling formula and the corresponding actual performance indicators to the initial sample set, and update the initial sample set. The loop iteration control module is used to coordinate the operation of each module, update the sample set, determine whether the preset termination condition is met, and output the optimal formula.

10. The small-sample Bayesian optimized sampling system based on cloud droplet data augmentation according to claim 9, characterized in that, The modules are connected via a data bus to form a closed-loop control automated recipe design system, which supports the automated execution of iterative optimization processes.

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