Adaptive active optimization method and device for small sample optimization problem

Abstract

The application relates to an adaptive active optimization method and device for a small sample optimization problem, wherein the method comprises the following steps: determining the real target function value of each sample of an initial sample set of a target optimization problem to construct an initial data set of the target optimization problem; training, based on the initial data set, a proxy model for predicting the target function value of any candidate solution of the target optimization problem, and generating an elite solution set of the target optimization problem based on the initial data set; constructing, based on the proxy model and the elite solution set, an adaptive exploration function of the target optimization problem, so as to solve the target optimization problem by using a genetic algorithm based on the adaptive exploration function to obtain a current candidate solution of the target optimization problem until the current candidate solution meets a preset termination condition, and determining an optimal solution of the target optimization problem. Thus, the problems of proxy model prediction error accumulation, single search direction and easy falling into local optimization in the related art are solved.

Description

Technical Field

[0001] This application relates to the fields of intelligent optimization and industrial parameter calibration technology, and in particular to an adaptive active optimization method and apparatus for small sample optimization problems. Background Technology

[0002] In engineering fields such as engine performance calibration and aerodynamic optimization, a single objective function evaluation (such as a three-dimensional numerical simulation or a bench test) is extremely time-consuming and costly. To reduce the number of actual evaluations, surrogate-assisted evolutionary algorithms are widely used.

[0003] In related technologies, surrogate-assisted evolutionary algorithms train a surrogate model (such as a neural network) using limited real data. This surrogate model is then combined with an evolutionary algorithm (such as a genetic algorithm) and a coupling method in which the predicted value of the surrogate model is directly used as the fitness of the genetic algorithm is adopted to guide the search toward the potential optimal region. The search direction of the surrogate-assisted evolutionary algorithm is highly dependent on the prediction accuracy of the surrogate model and relies on random mutation to maintain diversity.

[0004] However, in related technologies, the limited number of real samples in small-sample, high-dimensional, complex design spaces leads to prediction errors in surrogate models. These errors accumulate with each search iteration, causing genetic algorithms to over-concentrate on the local optima predicted by the surrogate model. This results in insufficient search space coverage, a rapid decline in population diversity, and a significant reduction in global optimization capabilities. On the other hand, the lack of an active exploration mechanism for unknown design spaces means that regions that have not yet been sampled but may contain global optima lack effective guidance. Relying solely on random mutation to maintain population diversity leads to low exploration efficiency. When the surrogate model's predictions deviate, the algorithm struggles to escape erroneous search directions in a timely manner, increasing the number of evaluations of the real function and reducing computational efficiency in expensive optimization problems. These issues urgently need improvement. Summary of the Invention

[0005] This application provides an adaptive active optimization method and apparatus for small sample optimization problems, in order to solve the problems in related technologies, such as insufficient search space coverage, rapid decline in population diversity and significant reduction in global optimization ability due to the accumulation of prediction errors in surrogate models, as well as the low exploration efficiency due to the lack of an active exploration mechanism for the unknown design space, and the difficulty in adjusting the direction in time when the prediction is biased, which leads to an increase in the number of evaluations of the true function and a decrease in computational efficiency.

[0006] The first aspect of this application provides an adaptive active optimization method for small-sample optimization problems, comprising the following steps: determining the true objective function value of each sample in an initial sample set of the target optimization problem to construct an initial dataset for the target optimization problem; training a surrogate model based on the initial dataset to predict the objective function value of any candidate solution of the target optimization problem, and generating an elite solution set for the target optimization problem based on the initial dataset; constructing an adaptive exploration function for the target optimization problem based on the surrogate model and the elite solution set, and using a genetic algorithm based on the adaptive exploration function to solve the target optimization problem to obtain a current candidate solution for the target optimization problem, until the current candidate solution satisfies a preset termination condition, and determining the optimal solution for the target optimization problem.

[0007] Through the above technical means, the embodiments of this application can complete the training of the surrogate model and the generation of the elite solution set based on the initial dataset to construct an adaptive exploration function, and use a genetic algorithm based on the adaptive exploration function to solve the target optimization problem until the preset termination condition is met and the optimal solution is output. Thus, even when there are errors in the surrogate model prediction, it can take into account both local development and global exploration, dynamically adjust the exploration intensity, and actively guide the search for unknown high-value areas, improve the global optimization capability, thereby reducing the real evaluation cost and improving the optimization efficiency.

[0008] Optionally, in one embodiment of this application, determining the true objective function value of each sample in the initial sample set of the objective optimization problem includes: generating the initial sample set within the design space using a preset sampling strategy; and obtaining the true objective function value of each sample based on real physical experiments or high-precision numerical simulations.

[0009] Through the above-mentioned technical means, the embodiments of this application can systematically generate initial samples within the design space through a preset sampling strategy, and obtain the real objective function values ​​through real physical experiments or high-precision numerical simulations, thereby ensuring the accuracy of the initial sample set and providing a reliable data foundation for subsequent training of the surrogate model and generation of elite solution sets, thus ensuring the credibility of the surrogate model's predictions.

[0010] Optionally, in one embodiment of this application, the step of using a genetic algorithm based on the adaptive exploration function to solve the target optimization problem to obtain the current candidate solution of the target optimization problem includes: calculating the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score of each individual in the current population; substituting the surrogate model prediction value, the adaptive exploration weight, and the normalized local novelty score into the adaptive exploration function to calculate the fitness of each individual; performing a preset operation of the genetic algorithm on the current population based on the fitness to generate a new generation of population until a preset number of iterations is reached to obtain the final population of the target optimization problem; and selecting elite solutions, high-novelty solutions, and random solutions from the final population to determine the current candidate solution.

[0011] Through the above-mentioned technical means, the embodiments of this application can design the fitness of the genetic algorithm as an adaptive exploration function that integrates surrogate predictions, local novelty, and adaptive exploration weights. This changes the coupling method between the surrogate model and the evolutionary algorithm, so that the search process no longer completely depends on the prediction accuracy of the surrogate model. Even if the surrogate model has limited accuracy in a sparse high-dimensional space, it can actively guide the population to access unknown regions that the model is uncertain through the novelty-driven mechanism, thereby effectively avoiding getting trapped in local optima and significantly improving the global optimization ability. It can stably converge to the global optimum with fewer evaluation times in high-dimensional complex problems, showing significant efficiency and robustness advantages.

[0012] Optionally, in one embodiment of this application, the expression of the adaptive exploration function is: , in, The adaptive exploration function is... The predicted value of the proxy model. The adaptive exploration weights, The normalized local novelty score is given.

[0013] Through the above-mentioned technical means, the embodiments of this application can simultaneously calculate the predicted objective function value and explore the unknown region. The normalization process eliminates the influence of dimensions, enabling the exploration term and the predicted value term to be added on the same scale. The adaptive exploration weight avoids the problem of over-exploration or under-exploration caused by fixed exploration parameters, thereby improving the global optimization capability of the surrogate model under the prediction error condition.

[0014] Optionally, in one embodiment of this application, the step of calculating the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score for each individual in the current population includes: inputting the feature vector of each individual into the surrogate model, predicting the objective function value of each individual, and outputting the surrogate model prediction value; calculating the adaptive exploration weight based on the absolute value of the preset maximum and minimum values ​​of the true objective function values ​​in the initial dataset; calculating the minimum distance between the elite solution set and all individuals in the current population, and calculating the locality factor of each individual based on the minimum distance; calculating the unvisited reward function and the duplication penalty factor for each individual; and calculating the original local novelty score of each individual based on the locality factor, the unvisited reward function, and the duplication penalty factor, so as to perform maximum and minimum value normalization on the original local novelty scores of all individuals in the current population to obtain the normalized local novelty score.

[0015] Through the above-mentioned technical means, the embodiments of this application can use three factors—locality factor, unvisited reward, and duplication penalty—to jointly construct a local novelty index. By comprehensively considering the local distance between candidate solutions and current elite solutions, historical visitation, and duplication degree of candidate solutions, different exploration values ​​are assigned to unknown regions, thereby ensuring that exploration resources are concentrated in new regions with higher potential returns.

[0016] Optionally, in one embodiment of this application, obtaining the current candidate solution of the target optimization problem until the current candidate solution satisfies a preset termination condition, and determining the optimal solution of the target optimization problem, includes: performing a real evaluation on the current candidate solution to obtain the real objective function value of the current candidate solution, and adding the current candidate solution and the real objective function value of the current candidate solution to an initial dataset to generate an expanded dataset; determining whether the expanded dataset satisfies the preset termination condition; in response to the expanded dataset not satisfying the preset termination condition, training the surrogate model based on the expanded dataset and entering the next iteration; in response to the expanded dataset satisfying the preset termination condition, determining the optimal solution of the target optimization problem, and outputting the optimal solution and the objective function value of the optimal solution as the final optimization result.

[0017] Through the above-mentioned technical means, the embodiments of this application can perform expensive real evaluations on the selected current candidate solutions and expand the dataset in each iteration, determine whether the preset termination conditions are met, and if the conditions are not met, retrain the surrogate model based on the expanded dataset to gradually correct the prediction bias. If the conditions are met, output the optimal solution and its objective function value as the final optimization result, thereby improving computational efficiency and robustness.

[0018] A second aspect of this application provides an adaptive active optimization device for small-sample optimization problems, comprising: a construction module for determining the true objective function value of each sample in an initial sample set of the target optimization problem to construct an initial dataset for the target optimization problem; a generation module for training a surrogate model for predicting the objective function value of any candidate solution of the target optimization problem based on the initial dataset, and generating an elite solution set of the target optimization problem based on the initial dataset; and an optimization module for constructing an adaptive exploration function for the target optimization problem based on the surrogate model and the elite solution set, and using a genetic algorithm based on the adaptive exploration function to solve the target optimization problem to obtain a current candidate solution of the target optimization problem, until the current candidate solution satisfies a preset termination condition, thereby determining the optimal solution of the target optimization problem.

[0019] Through the above technical means, the embodiments of this application can complete the training of the surrogate model and the generation of the elite solution set based on the initial dataset to construct an adaptive exploration function, and use a genetic algorithm based on the adaptive exploration function to solve the target optimization problem until the preset termination condition is met and the optimal solution is output. Thus, even when there are errors in the surrogate model prediction, it can take into account both local development and global exploration, dynamically adjust the exploration intensity, and actively guide the search for unknown high-value areas, improve the global optimization capability, thereby reducing the real evaluation cost and improving the optimization efficiency.

[0020] Optionally, in one embodiment of this application, the construction module includes: a first generation unit, used to generate the initial sample set within the design space using a preset sampling strategy; and an acquisition unit, used to acquire the true objective function value of each sample based on real physical experiments or high-precision numerical simulations.

[0021] Through the above-mentioned technical means, the embodiments of this application can systematically generate initial samples within the design space through a preset sampling strategy, and obtain the real objective function values ​​through real physical experiments or high-precision numerical simulations, thereby ensuring the accuracy of the initial sample set and providing a reliable data foundation for subsequent training of the surrogate model and generation of elite solution sets, thus ensuring the credibility of the surrogate model's predictions.

[0022] Optionally, in one embodiment of this application, the optimization module includes: a first calculation unit, configured to calculate the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score of each individual in the current population; a second calculation unit, configured to substitute the surrogate model prediction value, the adaptive exploration weight, and the normalized local novelty score into the adaptive exploration function to calculate the fitness of each individual; a second generation unit, configured to perform a preset operation of the genetic algorithm on the current population based on the fitness to generate a new generation population until a preset number of iterations is reached to obtain the final population of the target optimization problem; and a determination unit, configured to select elite solutions, high-novelty solutions, and random solutions from the final population to determine the current candidate solution.

[0023] Through the above-mentioned technical means, the embodiments of this application can design the fitness of the genetic algorithm as an adaptive exploration function that integrates surrogate predictions, local novelty, and adaptive exploration weights. This changes the coupling method between the surrogate model and the evolutionary algorithm, so that the search process no longer completely depends on the prediction accuracy of the surrogate model. Even if the surrogate model has limited accuracy in a sparse high-dimensional space, it can actively guide the population to access unknown regions that the model is uncertain through the novelty-driven mechanism, thereby effectively avoiding getting trapped in local optima and significantly improving the global optimization ability. It can stably converge to the global optimum with fewer evaluation times in high-dimensional complex problems, showing significant efficiency and robustness advantages.

[0024] Optionally, in one embodiment of this application, the expression of the adaptive exploration function is: , in, The adaptive exploration function is... The predicted value of the proxy model. The adaptive exploration weights, The normalized local novelty score is given.

[0025] Through the above-mentioned technical means, the embodiments of this application can simultaneously calculate the predicted objective function value and explore the unknown region. The normalization process eliminates the influence of dimensions, enabling the exploration term and the predicted value term to be added on the same scale. The adaptive exploration weight avoids the problem of over-exploration or under-exploration caused by fixed exploration parameters, thereby improving the global optimization capability of the surrogate model under the prediction error condition.

[0026] Optionally, in one embodiment of this application, the first calculation unit includes: a prediction subunit, configured to input the feature vector of each individual into the surrogate model, predict the objective function value of each individual, and output the predicted value of the surrogate model; a first calculation subunit, configured to calculate the adaptive exploration weights based on the absolute value of the preset maximum and minimum values ​​of the true objective function values ​​in the initial dataset; a second calculation subunit, configured to calculate the minimum distance between the elite solution set and all individuals in the current population, and calculate the locality factor of each individual based on the minimum distance; a third calculation subunit, configured to calculate the unvisited reward function and the duplication penalty factor of each individual; and a fourth calculation subunit, configured to calculate the original local novelty score of each individual based on the locality factor, the unvisited reward function, and the duplication penalty factor, and perform maximum and minimum value normalization on the original local novelty scores of all individuals in the current population to obtain the normalized local novelty score.

[0027] Through the above-mentioned technical means, the embodiments of this application can use three factors—locality factor, unvisited reward, and duplication penalty—to jointly construct a local novelty index. By comprehensively considering the local distance between candidate solutions and current elite solutions, historical visitation, and duplication degree of candidate solutions, different exploration values ​​are assigned to unknown regions, thereby ensuring that exploration resources are concentrated in new regions with higher potential returns.

[0028] Optionally, in one embodiment of this application, the optimization module includes: a real evaluation unit, configured to perform a real evaluation on the current candidate solution to obtain the real objective function value of the current candidate solution, and add the current candidate solution and the real objective function value of the current candidate solution to an initial dataset to generate an expanded dataset; a judgment unit, configured to judge whether the expanded dataset satisfies a preset termination condition; a training unit, configured to train the surrogate model based on the expanded dataset and enter the next iteration in response to the expanded dataset not satisfying the preset termination condition; and a determination unit, configured to determine the optimal solution of the objective optimization problem in response to the expanded dataset satisfying the preset termination condition, and output the optimal solution and the objective function value of the optimal solution as the final optimization result.

[0029] Through the above-mentioned technical means, the embodiments of this application can perform expensive real evaluations on the selected current candidate solutions and expand the dataset in each iteration, determine whether the preset termination conditions are met, and if the conditions are not met, retrain the surrogate model based on the expanded dataset to gradually correct the prediction bias. If the conditions are met, output the optimal solution and its objective function value as the final optimization result, thereby improving computational efficiency and robustness.

[0030] A third aspect of this application provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the adaptive active optimization method for few-sample optimization problems as described in the above embodiments.

[0031] A fourth aspect of this application provides a non-volatile computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described adaptive active optimization method for a few-sample optimization problem.

[0032] A fifth aspect of this application provides a computer program product that stores a computer program that, when executed by a processor, implements the adaptive active optimization method for few-sample optimization problems as described above.

[0033] This application's embodiments can complete surrogate model training and elite solution set generation based on an initial dataset to construct an adaptive exploration function. A genetic algorithm based on this adaptive exploration function is then used to solve the target optimization problem until a preset termination condition is met, outputting the optimal solution. This allows for a balance between local development and global exploration, dynamically adjusting the exploration intensity, and proactively guiding the search for unknown high-value regions, even when the surrogate model's predictions contain errors. This enhances global optimization capabilities, thereby reducing real evaluation costs and improving optimization efficiency. Therefore, this addresses the problems in related technologies where the accumulation of prediction errors in the surrogate model leads to insufficient search space coverage, a rapid decline in population diversity, and a significant reduction in global optimization capabilities. Furthermore, the lack of an active exploration mechanism for the unknown design space results in low exploration efficiency and difficulty in timely adjustment when predictions deviate, leading to increased real function evaluation times and reduced computational efficiency.

[0034] Additional aspects and advantages of this application will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of this application. Attached Figure Description

[0035] The above and / or additional aspects and advantages of this application will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein: Figure 1 This is a flowchart of an adaptive active optimization method for small-sample optimization problems provided according to an embodiment of this application; Figure 2 This is a flowchart illustrating the calculation of an adaptive exploratory fitness function according to an embodiment of this application; Figure 3 This is a flowchart of an adaptive active optimization method for a few-sample optimization problem according to an embodiment of this application; Figure 4 This is a schematic diagram of an adaptive active optimization device for small sample optimization problems provided in an embodiment of this application; Figure 5 This is a schematic diagram of the structure of an electronic device provided according to an embodiment of this application.

[0036] Figure label: 10 - Adaptive active optimization device for small sample optimization problems; 100 - Construction module, 200 - Generation module, 300 - Optimization module; 501 - Memory, 502 - Processor, 503 - Communication interface. Detailed Implementation

[0037] The embodiments of this application are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application.

[0038] The adaptive active optimization method and apparatus for few-sample optimization problems according to embodiments of this application are described below with reference to the accompanying drawings. In the related technologies mentioned in the background section, due to the accumulation of prediction errors in the surrogate model, the search space coverage is insufficient, the population diversity declines rapidly, and the global optimization capability is significantly reduced. Simultaneously, the lack of an active exploration mechanism for the unknown design space leads to low exploration efficiency and difficulty in timely adjustment of direction when prediction deviations occur, resulting in increased evaluation times of the true function and reduced computational efficiency. This application provides an adaptive active optimization method for few-sample optimization problems. In this method, surrogate model training and elite solution set generation can be completed based on an initial dataset to construct an adaptive exploration function. A genetic algorithm based on the adaptive exploration function is then used to solve the target optimization problem until a preset termination condition is met to output the optimal solution. This allows for the simultaneous consideration of local development and global exploration, dynamic adjustment of exploration intensity, and proactive guidance in searching unknown high-value regions, even when surrogate model predictions have errors. This improves global optimization capability, thereby reducing the cost of real evaluation and increasing optimization efficiency. This solves the problems in related technologies, such as insufficient search space coverage, rapid decline in population diversity, and significant reduction in global optimization ability due to the accumulation of prediction errors in surrogate models. At the same time, the lack of an active exploration mechanism for the unknown design space leads to low exploration efficiency, difficulty in timely adjustment of direction when prediction deviations occur, and increased number of evaluations of the true function and reduced computational efficiency.

[0039] Specifically, Figure 1This is a flowchart illustrating an adaptive active optimization method for small-sample optimization problems provided in an embodiment of this application.

[0040] like Figure 1 As shown, this adaptive active optimization method for small sample optimization problems includes the following steps: In step S101, the true objective function value of each sample in the initial sample set of the objective optimization problem is determined to construct the initial dataset of the objective optimization problem.

[0041] It is understood that the objective optimization problem in the embodiments of this application can be an engineering optimization task to be solved, such as aerospace aerodynamic design, chemical process optimization, and other small-sample optimization problems with high computational or experimental costs. The initial sample set can be understood as a finite number of sampling points selected in the design space at the beginning of the optimization stage. The initial dataset consists of these sampling points and their corresponding true objective function values.

[0042] In actual implementation, the embodiments of this application can obtain the objective function value of each sample through real physical experiments or high-precision numerical simulations to construct an initial dataset D. Each sample in dataset D consists of an input feature vector x and its corresponding true objective function value y. The size N of the initial sample set is between 100 and 500. If the size of the initial sample set is too small, the surrogate model will underfit and the prediction accuracy will be insufficient; if the size is too large, it will contradict the original intention of reducing evaluation costs and lose the significance of surrogate-assisted optimization.

[0043] For example, embodiments of this application may take minimizing the Ackley function as an example, where Ackley is defined as: , Where d represents the dimension of the independent variable, and the range of the independent variable is defined as [-5, 5].

[0044] This application's embodiments use minimizing 10-dimensional and 100-dimensional Ackley functions as examples. The Ackley function is a standard test function widely used in the field of optimization, with a global minimum of 0 located at the origin. This function has a large number of local minima, which can effectively test the global search capability of optimization algorithms.

[0045] Specifically, in this embodiment, the Latin hypercube sampling method can be used to randomly generate an initial sample set containing 200 samples, calculate the true Ackley function value for each sample, obtain the corresponding target function value set, and construct the initial dataset. In order to eliminate the influence of randomness on the comparison of experimental results, the Latin hypercube sampling in this embodiment can use a fixed random seed 42.

[0046] The embodiments of this application can complete the real evaluation of the initial samples and the construction of the initial dataset. It can obtain real benchmark data in the optimization and startup phase, providing a high-confidence data foundation for the subsequent training of the proxy model, and ensuring the initial prediction accuracy of the proxy model from the source.

[0047] Optionally, in one embodiment of this application, determining the true objective function value of each sample in the initial sample set of the objective optimization problem includes: generating an initial sample set within the design space using a preset sampling strategy; and obtaining the true objective function value of each sample based on real physical experiments or high-precision numerical simulations.

[0048] It is understood that the preset sampling strategy in the embodiments of this application can be an experimental design method, such as the Latin hypercube sampling method. The preset sampling strategy can be set by those skilled in the art according to the actual situation, and no specific restrictions are made here. Real physical experiments refer to actual measurements performed on physical prototypes or test benches, such as engine bench tests; high-precision numerical simulations refer to simulation experiments performed using high-fidelity numerical simulation methods such as computational fluid dynamics and finite element analysis.

[0049] In practical implementation, this embodiment of the application can use experimental design methods to sample and generate an initial sample set containing N samples within the design space. The objective function value of each sample is obtained through real physical experiments or high-precision numerical simulations, constructing an initial dataset D. Each sample in dataset D consists of an input feature vector x and its corresponding true objective function value y. The size N of the initial sample set is between 100 and 500. If the initial sample set is too small, the surrogate model will underfit, resulting in insufficient prediction accuracy; if the size is too large, it contradicts the initial intention of reducing evaluation costs and negates the significance of surrogate-assisted optimization.

[0050] For example, this application embodiment can take minimizing the Ackley function as an example, setting the design space as a 10-dimensional and 100-dimensional hypercube, with the value range of each dimension variable being [-5, 5]. Within this design space, an initial sample set containing 200 samples is randomly generated using the Latin hypercube sampling method, denoted as X0 = {x1, x2, ..., x...} 200}. Calculate the true Ackley function value for each sample to obtain the corresponding set of objective function values ​​Y0 = {y1, y2, ..., y}. 200 Combine X0 and Y0 to construct the initial dataset D = {(x i , y i ) | i = 1, 2, ..., 200}. To eliminate the influence of randomness on the comparison of experimental results, a fixed random seed 42 is used for Latin hypercube sampling in this embodiment.

[0051] The embodiments of this application can systematically generate initial samples within the design space through a preset sampling strategy, and obtain the real objective function values ​​through real physical experiments or high-precision numerical simulations, ensuring the accuracy of the initial sample set. This provides a reliable data foundation for subsequent training of the surrogate model and generation of elite solution sets, and guarantees the credibility of the surrogate model's predictions.

[0052] In step S102, a surrogate model is trained based on the initial dataset to predict the objective function values ​​of any candidate solutions to the objective optimization problem, and an elite solution set for the objective optimization problem is generated based on the initial dataset.

[0053] It is understood that, in the embodiments of this application, the surrogate model refers to a prediction model trained on the initial dataset, used to learn the mapping relationship between design variables and the objective function, and output the predicted value of the objective function corresponding to any candidate solution. Any candidate solution refers to any sample in the optimization process. The elite solution set can be used for subsequent calculation of local novelty scores. The role of the elite solution set is to provide a reference benchmark for the exploration behavior, guiding the algorithm to focus on the region near the currently known optimal solution that has not yet been fully explored.

[0054] In practical implementation, this embodiment can train a surrogate model based on the current dataset D to predict the objective function value of any candidate solution. The surrogate model learns the mapping relationship between design variables and the objective function, and outputs the predicted objective function value corresponding to the candidate solution. The surrogate model can employ neural networks, support vector regression models, random forest models, Gaussian process models, gradient boosting tree models, or other machine learning models capable of function approximation.

[0055] The surrogate model can employ a convolutional neural network (CNN). CNNs possess good non-linear feature extraction capabilities, which can improve the prediction accuracy of the surrogate model in this embodiment. The input to the surrogate model is the feature vector of the candidate solution. x The output is the predicted value of the objective function. (x) The model training aims to minimize the mean squared error between the predicted and actual values, and the model parameters are saved after training is complete.

[0056] The convolutional neural network consists of several convolutional layers and fully connected layers. The activation function can be ReLU, the optimizer can be Adam, and the loss function is mean squared error.

[0057] Furthermore, in this embodiment of the application, the top M solutions from the current dataset D, sorted by objective function value, are selected to form an elite solution set E. For a minimization problem, the M solutions with the smallest objective function value are selected; for a maximization problem, the M solutions with the largest objective function value are selected. The size M of the elite solution set E is a preset parameter, which can range from 10 to 20.

[0058] For example, this embodiment of the application can take minimizing the Ackley function as an example, and train a convolutional neural network as a surrogate model based on the current dataset D. In this embodiment, the structure of the convolutional neural network is shown in the figure. The number of nodes in the input layer is 10 (corresponding to 10 decision variables), followed by two one-dimensional convolutional layers (with 32 and 64 kernels respectively, a kernel size of 3, and the activation function of ReLU), one global average pooling layer, two fully connected layers (with 128 and 64 neurons respectively, and the activation function of ReLU), and the number of nodes in the output layer is 1 (outputting the predicted value of the objective function). The model training uses the Adam optimizer with a learning rate of 0.001, a loss function of mean squared error, a batch size of 32, and 200 training epochs. After training, the model parameters are saved for use in subsequent steps to calculate the predicted values ​​of candidate solutions.

[0059] Furthermore, in this embodiment, the top 5 solutions from the current dataset D, sorted by objective function value from smallest to largest (in this embodiment, it is a minimization problem), are selected to form an elite solution set E. In this embodiment, the size M of the elite solution set is 5. This set is dynamically updated as the dataset expands, and always represents the optimal region currently known to the algorithm.

[0060] The embodiments of this application can complete the training of the surrogate model and the construction of the elite solution set. The surrogate model can make predictions at low cost, and the selected elite solution set can guide the algorithm to focus on the region near the currently known optimal solution that has not been fully explored, providing a basis for subsequent adaptive exploration and supporting the balanced control of exploration and development.

[0061] In step S103, an adaptive exploration function for the objective optimization problem is constructed based on the surrogate model and the elite solution set. The genetic algorithm based on the adaptive exploration function is then used to solve the objective optimization problem to obtain the current candidate solution of the objective optimization problem. The process continues until the current candidate solution meets the preset termination condition, thereby determining the optimal solution of the objective optimization problem.

[0062] It is understood that the adaptive exploration function in this embodiment can be understood as a fitness function that weights the predicted value of the surrogate model with the local novelty score reflecting the exploration value of the individual's region. The preset termination condition may include, but is not limited to: the number of actual evaluations reaching a preset upper limit, the historical best objective function value satisfying the convergence threshold, or the prediction accuracy of the surrogate model reaching a preset standard. The preset termination condition can be set by those skilled in the art according to the actual situation, and no specific restrictions are imposed here.

[0063] In practical implementation, this application embodiment can use a genetic algorithm as a search engine to evolve a population. The genetic algorithm includes three basic operations: selection, crossover, and mutation. In this application embodiment, the fitness of an individual is no longer simply equivalent to the predicted value of the surrogate model, but is determined by an adaptive exploration function. This function includes the objective function predicted by the surrogate model for each individual in the population, adaptive exploration weights that dynamically adjust the proportion of exploration items in the fitness function, and a normalized local novelty score. For each individual in the population, its fitness is calculated, and the fitness function is the proposed adaptive exploration function.

[0064] After the genetic algorithm completes all G generations of iterations, a set of current candidate solutions is selected from the final population for the next step of real evaluation. The selection strategy is a hybrid strategy that combines greedy development and active exploration.

[0065] The embodiments of this application can perform expensive real evaluation on the selected current candidate solution to obtain the true objective function value, and add the newly obtained current candidate solution and its corresponding true objective function value to the dataset D to complete the expansion of the dataset.

[0066] The embodiments of this application can determine whether a preset termination condition is met. If the termination condition is met, the historical best solution in the current dataset is output as the final optimization result; otherwise, the proxy model is retrained based on the expanded dataset and the next iteration is entered.

[0067] The embodiments of this application can construct an adaptive exploration function and couple a genetic algorithm to carry out closed-loop iterative optimization, which breaks through the limitation of related technologies that rely solely on surrogate prediction values ​​to guide the search. Through the adaptive exploration mechanism, the population is actively guided to explore potential high-value areas, which effectively improves the solution efficiency and global optimization capability of small-sample and expensive optimization problems. It can be widely applied to engineering optimization scenarios with high computational or experimental costs, such as engine performance calibration, aerospace aerodynamic design, and chemical process optimization.

[0068] Optionally, in one embodiment of this application, a genetic algorithm based on an adaptive exploration function is used to solve the target optimization problem to obtain the current candidate solution of the target optimization problem. This includes: calculating the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score of each individual in the current population; substituting the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score into the adaptive exploration function to calculate the fitness of each individual; performing preset operations of the genetic algorithm on the current population based on the fitness to generate a new generation of population until a preset number of iterations is reached to obtain the final population of the target optimization problem; and selecting elite solutions, high-novelty solutions, and random solutions from the final population to determine the current candidate solution.

[0069] It is understood that in this embodiment, the current population can be understood as the set of all individuals in the current iteration, with each individual corresponding to a sample. The surrogate model prediction value refers to the approximate value of the objective function output after inputting the individual's feature vector into the surrogate model. The adaptive exploration weight can be used to dynamically adjust the proportion of exploration items in the fitness function. The normalized local novelty score refers to a quantitative indicator measuring the novelty of an individual. The preset operations of the genetic algorithm may include: selection operator, crossover operator, and mutation operator. The selection operator of the genetic algorithm can be tournament selection, the crossover operator can be simulated binary crossover or single-point crossover, and the mutation operator can be polynomial mutation or bit-flip mutation. The preset operations can be set by those skilled in the art according to the actual situation, and no specific restrictions are imposed here.

[0070] It is understood that the preset iteration count in the embodiments of this application refers to the termination generation of the genetic algorithm. For example, in minimizing a 10-dimensional Ackley function, the preset iteration count can be 50 generations. In minimizing a 100-dimensional Ackley function, given the exponential expansion of the search space for a 100-dimensional problem, the iteration count G of the genetic algorithm can be adjusted from 50 to 200 to provide more sufficient evolutionary search time. The preset iteration count can be set by those skilled in the art according to the actual situation, and no specific restrictions are imposed here. Elite solutions can be understood as the candidate solutions that the current surrogate model considers to have the most potential, high novelty solutions can be understood as candidate solutions with the highest exploration value for unknown regions, and random solutions can be used to introduce random perturbations.

[0071] In practical implementation, this embodiment can use a genetic algorithm as a search engine to evolve a population. The population size is set to P, and the number of generations is set to G. The genetic algorithm includes three basic operations: selection, crossover, and mutation. This embodiment can calculate the predicted objective function value of candidate solutions x based on a surrogate model. This embodiment can determine adaptive exploration weights, realizing a dynamic transition in the search process from an early development-oriented phase to a later phase that also considers exploration. This embodiment can calculate the original local novelty index of each individual based on three factors: locality factor, unvisited reward, and duplication penalty. The original local novelty scores of all individuals in the current batch are normalized by a minimum-maximum value to obtain a normalized local novelty score.

[0072] In this embodiment, after the genetic algorithm completes all G generations of iterations, a set of current candidate solutions is selected from the final population for the next step of real evaluation. The selection strategy is a hybrid strategy, which combines greedy development and active exploration.

[0073] Specifically, in the embodiments of this application, a predicted target value can be selected. (x) The A optimal solutions are considered the most promising candidate solutions by the current surrogate model. This application embodiment can select a local novelty score. The top B solutions are selected as candidate solutions with the highest exploration value for the unknown region; In this embodiment, C solutions can be randomly selected from the final population to introduce random perturbation and further enhance the diversity of the search.

[0074] Among them, A takes the value of 15, B takes the value of 3, and C takes the value of 2, for a total of 20 current candidate solutions; For example, in this embodiment of the application, minimizing the Ackley function can be used as an example, setting the population size P of the genetic algorithm to 20 and the number of iterations G to 50.

[0075] This application embodiment determines the initial population by selecting the 20 best samples from 200 randomly generated samples. For each individual in the population, the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score of each individual in the current population are calculated to determine its fitness. The fitness function is an adaptive exploration function; since this application embodiment is a minimization problem, the smaller the fitness value, the better the individual.

[0076] The embodiments of this application can perform the four basic operations of a genetic algorithm based on fitness values: elite retention, selection, crossover, and mutation.

[0077] Specifically, elite retention: the E individuals with the best fitness in the current population are directly retained to the next generation, without participating in crossover or mutation. E = P × elite_ratio = 20 × 0.1 = 2, meaning the two best individuals are retained. The elite retention strategy ensures that the optimal solution already found is not lost due to the randomness of genetic operations.

[0078] Selection: A tournament selection method is used, with a tournament size of 3. That is, 3 individuals are randomly selected each time, and the one with the best fitness is selected into the mating pool.

[0079] Crossover: Simulated binary crossover is used, with a crossover probability of 0.9 and a distribution index of 20.

[0080] Mutation: Multinomial mutation was used with a mutation probability of 0.1 (i.e., each variable has a 10% probability of mutation) and a distribution index of 20.

[0081] In this embodiment, a new generation of population can be generated through selection, crossover, and mutation, maintaining a size of 20. The fitness assessment and genetic operations described above are repeated until all 50 generations have been completed.

[0082] In this embodiment, after the genetic algorithm completes 50 generations of iterations, 20 current candidate solutions are selected from the entire population for evaluation of the true function. The selection strategy is a hybrid strategy.

[0083] Specifically, the elite solution: based on the predicted value of the surrogate model. (x) Sort the solutions from smallest to largest and select the 15 best solutions.

[0084] High novelty solution: Score based on normalized local novelty Sort the solutions from highest to lowest and select the top 3.

[0085] Random solution: Two solutions are randomly selected from the final population.

[0086] Supplementary solution: If there are duplicate solutions among the above three selection results, the second-best solution will be automatically selected in sequence to ensure that 20 unique candidate solutions are obtained each time.

[0087] In this study, given the exponential expansion of the search space in the 100-dimensional problem, the number of iterations G in the genetic algorithm was adjusted from 50 to 200 to provide more time for evolutionary search. Simultaneously, the adaptive exploration weight coefficient λ was adjusted from 0.25 to 0.02 to maintain a suitable balance between exploration and utilization. Other parameters, such as the initial sample set size N=200, the elite solution set size M=5, the population size P=20, the mutation probability 0.1, the elite retention rate 0.1, the tournament selection size 3, and the current candidate solution selection rate, remained consistent with the 10-dimensional test.

[0088] The embodiments of this application can design the fitness of the genetic algorithm as an adaptive exploration function that integrates surrogate predictions, local novelty, and adaptive exploration weights. This changes the coupling method between the surrogate model and the evolutionary algorithm, so that the search process no longer completely depends on the prediction accuracy of the surrogate model. Even if the surrogate model has limited accuracy in a sparse high-dimensional space, it can actively guide the population to visit unknown regions where the model is uncertain through a novelty-driven mechanism, thereby effectively avoiding getting trapped in local optima and significantly improving the global optimization capability. It can stably converge to the global optimum with fewer evaluations in high-dimensional complex problems, showing significant efficiency and robustness advantages.

[0089] Optionally, in one embodiment of this application, the expression for the adaptive exploration function is: , in, For adaptive exploration function, For the surrogate model's predicted values, To adaptively explore weights, This is used to normalize the local novelty score.

[0090] In actual implementation, the embodiments of this application can simultaneously pursue the objective function value and explore the unknown region, constructing an adaptive exploration function, the expression of which is: , in, For adaptive exploration function, For the surrogate model's predicted values, To adaptively explore weights, To normalize the local novelty score. In the embodiments of this application, the fitness of an individual is no longer simply equivalent to the predicted value of the surrogate model, but is determined by an adaptive exploration function F(x).

[0091] like Figure 2 As shown, in this embodiment of the application, an adaptive exploration function is used as the adaptive fitness function. Including utilization items and exploration items The exploration item is determined by adaptive exploration weights. and normalized local novelty score Composition, normalized local novelty score Including local novelty score No visit reward 0 or 1 Repeated penalty factor .

[0092] The embodiments of this application can simultaneously calculate the predicted objective function value and explore unknown regions. The normalization process eliminates the influence of dimensions, enabling the exploration term and the predicted value term to be added on the same scale. The adaptive exploration weight avoids the problems of over-exploration or under-exploration caused by fixed exploration parameters, thereby improving the global optimization capability of the surrogate model under the condition of prediction error.

[0093] Optionally, in one embodiment of this application, calculating the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score for each individual in the current population includes: inputting the feature vector of each individual into the surrogate model, predicting the objective function value of each individual, and outputting the surrogate model prediction value; calculating the adaptive exploration weight based on the absolute value of the preset maximum and minimum values ​​of the true objective function values ​​in the initial dataset; calculating the minimum distance between the elite solution set and all individuals in the current population, and calculating the locality factor of each individual based on the minimum distance; calculating the unvisited reward function and the duplication penalty factor for each individual; and calculating the original local novelty score of each individual based on the locality factor, the unvisited reward function, and the duplication penalty factor, and performing maximum and minimum value normalization on the original local novelty scores of all individuals in the current population to obtain the normalized local novelty score.

[0094] It is understood that, in the embodiments of this application, the locality factor can be used to measure the positional relationship of a candidate solution relative to the current elite solution region; the unvisited reward can be used to increase the priority of historically unsampled regions; and the duplication penalty can be used to reduce the fitness of duplicate candidate solutions. The preset maximum value can be the absolute value of the maximum value of all true objective function values ​​in the current dataset D. If the objective function is to be minimized, then the maximum value of the reciprocal of the objective function is taken. The preset maximum value can be set by those skilled in the art according to the actual situation, and no specific restrictions are imposed here.

[0095] In actual implementation, the embodiments of this application can input the feature vector of each individual into the surrogate model to predict the objective function value of each individual, and output the predicted value of the surrogate model. (x) Proxy model predicted value (x) This represents the objective function prediction of the surrogate model for the candidate solution x.

[0096] w(t) is the adaptive exploration weight, which dynamically adjusts the proportion of the exploration term in the fitness function. This weight is defined as: , in, This is a preset weighting coefficient, ranging from 0.01 to 1, used to control the exploration intensity; It is the absolute value of the maximum value of all true objective function values ​​in the current dataset D. If the objective function is to be minimized, then the maximum value of the reciprocal of the objective function is taken.

[0097] Since the surrogate model is trained on a limited number of real samples, in the early stages of optimization, the current historical optimal objective function value is relatively low. The surrogate model can better distinguish the superiority or inferiority relationship between candidate solutions, so the exploration weight remains low, making the search process mainly based on the surrogate model's prediction value, thus accelerating convergence towards the advantageous region. As the optimization process continues, the historical optimal objective function value gradually approaches the optimal value, and the influence of the surrogate model's prediction error on the search direction gradually increases. At this point, the adaptive exploration weight is increased synchronously, gradually increasing the contribution of local novelty to the fitness evaluation. This guides the algorithm to actively explore potential superior regions that have not yet been fully sampled, reducing the risk of getting trapped in local optima due to the surrogate model's prediction error. This achieves a dynamic transition in the search process from an early development-oriented approach to a later approach that also considers exploration. Furthermore, embodiments of this application can calculate candidate solutions. x Original local novelty score S ( x ), defined as: , , in, For candidate solutions x The minimum distance between the locality and all elite solutions in the elite solution set E. The distance metric can be Hamming distance or Euclidean distance, the choice depending on the encoding method. x The locality factor is used to focus the exploration behavior on the vicinity of the elite solution by weighting by the inverse distance, thus avoiding the blind diffusion of search resources to regions far from the known optimal region.

[0098] When the decision variable is a continuous variable, the distance metric Euclidean distance is used, and variables are normalized first; when the decision variable is a discrete variable or binary encoded, the distance metric is... Hamming distance is used.

[0099] Unvisited reward function ( ) and repeat penalty factor ( ) is defined as: , , in, For unvisited reward functions, if candidate solutions x If a value has not appeared in historical data, it is set to 1; if it has appeared, it is set to 0. This reward mechanism directly encourages the algorithm to access design space regions that have not yet been sampled. Let be the repetition penalty factor, where k For candidate solutions x The number of times a candidate solution appears in the current candidate pool. This factor inhibits the repeated generation of candidate solutions and helps maintain the diversity of the population.

[0100] This application's embodiments can construct a local novelty index using three factors: locality factor, unvisited reward, and duplication penalty, instead of relying on a single distance or random perturbation as the exploration basis. This is because these three factors respectively reflect the potential optimization value of candidate solutions, historical exploration status, and population diversity. Specifically, the locality factor measures the positional relationship of candidate solutions relative to the current elite solution region, prioritizing exploration near known superior regions and avoiding the excessive consumption of search resources in ineffective regions far from potential optimal regions. The unvisited reward increases the priority of historically unsampled regions, guiding the algorithm to actively acquire new real sample information and mitigating prediction bias caused by sample scarcity in the surrogate model. The duplication penalty reduces the fitness of duplicate candidate solutions, preventing the genetic algorithm from experiencing a decrease in search space coverage due to population duplication.

[0101] The embodiments of this application can use a product form instead of a simple weighted sum because there is a synergistic constraint relationship among the three factors. Only when the three conditions of "close to the current good area", "not yet sampled in history" and "having a high population uniqueness" are met simultaneously, will the candidate solution have a high exploration value; a poor result of any factor will reduce the overall exploration score, thereby ensuring that exploration resources are concentrated in new areas with higher potential returns.

[0102] Furthermore, embodiments of this application can score the original local novelty of all candidate solutions in the current batch. S(x) Perform maximum-minimum normalization to obtain the normalized local novelty score. : , Where max(S) and min(S) are the maximum and minimum original novelty scores among all candidate solutions in the current batch, respectively. Normalization eliminates the influence of dimensions, allowing the exploration and prediction terms to be summed on the same scale.

[0103] For example, embodiments of this application can take minimizing the Ackley function as an example, and embodiments of this application can calculate the predicted value of the surrogate model. (x) The feature vector of each individual in the population is input into a trained convolutional neural network surrogate model to obtain the corresponding predicted value. (x) Note that in minimization problems, a smaller predicted value indicates a better individual.

[0104] The embodiments of this application can calculate adaptive exploration weights: w(t): , Where λ takes the value of 0.25; It represents the maximum value of all true objective function values ​​in the current dataset D.

[0105] In particular, given that the search space of a 100-dimensional problem expands exponentially, the adaptive exploration weight coefficient λ can be adjusted from 0.25 to 0.02 to maintain a moderate balance between exploration and utilization.

[0106] The embodiments of this application can calculate a normalized local novelty score. Specifically, the embodiments of this application can calculate the distance between the elite solution set E and all individuals in the current population. In the embodiments of this application, the decision variables are continuous variables, the distance metric is Euclidean distance, and the variables are normalized to eliminate the influence of dimensions. For each individual x in the population, the minimum distance between it and all elite solutions in the elite solution set E is calculated. .

[0107] Furthermore, embodiments of this application can calculate the locality factor: , Next, it is determined whether each individual has appeared in historical data to identify unvisited rewards. If an individual has not appeared before, the value is 1; if it has appeared before, the value is 0. This counts the number of times each individual appears in the current population. k Calculate the repetition penalty factor .

[0108] In this embodiment, the above terms can be substituted to calculate the original local novelty score: , This application embodiment can perform S( ) on all 20 individuals in the current population. x Perform maximum and minimum value normalization to obtain .

[0109] The embodiments of this application can... (x) , w(t) and Substituting into formula (1), we obtain the final fitness F( ) for each individual. x Since the embodiments of this application address a minimization problem, the smaller the fitness value, the better the individual.

[0110] This application embodiment can use three factors—locality factor, unvisited reward, and duplication penalty—to jointly construct a local novelty index. By comprehensively considering the local distance between candidate solutions and current elite solutions, historical visitation, and the degree of duplication of candidate solutions, different exploration values ​​are assigned to unknown regions, thereby ensuring that exploration resources are concentrated in new regions with higher potential returns.

[0111] Optionally, in one embodiment of this application, obtaining a current candidate solution to the objective optimization problem, and determining the optimal solution to the objective optimization problem until the current candidate solution meets a preset termination condition, includes: performing a real evaluation on the current candidate solution to obtain the real objective function value of the current candidate solution, and adding the current candidate solution and the real objective function value of the current candidate solution to the initial dataset to generate an expanded dataset; determining whether the expanded dataset meets the preset termination condition; in response to the expanded dataset not meeting the preset termination condition, training a surrogate model based on the expanded dataset and entering the next iteration; in response to the expanded dataset meeting the preset termination condition, determining the optimal solution to the objective optimization problem, and outputting the optimal solution and the objective function value of the optimal solution as the final optimization result.

[0112] It is understood that, in the embodiments of this application, true evaluation refers to obtaining the true objective function value of the current candidate solution through real physical experiments or high-precision numerical simulation.

[0113] In actual implementation, the embodiments of this application can perform expensive real evaluations on the selected current candidate solutions to obtain the true objective function values. Real evaluations include, but are not limited to, high-precision numerical simulation calculations, benchtop physical experiments, or field measurements. The newly acquired current candidate solutions and their corresponding true objective function values ​​are added to dataset D, thus expanding the dataset.

[0114] Determine if the preset termination conditions are met. Termination conditions include, but are not limited to: the number of actual evaluations reaching a preset upper limit, the historical best objective function value meeting a convergence threshold, or the prediction accuracy of the surrogate model reaching a preset standard. If the termination conditions are met, output the historical best solution as the final optimization result; otherwise, retrain the surrogate model based on the expanded dataset and proceed to the next iteration.

[0115] For example, in this embodiment of the application, minimizing the Ackley function can be used as an example. For each of the 20 selected current candidate solutions, the true 10-dimensional Ackley function value is calculated. The 20 newly acquired samples and their corresponding true objective function values ​​are added to the dataset D, expanding the dataset size from the initial 200 to 220.

[0116] Determine whether the termination condition is met. The termination condition in this embodiment is: the historical best objective function value in the current dataset D is less than or equal to 1E-6 (i.e., it has converged to near the global optimum value of 0), or the total number of actual evaluations reaches a preset upper limit.

[0117] If the termination condition is met, the historical best solution and its objective function value are output as the final optimization result; otherwise, the convolutional neural network surrogate model is retrained based on the expanded dataset D, and the next iteration continues. Twenty new real evaluation samples are added to each iteration.

[0118] In each iteration, the selected candidate solution can be subjected to expensive real evaluation and the dataset can be expanded to determine whether the preset termination condition is met. If the condition is not met, the surrogate model is retrained based on the expanded dataset to gradually correct the prediction bias. If the condition is met, the optimal solution and its objective function value are output as the final optimization result, thereby improving computational efficiency and robustness.

[0119] To verify the technical effectiveness, this application's embodiments selected two genetic algorithms for comparison under the same conditions: First genetic algorithm: without using a surrogate model, directly use the true function value as fitness, the population size is 20, and 20 new individuals are evaluated in each generation.

[0120] The second genetic algorithm uses the same convolutional neural network model as the embodiments of this application, but the fitness is directly equal to the predicted value. (x), which does not include adaptive exploration terms.

[0121] All three methods were run independently and repeatedly 5 times, using the same initial sample set of 200 to eliminate interference introduced by differences in the initial data. The table presents the statistical results of the number of actual evaluations required for the three methods to converge to the global optimum (objective function value ≤ 1E-6) in 5 runs.

[0122] Table 1 compares the number of evaluations required for different algorithms on the Ackley-10d function. It can be seen that, compared to the first genetic algorithm, the average number of evaluations in this embodiment is reduced from 17828 to 365.4, a reduction of 97.9%, and the standard deviation is reduced from 2363.58 to 23.678, a reduction of two orders of magnitude. Compared to the second genetic algorithm, the average number of evaluations in this embodiment is further reduced from 449.2 to 365.4, a reduction of 18.7%, and the standard deviation is reduced from 83.73 to 23.678, a reduction of 71.7%. The test results for the 10-dimensional Ackley function show that the embodiments of this application achieve both efficient convergence and operational stability in medium-dimensional problems, demonstrating a clear advantage over existing surrogate-assisted evolutionary algorithms.

[0123] Table 1

[0124] The search space of a 100-dimensional Ackley function is 10 times larger than that of a 10-dimensional function, resulting in an exponential increase in space volume and a significant increase in computational cost per function evaluation. If the budget of nearly 20,000 evaluations required by traditional genetic algorithms in the 10-dimensional test is used, the overall time consumption would be excessive. Therefore, this part of the test sets the upper limit of the total number of real-world function evaluations to 2000 to simulate typical scenarios of limited computational resources in actual engineering optimization, and to examine the differences in optimization capabilities among various algorithms within this budget.

[0125] Table 2 compares the optimization results of different algorithms on the Ackley-100d function (evaluation budget: 2000 runs). As shown in Table 2, within the 2000 evaluation budget, neither the first nor the second genetic algorithm converged to the global optimum, with the final optimal values ​​remaining around 5.97 and 2.94, respectively. The second genetic algorithm significantly outperformed the first, but still fell far short of the global optimum of 0. This is because the prediction accuracy of the surrogate model decreases significantly in 100-dimensional space, making it difficult for the second genetic algorithm, which relies entirely on predicted values ​​to guide the search, to obtain a reliable search direction. In contrast, the embodiment of this application successfully converged to the global optimum in all five independent runs, requiring an average of only about 1721 real evaluations to reach the convergence criterion, far below the 2000-run budget limit. This result demonstrates that the embodiment of this application, by incorporating an adaptive exploration mechanism into the fitness function, enables the search process to actively explore unknown regions even when the surrogate model's accuracy is limited. This allows for stable convergence to the global optimum with fewer evaluations in high-dimensional complex problems, exhibiting significant efficiency and robustness advantages.

[0126] Table 2

[0127] Specifically, taking the performance optimization of an ammonia / diesel dual-fuel engine as an example, the effectiveness of the embodiments of this application in constrained engineering optimization problems is verified.

[0128] The objective function and constraints in this application embodiment are based on a polynomial response surface model regressed from engine experimental data. The optimization objective is to maximize BTE (Brake Thermal Efficiency), and the constraint is that nitrogen oxide (NOx) emissions do not exceed 750 ppm. Both BTE and NOx are predicted values ​​calculated by the response surface model based on the input parameters, and their specific functional form is a polynomial regression equation: , , Where A, B and C represent injection pressure, injection timing and ammonia premixing ratio, respectively, and their optimization range is shown in Table 3. Table 3 is a table of decision variables and their value ranges.

[0129] Table 3

[0130] The difference in this embodiment is the introduction of a penalty function to handle constraints in the fitness calculation. When the surrogate model predicts NOx emissions exceeding 750 ppm for a candidate solution, a maximum penalty value is applied to its fitness F(x), causing it to be naturally eliminated in genetic selection. The parameters are set as follows: initial sample set size N=50, population size P=20, number of iterations G=30, λ=0.02. Table 4 shows a comparison of the number of evaluations for different algorithms on the engine test function.

[0131] Table 4

[0132] As shown in Table 4, compared with the first genetic algorithm, the average number of evaluations in this embodiment decreased from 1584 to 207.8, a reduction of 86.9%, and the standard deviation decreased from 965.36 to 66.78, a reduction of more than an order of magnitude. Compared with the second genetic algorithm, the average number of evaluations decreased from 314 to 207.8, a reduction of 33.8%, and the standard deviation decreased from 99.55 to 66.78, a reduction of 32.9%. These results demonstrate that, in the constrained optimization scenario of an engine, this embodiment can stably converge to the global optimum with fewer evaluations and less operational fluctuation, while strictly meeting emission constraints, fully verifying its efficiency and robustness in practical engineering optimization problems.

[0133] Specifically, it can be combined with Figure 3 As shown, a specific embodiment is used to illustrate in detail the working principle of the adaptive active optimization method for small sample optimization problems in this application.

[0134] like Figure 3 As shown, embodiments of this application may include the following steps: Step S301: Randomly initialize 200 samples.

[0135] In this embodiment, an initial sample set containing 200 samples can be generated by using experimental design methods within the design space. The objective function value of each sample can be obtained through real physical experiments or high-precision numerical simulations to construct the initial dataset D.

[0136] Step S302: Deep neural network model training.

[0137] In this embodiment of the application, a surrogate model can be trained based on the current dataset D to predict the objective function value of any candidate solution.

[0138] Step S303: Genetic algorithm based on adaptive exploration function, population size 20, maximum number of iterations 50.

[0139] In this embodiment of the application, the genetic algorithm population size P is set to 20, the number of iterations G is 50, and the initial population is the 20 best samples selected from 200 randomly generated samples.

[0140] Step S304: Fitness function calculation and sorting.

[0141] In this embodiment, the fitness of each individual in the population can be calculated and sorted.

[0142] Step S305: Select the 15 individuals with the best predicted values ​​from the objective function.

[0143] Step S306: The three individuals with the highest local novelty.

[0144] Step S307: Randomly sample 2 individuals.

[0145] In this embodiment, individuals are sorted by their predicted objective function values ​​from smallest to largest, and the 15 best individuals are selected. Then, individuals are sorted by their normalized local novelty scores from highest to lowest, and the top 3 individuals are selected. Finally, 2 individuals are randomly selected from the final population. If duplicate solutions exist among the above three selection results, the next best solution is automatically selected, ensuring that 20 unique candidate solutions are obtained each time.

[0146] Step S308: Realistic assessment and data update.

[0147] In this embodiment, the true objective function value of each of the 20 selected candidate solutions can be calculated, and the 20 newly acquired samples and their corresponding true objective function values ​​can be added to the dataset D, expanding the dataset size from the initial 200 to 220.

[0148] Step S309: Determine whether convergence has occurred. If yes, proceed to step S310; otherwise, proceed to step S302.

[0149] In this embodiment, it can be determined whether the termination condition is met. The termination condition is: the historical best objective function value in the current dataset D is less than or equal to 1E-6 (i.e., it has converged to near the global optimum value of 0), or the total number of actual evaluations reaches a preset upper limit.

[0150] Step S310: Optimal solution.

[0151] In this embodiment, the historical best solution and its objective function value can be output as the final optimization result.

[0152] The adaptive active optimization method for small-sample optimization problems proposed in this application can complete surrogate model training and elite solution set generation based on an initial dataset to construct an adaptive exploration function. A genetic algorithm based on the adaptive exploration function is then used to solve the target optimization problem until a preset termination condition is met, outputting the optimal solution. This allows for a balance between local development and global exploration, dynamically adjusting the exploration intensity, and actively guiding the search for unknown high-value regions, even when the surrogate model has prediction errors. This enhances global optimization capabilities, reduces real evaluation costs, and improves optimization efficiency. Therefore, this method solves the problems in related technologies where the accumulation of prediction errors in the surrogate model leads to insufficient search space coverage, rapid decline in population diversity, and a significant reduction in global optimization capabilities. Furthermore, the lack of an active exploration mechanism for the unknown design space results in low exploration efficiency and difficulty in timely adjustment when predictions deviate, leading to increased evaluation times of the real function and reduced computational efficiency.

[0153] Next, with reference to the accompanying drawings, an adaptive active optimization apparatus for small-sample optimization problems proposed according to embodiments of this application is described.

[0154] Figure 4 This is a schematic diagram of the structure of an adaptive active optimization device for small sample optimization problems according to an embodiment of this application.

[0155] like Figure 4 As shown, the adaptive active optimization device 10 for small sample optimization problems includes: a construction module 100, a generation module 200, and an optimization module 300.

[0156] The construction module 100 is used to determine the true objective function value of each sample in the initial sample set of the objective optimization problem, so as to construct the initial dataset of the objective optimization problem.

[0157] The generation module 200 is used to train a surrogate model for predicting the objective function values ​​of arbitrary candidate solutions to the objective optimization problem based on the initial dataset, and to generate an elite set of solutions to the objective optimization problem based on the initial dataset.

[0158] The optimization module 300 is used to construct an adaptive exploration function for the target optimization problem based on the surrogate model and the elite solution set, so as to solve the target optimization problem using a genetic algorithm based on the adaptive exploration function, to obtain the current candidate solution of the target optimization problem, until the current candidate solution meets the preset termination condition, and to determine the optimal solution of the target optimization problem.

[0159] Optionally, in one embodiment of this application, the construction module 100 includes a first generation unit and an acquisition unit.

[0160] The first generation unit is used to generate an initial sample set within the design space using a preset sampling strategy.

[0161] The acquisition unit is used to acquire the true objective function value of each sample based on real physical experiments or high-precision numerical simulations.

[0162] Optionally, in one embodiment of this application, the optimization module 300 includes: a first calculation unit, a second calculation unit, a second generation unit, and a determination unit.

[0163] The first computing unit is used to calculate the surrogate model prediction value, adaptive exploration weight, and normalized local novelty score for each individual in the current population.

[0164] The second computational unit is used to substitute the surrogate model predictions, adaptive exploration weights, and normalized local novelty scores into the adaptive exploration function to calculate the fitness of each individual.

[0165] The second generation unit is used to perform preset operations of the genetic algorithm on the current population based on fitness to generate a new generation of population until a preset number of iterations is reached to obtain the final population of the target optimization problem.

[0166] The determination unit is used to select elite solutions, highly novel solutions, and random solutions from the final population to determine candidate solutions.

[0167] Optionally, in one embodiment of this application, the expression for the adaptive exploration function is: , in, For adaptive exploration function, For the surrogate model's predicted values, To adaptively explore weights, This is used to normalize the local novelty score.

[0168] Optionally, in one embodiment of this application, the first calculation unit includes: a prediction subunit, a first calculation subunit, a second calculation subunit, a third calculation subunit, and a fourth calculation subunit.

[0169] The prediction subunit is used to input the feature vector of each individual into the surrogate model, predict the objective function value of each individual, and output the predicted value of the surrogate model.

[0170] The first calculation subunit is used to calculate the adaptive exploration weights based on the absolute value of the preset maximum and minimum values ​​of the adaptive exploration weights and the true objective function values ​​in the initial dataset.

[0171] The second computational subunit is used to calculate the minimum distance between the elite solution set and all individuals in the current population, and to calculate the locality factor of each individual based on the minimum distance.

[0172] The third computational subunit is used to calculate the unvisited reward function and the repetition penalty factor for each individual.

[0173] The fourth computational subunit is used to calculate the original local novelty score of each individual based on the locality factor, the unvisited reward function, and the duplication penalty factor, so as to perform maximum and minimum value normalization on the original local novelty scores of all individuals in the current population to obtain the normalized local novelty score.

[0174] Optionally, in one embodiment of this application, the optimization module 300 includes: a real evaluation unit, a judgment unit, a training unit, and a determination unit.

[0175] The real evaluation unit is used to perform a real evaluation on the current candidate solution, obtain the real objective function value of the current candidate solution, and add the current candidate solution and the real objective function value of the current candidate solution to the initial dataset to generate the expanded dataset.

[0176] The judgment unit is used to determine whether the expanded dataset meets the preset termination conditions.

[0177] The training unit is used to train the surrogate model based on the expanded dataset in response to the expanded dataset not meeting the preset termination condition, and then proceed to the next iteration.

[0178] The determination unit is used to determine the optimal solution to the objective optimization problem in response to the expanded dataset satisfying the preset termination condition, and outputs the optimal solution and the objective function value of the optimal solution as the final optimization result.

[0179] It should be noted that the foregoing explanation of the adaptive active optimization method embodiment for few-sample optimization problems also applies to the adaptive active optimization device for few-sample optimization problems in this embodiment, and will not be repeated here.

[0180] The adaptive active optimization device for small-sample optimization problems proposed in this application can complete surrogate model training and elite solution set generation based on an initial dataset to construct an adaptive exploration function. A genetic algorithm based on the adaptive exploration function is then used to solve the target optimization problem until a preset termination condition is met, outputting the optimal solution. This allows for a balance between local development and global exploration, dynamically adjusting the exploration intensity, and actively guiding the search for unknown high-value regions, even when the surrogate model has prediction errors. This enhances global optimization capabilities, reduces real evaluation costs, and improves optimization efficiency. Therefore, this addresses the problems in related technologies where the accumulation of prediction errors in the surrogate model leads to insufficient search space coverage, a rapid decline in population diversity, and a significant reduction in global optimization capabilities. Furthermore, the lack of an active exploration mechanism for the unknown design space results in low exploration efficiency and difficulty in timely adjustment when predictions deviate, leading to increased evaluation times of the real function and reduced computational efficiency.

[0181] Figure 5 A schematic diagram of the structure of an electronic device provided in an embodiment of this application. The electronic device may include: The memory 501, the processor 502, and the computer program stored on the memory 501 and capable of running on the processor 502.

[0182] When the processor 502 executes the program, it implements the adaptive active optimization method for few-sample optimization problems provided in the above embodiments.

[0183] Furthermore, electronic devices also include: Communication interface 503 is used for communication between memory 501 and processor 502.

[0184] The memory 501 is used to store computer programs that can run on the processor 502.

[0185] Memory 501 may include high-speed RAM memory, and may also include non-volatile memory, such as at least one disk storage device.

[0186] If the memory 501, processor 502, and communication interface 503 are implemented independently, then the communication interface 503, memory 501, and processor 502 can be interconnected via a bus to complete communication between them. The bus can be an Industry Standard Architecture (ISA) bus, a Peripheral Component Interconnect (PCI) bus, or an Extended Industry Standard Architecture (EISA) bus, etc. The bus can be divided into address bus, data bus, control bus, etc. For ease of representation, Figure 5 The bus is represented by a single thick line, but this does not mean that there is only one bus or one type of bus.

[0187] Optionally, in a specific implementation, if the memory 501, processor 502, and communication interface 503 are integrated on a single chip, then the memory 501, processor 502, and communication interface 503 can communicate with each other through an internal interface.

[0188] Processor 502 may be a central processing unit (CPU), an application specific integrated circuit (ASIC), or one or more integrated circuits configured to implement the embodiments of this application.

[0189] This application also provides a non-volatile computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described adaptive active optimization method for few-sample optimization problems.

[0190] This application also provides a computer program product storing a computer program that, when executed by a processor, implements the above-described adaptive active optimization method for few-sample optimization problems.

[0191] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0192] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0193] Any process or method described in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or N executable instructions for implementing custom logic functions or processes, and the scope of the preferred embodiments of this application includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functions involved, as should be understood by those skilled in the art to which embodiments of this application pertain.

[0194] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device. More specific examples (a non-exhaustive list) of computer-readable media include: an electrical connection having one or more wires (electronic device), a portable computer disk drive (magnetic device), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Alternatively, the computer-readable medium may be paper or other suitable media on which the program can be printed, since the program can be obtained electronically by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in a computer memory.

[0195] It should be understood that the various parts of this application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, the N steps or methods can be implemented using software or firmware stored in memory and executed by a suitable instruction execution system. If implemented in hardware, as in another embodiment, it can be implemented using any one or more of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0196] Those skilled in the art will understand that all or part of the steps of the methods in the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, the program includes one or a combination of the steps of the method embodiments.

[0197] Furthermore, the functional units in the various embodiments of this application can be integrated into a processing module, or each unit can exist physically separately, or two or more units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium.

[0198] The storage medium mentioned above can be a read-only memory, a disk, or an optical disk, etc. Although embodiments of this application have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting this application. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of this application.

Claims

1. An adaptive active optimization method for small sample optimization problems, characterized in that, Includes the following steps: Determine the true objective function value for each sample in the initial sample set of the objective optimization problem to construct the initial dataset for the objective optimization problem; Based on the initial dataset, a surrogate model is trained to predict the objective function values ​​of any candidate solutions to the objective optimization problem, and an elite solution set for the objective optimization problem is generated based on the initial dataset. Based on the surrogate model and the elite solution set, an adaptive exploration function for the objective optimization problem is constructed. A genetic algorithm based on the adaptive exploration function is then used to solve the objective optimization problem to obtain the current candidate solution of the objective optimization problem. The optimal solution of the objective optimization problem is determined when the current candidate solution meets the preset termination condition.

2. The method according to claim 1, characterized in that, The determination of the true objective function value for each sample in the initial sample set of the objective optimization problem includes: Within the design space, the initial sample set is generated using a preset sampling strategy; The true objective function value for each sample is obtained based on real physical experiments or high-precision numerical simulations.

3. The method according to claim 1, characterized in that, The step of using a genetic algorithm based on the adaptive exploration function to solve the objective optimization problem to obtain the current candidate solution to the objective optimization problem includes: Calculate the surrogate model prediction, adaptive exploration weight, and normalized local novelty score for each individual in the current population; The surrogate model prediction, the adaptive exploration weight, and the normalized local novelty score are substituted into the adaptive exploration function to calculate the fitness of each individual. Based on the fitness, the genetic algorithm is performed on the current population to generate a new generation of population until a preset number of iterations is reached, and the final population of the target optimization problem is obtained. Elite solutions, highly novel solutions, and random solutions are selected from the final population to determine the current candidate solutions.

4. The method according to claim 3, characterized in that, The expression for the adaptive exploration function is: ， in, The adaptive exploration function is... The predicted value of the proxy model. The adaptive exploration weights, The normalized local novelty score is given.

5. The method according to claim 3, characterized in that, The calculation of the surrogate model prediction, adaptive exploration weights, and normalized local novelty score for each individual in the current population includes: The feature vector of each individual is input into the surrogate model to predict the objective function value of each individual, and the predicted value of the surrogate model is output. The adaptive exploration weights are calculated based on the absolute values ​​of the adaptive exploration weights and the preset maximum and minimum values ​​of the true objective function values ​​in the initial dataset. Calculate the minimum distance between the elite solution set and all individuals in the current population, and calculate the locality factor of each individual based on the minimum distance; Calculate the unvisited reward function and the repeat penalty factor for each individual; Based on the locality factor, the unvisited reward function, and the duplication penalty factor, the original local novelty score of each individual is calculated, and the original local novelty scores of all individuals in the current population are normalized by a minimum-maximum method to obtain the normalized local novelty score.

6. The method according to claim 1, characterized in that, The process of obtaining the current candidate solution to the objective optimization problem, and determining the optimal solution to the objective optimization problem until the current candidate solution satisfies a preset termination condition, includes: The current candidate solution is evaluated to obtain the true objective function value of the current candidate solution. The current candidate solution and the true objective function value of the current candidate solution are then added to the initial dataset to generate the expanded dataset. Determine whether the expanded dataset meets the preset termination condition; If the expanded dataset does not meet the preset termination condition, the proxy model is trained based on the expanded dataset, and the next iteration begins. In response to the expanded dataset satisfying a preset termination condition, the optimal solution to the objective optimization problem is determined, and the optimal solution and the objective function value of the optimal solution are output as the final optimization result.

7. An adaptive active optimization device for small sample optimization problems, characterized in that, include: A construction module is used to determine the true objective function value for each sample in the initial sample set of the objective optimization problem, so as to construct the initial dataset of the objective optimization problem; The generation module is used to train a proxy model for predicting the objective function values ​​of any candidate solutions to the objective optimization problem based on the initial dataset, and to generate an elite solution set for the objective optimization problem based on the initial dataset. An optimization module is used to construct an adaptive exploration function for the target optimization problem based on the surrogate model and the elite solution set, and to solve the target optimization problem using a genetic algorithm based on the adaptive exploration function to obtain the current candidate solution of the target optimization problem until the current candidate solution satisfies a preset termination condition, thereby determining the optimal solution of the target optimization problem.

8. An electronic device, characterized in that, include: A memory, a processor, and a computer program stored in the memory and capable of running on the processor, the processor executing the program to implement the adaptive active optimization method for few-sample optimization problems as described in any one of claims 1-6.

9. A non-volatile computer-readable storage medium having a computer program stored thereon, characterized in that, The program is executed by the processor to implement the adaptive active optimization method for few-sample optimization problems as described in any one of claims 1-6.

10. A computer program product, comprising a computer program, characterized in that, The computer program is executed to implement the adaptive active optimization method for few-sample optimization problems as described in any one of claims 1-6.

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