Multi-fidelity particle swarm optimization method for a mirror support system

Abstract

The present application belongs to the technical field of optical-mechanical structure optimization design, and relates to a multi-fidelity particle swarm optimization method for a mirror support system, comprising determining design variables, variable upper and lower bounds, optimization objectives and constraint conditions of the mirror support system; using a Latin hypercube sampling method to generate an initial low-fidelity sample set and an initial high-fidelity sample set; respectively calculating three types of sensitivity indexes of each design variable on a performance index of the mirror support system; using a Co-Kriging multi-fidelity surrogate model to establish a correlation between low-fidelity responses and high-fidelity responses; based on minimum values in the high-fidelity samples, constructing an expected improvement criterion as a fitness function of a particle swarm optimization algorithm, finding a candidate sample point that maximizes the expected improvement criterion; and selecting a sample with the minimum objective function value from the current high-fidelity sample set as a final optimization result. PV EI EI The present application is applicable to optimization design of the mirror support system under high-dimensional, expensive and small-sample conditions.​​

Description

Technical Field

[0001] This invention belongs to the field of optomechanical structure optimization design technology, and relates to a multi-fidelity particle swarm optimization method for a mirror support system. Background Technology

[0002] In large-aperture optical systems, the structural performance of the mirror support system directly affects the mirror's surface accuracy, imaging quality, and system stability. Mirror support structures typically include multiple geometric design variables, such as support location, support radius, flexible joint dimensions, support height, and connection structure dimensions. These parameters have complex coupling relationships, and the structural response usually needs to be obtained through finite element simulation. Therefore, optimizing the design of mirror support systems often presents challenges such as high dimensionality of design variables, long time consumption for single high-precision simulations, and a limited number of available high-fidelity samples.

[0003] Traditional intelligent optimization methods such as particle swarm optimization and genetic algorithms require extensive evaluation of real functions to obtain a satisfactory solution. Directly calling high-fidelity finite element models for iterative optimization would incur extremely high computational costs. To reduce computational burden, existing methods typically employ surrogate models such as Kriging, RBF, or response surfaces to replace expensive simulation models. However, single high-fidelity surrogate models are prone to insufficient prediction accuracy under high-dimensional and small-sample conditions, making it difficult to accurately describe the complex nonlinear response relationships in the mirror support system.

[0004] Multi-fidelity surrogate models can simultaneously leverage the low-cost advantage of low-fidelity models and the high-precision advantage of high-fidelity models. Among them, the Co-Kriging model can establish correlations between low-fidelity and high-fidelity data, improving the predictive ability of the surrogate model under limited high-fidelity sample conditions. However, for the optimization problem of high-dimensional mirror support systems, the Co-Kriging model requires optimizing a large number of correlation hyperparameters, and the maximum likelihood function often exhibits multi-modal characteristics, leading to difficulties in hyperparameter optimization and reduced modeling efficiency.

[0005] Therefore, it is necessary to propose a multi-fidelity optimization method for mirror support systems that can identify key design variables, reduce the difficulty of multi-fidelity proxy modeling, and improve optimization search efficiency. Summary of the Invention

[0006] To address the problems of low modeling efficiency, strong search bias, and numerous high-fidelity simulation calls in existing optimization methods for mirror support systems under high-dimensional, expensive, and small-sample conditions, this invention provides a multi-fidelity particle swarm optimization method for mirror support systems. This method obtains a combined sensitivity index by fusing multiple sensitivity analysis results and simultaneously incorporates this combined sensitivity information into the construction of the multi-fidelity surrogate model and the particle swarm search process, thereby improving the efficiency and accuracy of the optimization design of mirror support systems.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A multi-fidelity particle swarm optimization method for a mirror support system includes the following steps: Step 1: Determine the design variables, upper and lower bounds of the variables, optimization objectives and constraints of the reflector support system, and establish low-fidelity and high-fidelity models.

[0008] The design variables include, but are not limited to, the support position parameters, flexible joint size parameters, support column size parameters, support ring diameter parameters, and local structural size parameters of the reflector support structure.

[0009] Let the design variables of the mirror support system be: in, For design variable dimensions. Each design variable satisfies: in, and The first The lower and upper bounds of each design variable.

[0010] The objective function for optimizing the mirror support system is defined as follows: in, This represents the performance index of the mirror support system calculated using a high-fidelity finite element model. The performance index is the peak-to-valley value of the mirror surface shape. For the surface optimization problem, we can define: in, The peak and valley values ​​of the surface shape of the mirror under given operating conditions.

[0011] Simultaneously establish a low-fidelity model and high-fidelity models Low-fidelity models can employ coarse-mesh finite element models, simplified boundary condition models, reduced-order models, or low-precision simulation models; high-fidelity models can employ fine-mesh finite element models or high-precision optomechanical coupling simulation models.

[0012] Step 2: Within the design variable space, use the Latin hypercube sampling method to generate an initial low-fidelity sample set and an initial high-fidelity sample set, typically satisfying the requirement for a certain number of low-fidelity samples. >Number of high-fidelity samples The low-fidelity response is calculated by calling the low-fidelity model, and the high-fidelity response is calculated by calling the high-fidelity finite element model.

[0013] Step 3: Based on the current sample set, calculate the distance correlation sensitivity index of each design variable to the performance index of the mirror support system. Hilbert-Schmidt Independence Criterion Sensitivity Index And Sobol sensitivity index .

[0014] To avoid identification bias in complex nonlinear problems due to the use of a single sensitivity index, the entropy weight method is employed to objectively assign weights to the three types of sensitivity indices mentioned above. First, the first... Class sensitivity index in the first Relative contribution of each design variable : in, , represent , represent , represent .

[0015] Calculate the first Information entropy of class sensitivity index : in, The number of design variables for the mirror support system.

[0016] Further calculate the information utility value : Get the first Weights of class sensitivity indicators : Finally, we get the first Combined sensitivity index of design variables : Combined sensitivity index It is used to characterize the combined influence of each design variable in the reflector support system on the objective function.

[0017] Step 4: Establish the correlation between low-fidelity and high-fidelity responses using the Co-Kriging multifidelity proxy model. The high-fidelity response is represented as: in, This is the scaling factor between the low-fidelity model and the high-fidelity model. It is a residual random process.

[0018] First, a low-fidelity Kriging sub-model is constructed using a low-fidelity sample set. To reduce the hyperparameter optimization difficulty of the high-dimensional Co-Kriging model, a low-fidelity combinatorial sensitivity index is introduced into the correlation hyperparameter of the low-fidelity Kriging sub-model, establishing the following relationship: in, For the low-fidelity Kriging sub-model The correlation hyperparameters of the design variables. For low-fidelity samples, the combined sensitivity index is... This is the scaling factor to be optimized in the low-fidelity Kriging sub-model.

[0019] Through the above mapping relationship, the original The dimensional hyperparameter optimization problem is transformed into a low-dimensional scaling factor optimization problem, thereby improving the modeling efficiency of the low-fidelity Kriging sub-model.

[0020] Then, the residuals between the low-fidelity and high-fidelity responses are calculated at the high-fidelity sample points: The residual portfolio sensitivity index is calculated based on residual samples, and a residual Kriging sub-model is constructed. The correlation hyperparameters of the residual sub-model are expressed as follows: in, For the residual Kriging sub-model The correlation hyperparameters of the design variables. The combined sensitivity index corresponding to the residual samples. This is the scaling factor to be optimized in the residual Kriging sub-model.

[0021] Finally, based on the calculation formula of high-fidelity response, the low-fidelity Kriging sub-model and the residual Kriging sub-model are fused to obtain a sensitivity information-driven multi-fidelity proxy model.

[0022] in, These are high-fidelity model predictions. These are predictions from a low-fidelity model. These are the predicted values ​​from the residual Kriging submodel.

[0023] This multifidelity proxy model applies to any set of design variables. Output predicted mean and prediction variance .

[0024] Step 5: Based on the minimum value among the currently obtained high-fidelity samples value Construct the expected improvement criterion It is used to balance global exploration and local development.

[0025] For the minimization problem, the improvement amount Defined as: Expected Improvement Function for: in, It is the standard normal distribution function. It is the standard normal probability density function. This represents the prediction standard deviation of the high-fidelity response.

[0026] By maximizing the expected improvement function This determines the current optimal candidate high-fidelity sampling point. Because... Since the function is usually non-convex in the design space, a particle swarm optimization algorithm is used to perform a global search to obtain the optimal candidate sample points.

[0027] Step 6: In each iteration, adjust the function to be improved. As a fitness function in particle swarm optimization, it guides the search direction of the particle swarm in the design space, thereby finding a suitable target. The largest candidate sample point. The model is optimized using the particle swarm optimization algorithm, and the optimization process is judged to meet the stopping conditions. The stopping conditions include any one or more of the following: the number of high-fidelity samples reaches a preset maximum value; the change in the optimal objective function value is less than a set threshold in several consecutive iterations; the expected improvement value is lower than a set threshold; the maximum number of iterations is reached.

[0028] If the stopping condition is not met, then for the candidate sample points Low-fidelity and high-fidelity evaluations were performed separately to obtain low-fidelity predicted values ​​at candidate sample points. High-fidelity prediction The new sample will be added to both the low-fidelity sample set and the high-fidelity sample set. , .

[0029] Then return to step three, recalculate the combined sensitivity index, update the sensitivity information-driven multi-fidelity agent model, and continue the particle swarm optimization search.

[0030] If the stopping condition is met, the optimization process is terminated.

[0031] Step 7: Select the sample with the smallest objective function value from the current high-fidelity sample set as the final optimization result. This is the optimal design variable for the mirror support system. and output the corresponding optimal performance index. .

[0032] The effective gain effect of the present invention is as follows: This invention leverages the low computational cost of low-fidelity models and the high accuracy of high-fidelity models to construct a multi-fidelity surrogate model, which improves the accuracy of mirror support system optimization design under limited high-fidelity sample conditions. By fusing three sensitivity indices—DIC, HSIC, and Sobol—a combined sensitivity index is obtained, which more robustly identifies key design variables in the mirror support system and reduces identification bias caused by single sensitivity indices. Embedding the combined sensitivity index into the velocity update formula of the particle swarm optimization algorithm ensures that particles maintain strong search capabilities in high-sensitivity variable directions and reduce invalid perturbations in low-sensitivity variable directions, thereby improving search efficiency in high-dimensional design spaces. Using the expected improvement criterion as the sequential addition objective achieves a balance between the surrogate model's predicted mean and prediction uncertainty, contributing to improved global search and local development capabilities in the optimization process. This invention is applicable to high-dimensional, expensive, and small-sample engineering optimization problems of mirror support systems, reducing the number of high-fidelity finite element simulation calls, lowering optimization design computational costs, and improving the performance of the final design scheme. Attached Figure Description

[0033] Figure 1 This is a flowchart of the multi-fidelity particle swarm optimization method for a mirror support system according to the present invention.

[0034] Figure 2 This is a schematic diagram of the reflector support system according to an embodiment of the present invention.

[0035] Figure 3 Convergence curves of different optimization methods in the embodiments of the present invention. Detailed Implementation

[0036] The present invention will be further described below with reference to specific embodiments.

[0037] like Figure 1As shown, this invention provides a multi-fidelity particle swarm optimization method for a mirror support system. First, the design variable range, objective function, and constraints of the mirror support system optimization problem are obtained, and low-fidelity and high-fidelity models are established. Then, an initial low-fidelity and initial high-fidelity sample set are generated using the Latin hypercube sampling method, and the low-fidelity and high-fidelity models are respectively invoked to complete response evaluation. Next, based on the current sample set, three types of sensitivity indices—distance correlation, Hilbert-Schmidt independence criterion, and Sobol exponent—are calculated, and the entropy weight method is used to fuse them to obtain a combined sensitivity index. Further, the combined sensitivity index is introduced into the hyperparameter optimization process of the Co-Kriging multi-fidelity surrogate model, constructing a low-fidelity Kriging sub-model and a residual Kriging sub-model respectively, to obtain a sensitivity information-driven multi-fidelity surrogate model. Then, using the expected improvement criterion as the objective, new candidate sample points are searched. After high-fidelity and low-fidelity evaluations of the candidate sample points, they are added to the sample set, and the above process is repeated until the maximum number of high-fidelity samples or the optimization convergence condition is met. Finally, the optimal design parameters and corresponding optimal performance indicators of the output mirror support system are presented.

[0038] Figure 2 The structure of the mirror support system consists of a mirror chamber, a mirror, and a mirror support column. The mirror chamber is the fixing device for the mirror system, used to maintain the stability of the mirror; one end of the mirror support column is fixed to the mirror chamber, and the other end is fixed to and supports the mirror. As the core component of the optical system, the surface accuracy of the mirror directly affects the imaging quality and working performance of the optical system. Therefore, this embodiment uses the geometric parameters of the mirror support structure as design variables, and optimizes the structural parameters of the support system to reduce the surface error of the mirror under gravity.

[0039] The optimization objective of this embodiment is to minimize the peak-to-valley value of the reflector under gravity, denoted as . . The value represents the difference between the maximum and minimum values ​​of mirror deformation, and is an important indicator for measuring the accuracy of the reflective mirror's shape. The objective function is defined as: in, For the design variable vector of the mirror support system, The peak and valley values ​​of the reflective mirror surface obtained through finite element analysis under given design variables. The smaller the value, the higher the accuracy of the reflector surface and the better the design effect of the support system.

[0040] Since the mechanical response of the mirror support system needs to be obtained through finite element analysis, directly using a high-fidelity finite element model for extensive optimization iterations would result in high computational costs. Therefore, this embodiment establishes both a low-fidelity model and a high-fidelity model. The low-fidelity model uses a relatively coarse mesh and lower computational precision to quickly obtain the response trend in the design space; the high-fidelity model uses a finer mesh and higher-precision solution settings to obtain accurate surface response results.

[0041] First, initial training samples are generated within the design space using the Latin hypercube sampling method. In this embodiment, the initial number of sample points is set to 40, and the maximum number of high-fidelity samples is set to 70. By calling the low-fidelity model and the high-fidelity model, the low-fidelity response and high-fidelity response corresponding to the sample points are calculated respectively, thus constructing the initial low-fidelity sample set and the initial high-fidelity sample set.

[0042] Then, based on the current sample set, the DIC, HSIC, and Sobol sensitivity indices for each design variable are calculated separately. Since a single sensitivity index is easily affected by model assumptions and sample distribution, potentially leading to bias in variable importance identification, this embodiment uses the entropy weight method to fuse the three types of sensitivity indices, obtaining a combined sensitivity index. This combined sensitivity index is used to characterize the influence of each design variable on the mirror support system. The overall impact of the value.

[0043] After obtaining the combined sensitivity index, a sensitivity-driven multi-fidelity surrogate model is constructed. First, a low-fidelity Kriging sub-model is built using low-fidelity samples, and its hyperparameters are optimized for dimensionality reduction using the combined low-fidelity sensitivity index. Next, the residuals of the high-fidelity samples relative to the low-fidelity model are calculated, and a residual Kriging sub-model is constructed based on these residual samples. The hyperparameters of the residual sub-model are then optimized for dimensionality reduction using residual sensitivity information. Finally, the low-fidelity Kriging sub-model and the residual Kriging sub-model are fused to obtain the Co-Kriging multi-fidelity surrogate model. This model can provide high-fidelity results at any candidate design point. The predicted mean and predicted variance of the values.

[0044] In each round of sequential optimization, the expected improvement criterion is used as the fitness function for particle swarm optimization, and candidate sample points with the largest expected improvement values ​​are searched. After obtaining the candidate sample points, the low-fidelity model and the high-fidelity model are used to evaluate the response, and the sample point and its response value are added to the current sample set. Then, the combined sensitivity index is recalculated, the multi-fidelity surrogate model is updated, and the next round of particle swarm search continues. The above process is repeated until the number of high-fidelity samples reaches a preset maximum value or the optimization convergence condition is met.

[0045] Finally, select from all high-fidelity samples The design scheme with the minimum value is taken as the final optimization result. Figure 3 These are the convergence curves for different optimization methods in this embodiment. Figure 3 It can be seen that the method of the present invention has a faster convergence speed during the optimization process, and achieves lower performance under the same high-fidelity sample budget. The value indicates that the method of the present invention can more effectively handle high-dimensional and expensive engineering optimization problems such as mirror support systems.

[0046] Finally, the optimized result was obtained, showing the original design's surface shape. Value mm. After optimization using the method of this invention, The value decreased to The accuracy is improved by 85.82% compared to the original design. This result shows that the method of the present invention can effectively improve the surface accuracy of the mirror support system under limited high-fidelity sample conditions and reduce the computational cost of optimizing complex optomechanical structures.

Claims

1. A multi-fidelity particle swarm optimization method for a mirror support system, characterized in that, Includes the following steps: Step 1: Determine the design variables, upper and lower bounds of the variables, optimization objectives and constraints of the reflector support system, and establish low-fidelity and high-fidelity models; In Step 1, the design variables are the support position parameters, flexible joint size parameters, support column size parameters, support ring diameter parameters or local structural size parameters of the reflector support structure; Step 2: Within the design variable space, the Latin hypercube sampling method is used to generate the initial low-fidelity sample set and the initial high-fidelity sample set; The response evaluation is completed by calling the low-fidelity model to calculate the low-fidelity response and calling the high-fidelity finite element model to calculate the high-fidelity response. Step 3: Based on the current sample set, calculate the distance correlation sensitivity index of each design variable to the performance index of the mirror support system. Hilbert-Schmidt Independence Criterion Sensitivity Index And Sobol sensitivity index ; Step 4: Establish the correlation between low-fidelity and high-fidelity responses using the Co-Kriging multifidelity proxy model; In the fourth step, A high-fidelity response is represented as: in, This is the scaling factor between the low-fidelity model and the high-fidelity model. It is a residual stochastic process; The specific steps are as follows: First, construct a low-fidelity Kriging sub-model using a low-fidelity sample set; then, introduce the low-fidelity combined sensitivity index into the correlation hyperparameters of the low-fidelity Kriging sub-model to establish the following relationship: in, For the low-fidelity Kriging sub-model The correlation hyperparameters of the design variables. For low-fidelity samples, the combined sensitivity index is... The scaling factor to be optimized in the low-fidelity Kriging sub-model; Then, the residuals between the low-fidelity and high-fidelity responses are calculated at the high-fidelity sample points: The residual portfolio sensitivity index is calculated based on the residual samples, and a residual Kriging sub-model is constructed; the correlation hyperparameters of the residual sub-model are expressed as follows: in, For the residual Kriging sub-model The correlation hyperparameters of the design variables. The combined sensitivity index corresponding to the residual samples. The scaling factor to be optimized in the residual Kriging sub-model; Finally, based on the calculation formula of high-fidelity response, the low-fidelity Kriging sub-model and the residual Kriging sub-model are fused to obtain a sensitivity information-driven multi-fidelity proxy model; in, These are high-fidelity model predictions. These are predictions from a low-fidelity model. These are the predicted values ​​from the residual Kriging submodel; Multifidelity proxy model in any set of design variables Output predicted mean and prediction variance ; Step 5: Based on the minimum value among the currently obtained high-fidelity samples value Construct the expected improvement criterion ; Step 6: In each iteration, adjust the function to be improved. As a fitness function in particle swarm optimization, it guides the search direction of the particle swarm in the design space, thereby finding a suitable target. The largest candidate sample point; the model is optimized using the particle swarm optimization algorithm, and it is determined whether the optimization process meets the stopping conditions; the stopping conditions include any one or more of the following: the number of high-fidelity samples reaches a preset maximum value; the change in the optimal objective function value is less than a set threshold in several consecutive iterations; the expected improvement value is lower than a set threshold; the maximum number of iterations is reached; Step 7: Select the sample with the smallest objective function value from the current high-fidelity sample set as the final optimization result.

2. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, Let the design variables of the mirror support system be: in, For design variable dimensions; each design variable satisfies: in, and The first Lower and upper bounds of a design variable; The objective function for optimizing the mirror support system is defined as follows: in, This represents the performance index of the mirror support system calculated by a high-fidelity finite element model; the performance index is the peak-to-valley value of the mirror surface shape. For the surface optimization problem, the definition is: in, The peak and valley values ​​of the surface shape of the mirror under a given operating condition; Simultaneously establish a low-fidelity model and high-fidelity models .

3. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 2, characterized in that, Low-fidelity models employ coarse-grid finite element models, simplified boundary condition models, or reduced-order models; high-fidelity models employ fine-grid finite element models or optomechanical coupling simulation models.

4. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, In the second step, the number of low-fidelity samples >Number of high-fidelity samples .

5. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, In the third step, the entropy weight method is used to objectively assign weights to the three types of sensitivity indicators; firstly, the first... Class sensitivity index in the first Relative contribution of each design variable : in, , represent , represent , represent ; Calculate the first Information entropy of class sensitivity index : in, The number of design variables for the mirror support system; Further calculate the information utility value : Get the first Weights of class sensitivity indicators : Finally, we get the first Combined sensitivity index of design variables : 。 6. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, In step five, for the minimization problem, the improvement amount Defined as: Expected Improvement Function for: in, It is the standard normal distribution function. It is the standard normal probability density function. The standard deviation of the high-fidelity response prediction.

7. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, In step six, if the stopping condition is not met, then for the candidate sample points... Low-fidelity and high-fidelity evaluations were performed separately to obtain low-fidelity predicted values ​​at candidate sample points. High-fidelity prediction ; The new sample will be added to both the low-fidelity sample set and the high-fidelity sample set. , ; Then return to step 3, recalculate the combined sensitivity index, update the sensitivity information-driven multi-fidelity agent model, and continue the particle swarm optimization search; If the stopping condition is met, the optimization process is terminated.

8. The multi-fidelity particle swarm optimization method for a mirror support system according to claim 1, characterized in that, In step seven, the final optimization result is: This is the optimal design variable for the mirror support system. and output the corresponding optimal performance index. .

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