Method for adaptive expansion of design space for ship hull form optimization based on boundary perception

Abstract

This application discloses a boundary-aware adaptive expansion method for ship morphology optimization design space. It generates samples in the current design space and maps them to a normalized space; constructs and trains a surrogate model integrating XGBoost, stochastic Fourier characteristic ridge regression, and support vector regression, outputting the predicted mean and standard deviation; calculates the boundary signal of elite samples, the model's uncertainty calibration error, and the direction signal; and performs design space updates according to a hierarchical decision rule prioritizing boundary geometry updates, followed by fitting quality, and then risk expansion as a fallback, mapping the normalized space boundary back to the physical space to obtain the expanded design space. This application adaptively expands the design space through multi-signal collaborative diagnosis and hierarchical expansion, avoiding optimal solution truncation due to improper initial space pre-setting, thus improving optimization efficiency and stability, and is particularly suitable for complex engineering problems such as ship morphology optimization.

Description

Technical Field

[0001] This application relates to the field of engineering optimization design technology, specifically to a boundary-aware space adaptive expansion method for ship morphology optimization design. Background Technology

[0002] With the continuous maturation of computer simulation technology, optimization design based on high-fidelity numerical analysis has become a key technical approach for the development of high-efficiency, low-drag products in fields such as shipbuilding, aviation, and automobiles. Its core idea is to sample within the design space, evaluate the performance of the proposed solutions using simulation tools, approximate the objective function using a surrogate model, and guide the optimization process to obtain the optimal design solution with limited computational cost.

[0003] In engineering practice, the initial range of the design space is usually determined by the designer's experience. If the design space is set too narrowly, the true optimal solution may lie outside this domain, resulting in only a suboptimal solution truncated at the boundary, regardless of the optimization strategy employed. Conversely, if the design space is set too wide, it significantly increases the difficulty of sufficient sampling and accurate model fitting, leading to a surge in optimization costs. This contradiction is particularly pronounced in ship hull optimization, where the large number of parameterized variables on the hull surface and their strong nonlinearity make it difficult to determine a reasonable range of variables based on prior knowledge.

[0004] To address the aforementioned issues, current research in academia and industry primarily focuses on "reducing" the design space. For example, Shan and Wang proposed a high-potential subspace extraction method based on rough set theory; Wang and Simpson, as well as Qiu et al., utilized fuzzy clustering and self-organizing maps to achieve multi-stage design space reduction and surrogate model optimization. These methods use data mining techniques to eliminate secondary regions, gradually focusing the search scope on more promising subspaces, thereby improving optimization efficiency. However, these methods implicitly assume that the initial design space already contains the global optimum or its sufficiently good neighborhood. If the initial space is set too small, the space reduction strategy not only fails to compensate for this deficiency but may also further accelerate convergence to a suboptimal boundary solution.

[0005] A few studies have attempted to overcome the limitations of fixed design spaces from different perspectives. For example, Wang et al. proposed an adaptive design space expansion strategy based on a Kriging surrogate model and expected improvement criterion in aerodynamic optimization, which can expand the search area in the potential dimension according to the point-addition criterion. US Patent Application US 2021 / 0141954 discloses an adaptive design space adjustment method based on a sequential Kriging surrogate model, applied to structural topology optimization, which scales the design domain boundary according to the surrogate model accuracy and the point-addition process. Furthermore, in the field of machine learning, some methods actively explore unbounded design spaces by identifying feasible domain boundaries.

[0006] While the aforementioned methods introduce the concept of dynamic adjustment of the design space, they still suffer from at least the following shortcomings when applied to ship hydrodynamic optimization: First, the triggering mechanism for spatial adjustment is relatively simple. Most methods rely solely on the improvement indicators or model accuracy predicted by the surrogate model, lacking direct geometric criteria for determining whether the current design domain truncates potential optimal areas. When elite samples are concentrated at the boundary, these indirect indicators may lag or become ineffective. Second, effective utilization and calibration of the uncertainty of the surrogate model are not achieved. The surrogate models used are usually of a single type, making it difficult to simultaneously address nonlinear expression, smoothness characterization, and small-sample generalization. Furthermore, there is a lack of measurement for the reliability of uncertainty estimation, making it difficult to adopt reasonable expansion strategies when the model is unreliable. Third, a hierarchical, multi-signal collaborative decision-making framework is lacking. When multiple adjustment needs conflict, it is impossible to ensure the robustness of expansion actions according to clear priority rules. Fourth, existing surrogate models in aircraft or structural optimization are mostly Kriging models. However, ship optimization often faces situations with high variable dimensionality, multiple peaks, and complex function shapes. A single Kriging model is insufficient to stably handle these situations, requiring the integration of heterogeneous models more adaptable to complex terrains. Summary of the Invention

[0007] The main objective of this application is to provide a boundary-aware adaptive expansion method for ship morphology optimization design space, comprising the following steps:

[0008] Step S1: Generate sample points within the current ship type physical design domain, calculate the objective function value corresponding to each sample point, and map the coordinates of all sample points to the normalized space;

[0009] Step S2: Construct and train an integrated surrogate model, which includes at least two different types of regression models. Through resampling training and weighted fusion based on out-of-bag prediction evaluation, the predicted mean and predicted standard deviation of any point in the normalized space are obtained.

[0010] Step S3: Select elite samples based on the objective function values ​​of the sample points, calculate the boundary signals of the elite samples in each dimension of the normalized space; calculate the uncertainty calibration error of the model based on the predicted mean and the predicted standard deviation, as a risk signal; and calculate the gradient direction of the model in each dimension based on the predicted mean, as a direction signal.

[0011] Step S4: Perform hierarchical decision-making according to preset priority to update the design space boundary. If the boundary signal indicates that the elite region touches the boundary of the current normalized design space, then perform geometric expansion based on the safety boundary; otherwise, if the holdout set determination coefficient of the surrogate model satisfies the first condition, then perform fitting-driven adjustment based on the direction signal; otherwise, if the uncertainty calibration error exceeds the set risk threshold, then perform risk-driven global expansion.

[0012] Step S5: Map the updated normalized spatial boundary back to the physical space to obtain the expanded design space.

[0013] In one embodiment, the step of generating sample points within the current physical design domain, calculating the objective function value corresponding to each sample point, and mapping the coordinates of all sample points to the normalized space includes:

[0014] A set of space-filling sample points is generated in the physical domain of the current design space using a quasi-random sequence;

[0015] For the sample points, the corresponding true objective function values ​​are obtained through numerical simulation;

[0016] Based on the lower and upper bounds of the current physical domain, the coordinates of each sample point are mapped to [0,1] through a linear transformation. d The normalized space, where d is the dimension of the design variables.

[0017] In one embodiment, the step of constructing and training an ensemble surrogate model, which includes at least two different types of regression models, and obtaining the predicted mean and predicted standard deviation of any point in the normalized space through resampling training and weighted fusion based on out-of-bag prediction evaluation, includes:

[0018] Each regression model is trained multiple times using Bootstrap resampling. Each training iteration yields a sub-model, and the unsampled samples constitute the out-of-bag validation set for that training iteration.

[0019] The out-of-bag determination coefficients are calculated using the out-of-bag predicted values ​​of each regression model, and the fusion weights of each regression model are determined based on the out-of-bag determination coefficients.

[0020] The predicted values ​​of all sub-models of each regression model are weighted and averaged according to the fusion weights to obtain the predicted mean, and the predicted standard deviation is obtained after calibration based on the dispersion of the predicted values ​​of each sub-model.

[0021] In one embodiment, the integrated proxy model includes an XGBoost model, a stochastic Fourier feature ridge regression model, and a support vector regression model; the resampling training is Bootstrap resampling, which obtains multiple sub-models for each model, calculates the fusion weight of each model using the out-of-bag prediction determination coefficient of each sub-model, and weights the prediction mean based on the fusion weight; the prediction standard deviation is obtained based on the dispersion of the prediction values ​​of each sub-model or after calibration.

[0022] In one embodiment, the steps of screening elite samples based on the objective function value of sample points, calculating the boundary signals of the elite samples in each dimension of the normalized space, calculating the uncertainty calibration error of the model based on the predicted mean and the predicted standard deviation as a risk signal, and calculating the gradient direction of the model in each dimension based on the predicted mean as a direction signal, include:

[0023] After sorting the samples in ascending order of the objective function value, the top preset proportion of samples are taken to form an elite set. The minimum and maximum values ​​of the elite set in each dimension are calculated, and it is determined whether the minimum value is lower than the lower edge threshold or whether the maximum value is higher than the upper edge threshold, so as to generate a boundary signal to indicate whether the elite region touches the normalized spatial boundary.

[0024] The standardized residuals of each sample point are constructed using the predicted mean and predicted standard deviation. The range of values ​​of the standardized residuals is divided into multiple stratified intervals. The theoretical coverage probability and actual observation frequency of each stratified interval are calculated respectively, and the absolute values ​​of the difference between the two are summed to obtain a risk signal used to measure the degree of model uncertainty calibration.

[0025] For each dimension, based on the difference between adjacent samples arranged in ascending order along the coordinates of that dimension according to the predicted mean, the average gradient of that dimension is estimated to obtain a direction signal used to indicate the extension direction of the potential superior region.

[0026] In one embodiment, the calculation of the boundary signal includes: taking the top k samples with the best objective function values ​​as the elite set, calculating the minimum and maximum values ​​of the elite set in each dimension; when the minimum value in a certain dimension is less than or equal to the lower edge threshold, or the maximum value is greater than or equal to 1 minus the upper edge threshold, it is determined to be an edge; and calculating the safety boundary based on the span of the elite set in each dimension and the preset safety margin.

[0027] The calculation of the uncertainty calibration error includes: constructing a standardized residual using the predicted mean and predicted standard deviation, dividing the value range of the standardized residual into multiple stratified intervals, calculating the theoretical coverage probability and actual coverage frequency of each stratified interval, summing the absolute values ​​of the differences between the two to obtain the stratified calibration error, which serves as the risk signal.

[0028] In one embodiment, the step of updating the design space boundary by performing hierarchical decision-making according to a preset priority includes: if the boundary signal indicates that the elite region touches the boundary of the current normalized design space, then geometric expansion is performed based on the safety boundary; otherwise, if the holdout set determination coefficient of the surrogate model satisfies a first condition, then fitting-driven adjustment is performed based on the direction signal; otherwise, if the uncertainty calibration error exceeds a set risk threshold, then risk-driven global expansion is performed.

[0029] The boundary signal is checked. If the elite region in any dimension touches the normalized space boundary, the safety boundary is calculated based on the span of the elite set in each dimension and the preset safety margin. After modulating the expansion amplitude using the uncertainty calibration error, geometric expansion is performed, and the current round of hierarchical decision-making ends.

[0030] If none of the boundary signals are triggered, it is determined whether the retention set determination coefficient meets the preset fitting reliability threshold. If it does, the boundary of each dimension is fine-tuned by a preset step size near the current optimal sample position according to the gradient direction indicated by the direction signal, and the current round of hierarchical decision-making ends.

[0031] If the holdout set determination coefficient does not meet the fitting reliability threshold, then it is determined whether the uncertainty calibration error exceeds the preset risk trigger threshold. If it does, then the global expansion ratio is linearly determined according to the degree to which the uncertainty calibration error exceeds the risk trigger threshold, and risk-driven global expansion is performed.

[0032] In one embodiment, the step of mapping the updated normalized spatial boundary back to physical space to obtain the expanded design space includes:

[0033] In the normalized space, a minimum width constraint is applied to the updated target interval to prevent interval collapse in any dimension, and a step size constraint is applied to limit the maximum change of the boundary relative to the original boundary in a single update, thus obtaining a valid new normalized interval.

[0034] Based on the lower and upper bounds of the original physical domain in this round, the lower and upper bounds of the normalized new interval are mapped back to the physical space through inverse linear transformation to obtain the expanded design space.

[0035] In one embodiment, the first condition is that the coefficient of determination R²_oof of the holdout set is ≥0.80; the fitting-driven adjustment includes adjusting the boundary of the position corresponding to the current optimal sample point along each dimension according to the gradient direction indicated by the direction signal.

[0036] In one embodiment, the method is applied to ship hull optimization design, wherein the objective function is a ship resistance performance index, and the design space is the space spanned by the deformation parameters of the hull surface.

[0037] Therefore, this application has the following beneficial effects:

[0038] (1) This application directly diagnoses whether the current design space has truncated the potential optimal area by using elite sample edge detection, and uses geometric information as the first criterion for spatial expansion, which overcomes the delay or misjudgment that may be caused by the existing method relying solely on the statistical indicators of the proxy model; combined with safety boundary and uncertainty calibration error modulation, it realizes "evidence-based and risk-controllable" directional expansion, avoids indiscriminate global amplification of the search domain, and effectively prevents the truncation of the optimal solution while ensuring the targeted nature of the optimization.

[0039] (2) This application uses three heterogeneous models, namely XGBoost, stochastic Fourier feature ridge regression and support vector regression, to construct an integrated surrogate model. It captures the complex features of the objective function from three perspectives: tree model integration, Fourier feature mapping and kernel method. It is supplemented by Bootstrap resampling and adaptive weighted fusion based on out-of-bag evaluation. Compared with existing solutions such as single Kriging model, it effectively improves the prediction accuracy and uncertainty estimation quality under complex terrains such as limited samples, strong nonlinearity of objective function and multi-peak, and provides a more reliable foundation for subsequent risk diagnosis.

[0040] (3) This application introduces uncertainty calibration error as a risk signal, which not only evaluates the model prediction accuracy, but also measures the degree of matching between the prediction uncertainty estimate and the actual residual distribution. It can effectively identify the state of overconfidence or overconservatism of the model, and transform the risk state into the quantitative modulation parameter of the expansion amplitude, thus realizing the adaptive matching between the spatial expansion strength and the model reliability.

[0041] (4) This application establishes a hierarchical decision-making mechanism of “geometric boundary priority, fitting quality second, and risk extension as a safety net”, which coordinates multiple extension needs with fixed priority, avoids decision conflicts when multiple signals are triggered, ensures that the diagnostic basis for each spatial update is clear and the action logic is clear, and significantly improves the robustness and interpretability of the design space adaptive adjustment. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a system flowchart of a boundary-aware space adaptive expansion method for ship hull optimization design;

[0044] Figure 2 It is a design space expansion decision path based on the boundary-aware ship morphology optimization design space adaptive expansion method;

[0045] Figure 3 It is the distribution of the shape value points on the surface of the ship model based on the boundary-aware adaptive expansion method of ship shape optimization design space;

[0046] Figure 4 This is a cross-sectional comparison diagram of the space adaptive expansion method for ship hull optimization design based on boundary awareness;

[0047] Figure 5 This is a longitudinal section comparison diagram of a boundary-aware, space-adaptive expansion method for ship hull optimization design.

[0048] Figure 6 It is a hull surface pressure map based on the boundary-aware space adaptive expansion method for hull optimization design;

[0049] Figure 7 This is a waveform diagram of the space adaptive expansion method for ship hull optimization design based on boundary awareness. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0051] It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of this application.

[0052] To address the shortcomings of existing technologies, this application provides a boundary-aware space-adaptive expansion method for ship hull optimization design, comprising steps S1-S5, as described above. Figure 1 , Figure 1 This is a system flowchart of a boundary-aware space adaptive expansion method for ship hull optimization design.

[0053] Step S1: Generate sample points within the current ship type physical design domain, calculate the objective function value corresponding to each sample point, and map the coordinates of all sample points to the normalized space;

[0054] Step S2: Construct and train an integrated surrogate model, which includes at least two different types of regression models. Through resampling training and weighted fusion based on out-of-bag prediction evaluation, the predicted mean and predicted standard deviation of any point in the normalized space are obtained.

[0055] Step S3: Select elite samples based on the objective function values ​​of the sample points, calculate the boundary signals of the elite samples in each dimension of the normalized space; calculate the uncertainty calibration error of the model based on the predicted mean and the predicted standard deviation, as a risk signal; and calculate the gradient direction of the model in each dimension based on the predicted mean, as a direction signal.

[0056] Step S4: Perform hierarchical decision-making according to preset priority to update the design space boundary. If the boundary signal indicates that the elite region touches the boundary of the current normalized design space, then perform geometric expansion based on the safety boundary; otherwise, if the holdout set determination coefficient of the surrogate model satisfies the first condition, then perform fitting-driven adjustment based on the direction signal; otherwise, if the uncertainty calibration error exceeds the set risk threshold, then perform risk-driven global expansion.

[0057] Step S5: Map the updated normalized spatial boundary back to the physical space to obtain the expanded design space.

[0058] Specifically, in this embodiment, the present application provides a boundary-aware adaptive expansion method for ship morphology optimization design space, which includes steps S1 to S5, referring to... Figure 1 The system flowchart shown below provides a detailed description of the implementation process for each step.

[0059] Step S1: Generate sample points within the current physical design domain, calculate the objective function value corresponding to each sample point, and map the coordinates of all sample points to the normalized space. Specifically, assuming the current iteration is in the t-th iteration, the physical design space is defined by the lower bound vector L and the upper bound vector U, and the search space dimension is d. First, a set of initial sample points with good space-filling properties is generated within the current physical domain using a Sobol quasi-random sequence. In this embodiment, the number of samples n is 400. For each sample point, its corresponding objective function value is calculated using a high-fidelity CFD simulation tool or an analytical expression of the objective function. For example, in the ship hull optimization scenario, the objective function is the wave-making resistance coefficient. Subsequently, the physical coordinates of each sample point are mapped to [0,1] through a linear transformation. d The normalized space is used to ensure that subsequent analyses are performed on a uniform scale, eliminating the impact of differences in the dimensions of various variables on the algorithm.

[0060] Step S2: Construct and train an ensemble surrogate model. The ensemble surrogate model includes at least two different types of regression models. Through resampling training and weighted fusion based on out-of-bag prediction evaluation, the predicted mean and standard deviation at any point in the normalized space are obtained. In this embodiment, the ensemble surrogate model consists of three heterogeneous models: an XGBoost gradient boosting tree model, a stochastic Fourier feature ridge regression model, and a support vector regression model. The three models are trained by Bootstrap resampling on the normalized sample set. The number of resampling times N_bag is 50. Each training iteration yields a set of sub-models, and the unsampled samples constitute the out-of-bag validation set for that training iteration. The out-of-bag determination coefficient R²_arm is calculated using the predicted values ​​of each model for the out-of-bag samples, and the fusion weights of each model are determined based on R²_arm, which is then normalized to obtain w'_arm. Finally, the ensemble surrogate model outputs the fused predicted mean at any point z, and the predicted standard deviation after calibration based on the dispersion of the predicted values ​​of each sub-model. This ensemble mechanism takes into account the ability to express complex nonlinearities, the ability to capture smoothness, and the generalization performance of small samples.

[0061] Step S3: Select elite samples based on the objective function values ​​of the sample points, and calculate the boundary signals of the elite samples in each dimension of the normalized space; calculate the uncertainty calibration error of the model based on the predicted mean and predicted standard deviation, as a risk signal; and calculate the gradient direction of the model in each dimension based on the predicted mean, as a direction signal. This step calculates three types of state signals simultaneously, providing a basis for subsequent hierarchical decision-making.

[0062] The boundary signal is calculated by sorting the objective function values ​​in ascending order and taking the top k samples to form an elite set E. The minimum and maximum values ​​of the elite set in each dimension are calculated to determine whether the lower or upper boundary conditions are met. If they are met, it indicates that the elite region has approached the normalized boundary and the current design space may be too narrow.

[0063] The risk signal employs Uncertainty Calibration Error (UCE). This is achieved by constructing standardized residuals and dividing the interval into K=10 strata. The sum of the absolute values ​​of the differences between the theoretical coverage probability and the actual coverage frequency of each stratum is calculated. A smaller UCE indicates better model calibration. The direction signal is based on the average gradient vector g=[g1,…,g], which is the difference estimate of the predicted values ​​from adjacent samples in each dimension by the surrogate model. d ] T , used to indicate the direction of extension of potential preferred areas.

[0064] Step S4: Perform hierarchical decision-making according to preset priorities to update the design space boundary. This step is the core decision-making link of the method, judging and executing corresponding actions in sequence according to fixed priorities. The first layer is boundary geometry update: If the boundary signal in step S3 indicates that any dimension touches the edge, then directly enter the geometry update branch. Calculate the safety boundary based on the elite span and safety margin μ, and introduce a modulation coefficient α to perform risk modulation on the expansion amplitude. When the UCE is small, it tends to adopt the safety boundary expansion; when the UCE is large, it tends to be conservative or even freeze the expansion, obtaining the normalized target interval after modulation. The second layer is fitting-driven adjustment: If the boundary signal is not triggered, further check whether the hold-out set determination coefficient R²_oof of the surrogate model meets the condition R²_oof≥0.80. If it does, the model fitting quality is considered reliable. Based on the gradient direction indicated by the direction signal, perform boundary fine-tuning with a preset step size near the current optimal sample position to ensure that the optimal region is completely covered by the current boundary. The third layer is risk-driven global expansion: if the model reliability is insufficient and the UCE exceeds a preset risk threshold, it indicates that the current model uncertainty mismatch is severe, requiring a larger exploration range to restore calibration capability. The expansion ratio is linearly determined according to the degree to which the UCE exceeds the threshold. If none of the above conditions are met, the current design space remains unchanged.

[0065] Step S5: Map the updated normalized space boundary back to the physical space to obtain the expanded design space. After obtaining the new normalized space interval through hierarchical decision-making in Step S4, a legalization process is first performed, allowing small out-of-bounds limits and imposing minimum width constraints to prevent domain collapse, as well as step size constraints to limit the maximum change in a single update. Subsequently, the new normalized interval is mapped back to the physical space through an inverse linear transformation. This yields the expanded physical design space, which is used for sampling and optimization in subsequent optimization rounds. This method, through multi-signal collaborative diagnosis and hierarchical decision-making adaptive expansion of the design space, effectively avoids the optimal solution truncation problem caused by improper initial design space pre-setting, significantly improving the solution quality and stability of complex engineering optimization problems.

[0066] In one embodiment, the step of generating sample points within the current physical design domain, calculating the objective function value corresponding to each sample point, and mapping the coordinates of all sample points to the normalized space includes:

[0067] A set of space-filling sample points is generated in the physical domain of the current design space using a quasi-random sequence;

[0068] For the sample points, the corresponding true objective function values ​​are obtained through numerical simulation;

[0069] Based on the lower and upper bounds of the current physical domain, the coordinates of each sample point are mapped to [0,1] through a linear transformation. d The normalized space, where d is the dimension of the design variables.

[0070] Specifically, in this embodiment, a set of well-filled sample points is first generated within the current physical design domain using a Sobol quasi-random sequence. Taking ship hull optimization as an example, assuming the current optimization cycle is in its initial stage, the design variable dimension d=8, and the values ​​of the lower bound L and upper bound U of the physical domain are shown in the table, corresponding to the deformation range of 8 shape value points at the bow and stern of the hull, respectively. A low-discrepancy sequence is generated within a d-dimensional unit hypercube using a Sobol sequence, and mapped to the physical domain through an inverse linear transformation, generating a total of n=400 initial sample points. The Sobol sequence has good space-filling uniformity and low projection correlation, enabling comprehensive coverage of the design space with a limited number of samples, laying the foundation for subsequent surrogate model training.

[0071] Secondly, for each generated sample point, its corresponding true objective function value is obtained using numerical simulation tools. In the ship morphology optimization scenario of this embodiment, the objective function is the ship's performance at the Froude number F. n The wave-making drag coefficient C at 0.305 w For each sample point, the deformation scheme of the hull surface is first generated using radial basis function interpolation. Then, the Shipflow solver is called to calculate wave-making resistance. The mesh density is determined to be 5653 elements after independence verification. For sample schemes that do not meet the constraints (such as displacement volume or longitudinal position of the center of buoyancy exceeding the preset allowable range), a penalty function value can be directly assigned or the sample can be discarded. Through the above simulation process, the objective function values ​​corresponding to n valid samples are obtained.

[0072] Finally, based on the lower bound L and upper bound U of the current physical domain, the physical coordinates of each sample point are mapped to [0,1] through a linear transformation. d The normalized space. For the j-th dimension of the i-th sample point, the formula for calculating the normalized coordinates is:

[0073]

[0074] As a result, the coordinates of all samples are uniformly scaled to the same [0,1] interval, eliminating the adverse effects of differences in the dimensions and value ranges of each variable, ensuring that subsequent surrogate model training, signal calculation and hierarchical decision-making are carried out on the same scale, forming a unified data basis for each subsequent processing step.

[0075] In one embodiment, the step of constructing and training an ensemble surrogate model, which includes at least two different types of regression models, and obtaining the predicted mean and predicted standard deviation of any point in the normalized space through resampling training and weighted fusion based on out-of-bag prediction evaluation, includes:

[0076] Each regression model is trained multiple times using Bootstrap resampling. Each training iteration yields a sub-model, and the unsampled samples constitute the out-of-bag validation set for that training iteration.

[0077] The out-of-bag determination coefficients are calculated using the out-of-bag predicted values ​​of each regression model, and the fusion weights of each regression model are determined based on the out-of-bag determination coefficients.

[0078] The predicted values ​​of all sub-models of each regression model are weighted and averaged according to the fusion weights to obtain the predicted mean, and the predicted standard deviation is obtained after calibration based on the dispersion of the predicted values ​​of each sub-model.

[0079] Specifically, in this embodiment, the construction and training process of the integrated agent model in step S2 is implemented as follows:

[0080] First, each regression model undergoes multiple bootstrap resampling training iterations. This embodiment employs three heterogeneous regression models: XGBoost, Stochastic Fourier Feature Ridge Regression (RFF-Ridge), and Support Vector Regression (SVR). For each model, N_bag = 50 bootstrap resampling iterations are performed independently: each iteration randomly selects n samples with replacement from a normalized sample set D containing n samples to form a training subset, with a target value of The training process yields a sub-model; samples not selected in this sampling automatically form the out-of-bag validation set. Bootstrap resampling allows each sub-model to be trained on a different subset of data, increasing the diversity between sub-models and providing a foundation for subsequent ensemble. Each of the three models has its advantages: XGBoost excels at capturing strong nonlinear relationships and multivariate interactions; RFF-Ridge effectively characterizes smooth and periodic structures through Fourier feature mapping; and SVR, based on the principle of structural risk minimization, has good small-sample generalization ability. The three models complement each other synergistically.

[0081] Secondly, the coefficients of determination are calculated using the out-of-bag predictions of each regression model, and the fusion weights are determined based on these coefficients. For each class of model arm∈{xgb, rff, svr}, the coefficients of determination are calculated using the predictions from the out-of-bag validation set samples of each Bootstrap training iteration.

[0082]

[0083] This metric reflects the model's predictive performance on samples that were not used in training and can objectively measure the model's generalization ability.

[0084] For each model Construct an original weight :

[0085]

[0086] And normalize it:

[0087]

[0088] The final fusion prediction value is obtained as follows:

[0089]

[0090] Calculate the original weights w_arm, with ε set to 10. -2 To avoid the division-by-zero problem, the weights are then normalized to obtain the final fusion weights w'_arm. This weighting mechanism assigns higher fusion weights to models with better generalization performance and automatically suppresses poorly performing models.

[0091] Finally, weighted fusion is performed based on the aforementioned fusion weights, outputting the predicted mean and predicted standard deviation. For any input point z in the normalized space, the predicted mean is obtained by weighted averaging of the predicted values ​​of all sub-models in each model. The predicted standard deviation is obtained by calibrating the dispersion of the predicted values ​​of each sub-model at that point. The calibration process uses the empirical distribution of the out-of-bag standardized residuals to scale the original standard deviation, ensuring that the output uncertainty estimate matches the actual error statistical characteristics, providing a reliable basis for the subsequent calculation of the uncertainty calibration error UCE.

[0092] In one embodiment, the integrated proxy model includes an XGBoost model, a stochastic Fourier feature ridge regression model, and a support vector regression model; the resampling training is Bootstrap resampling, which obtains multiple sub-models for each model, calculates the fusion weight of each model using the out-of-bag prediction determination coefficient of each sub-model, and weights the prediction mean based on the fusion weight; the prediction standard deviation is obtained based on the dispersion of the prediction values ​​of each sub-model or after calibration.

[0093] Specifically, in this embodiment, the construction and training process of the integrated agent model is as follows:

[0094] The integrated surrogate model in this embodiment consists of three heterogeneous regression models: XGBoost, Stochastic Fourier Feature Ridge Regression, and Support Vector Regression. The XGBoost model, based on an additive tree structure, captures complex nonlinear relationships by iteratively fitting residuals. The Stochastic Fourier Feature Ridge Regression model performs ridge regression after mapping the input to a high-dimensional stochastic Fourier feature space, excelling at characterizing the smoothness and periodic structure of the objective function. The Support Vector Regression model employs a Gaussian radial basis function kernel and an ε-insensitive loss function, achieving good small-sample generalization ability based on the principle of structural risk minimization. These three models synergistically complement each other from different perspectives, jointly improving the predictive performance of the surrogate model in complex terrain.

[0095] During the training phase, the three models described above undergo multiple Bootstrap resampling training iterations. Each Bootstrap iteration randomly selects an equal number of samples with replacement from the normalized sample set to form a training subset, training a sub-model. The samples not selected form the out-of-bag validation set for that training iteration. Each model type independently completes multiple resampling training iterations, resulting in the corresponding set of sub-models.

[0096] After training, the out-of-bag determination coefficients (ODCs) of each model are calculated using the predicted values ​​from the out-of-bag validation set, objectively measuring the model's generalization ability. The fusion weights of each model are then calculated based on the ODCs, assigning higher weights to models with better generalization performance. During final prediction, for any input point in the normalized space, the prediction mean is obtained by weighting the predicted values ​​of all sub-models according to the fusion weights, while the prediction standard deviation is obtained by calibrating the dispersion of the predicted values ​​of each sub-model, ensuring that the uncertainty estimate matches the actual error distribution.

[0097] In one embodiment, the steps of screening elite samples based on the objective function value of sample points, calculating the boundary signals of the elite samples in each dimension of the normalized space, calculating the uncertainty calibration error of the model based on the predicted mean and the predicted standard deviation as a risk signal, and calculating the gradient direction of the model in each dimension based on the predicted mean as a direction signal, include:

[0098] After sorting the samples in ascending order of the objective function value, the top preset proportion of samples are taken to form an elite set. The minimum and maximum values ​​of the elite set in each dimension are calculated, and it is determined whether the minimum value is lower than the lower edge threshold or whether the maximum value is higher than the upper edge threshold, so as to generate a boundary signal to indicate whether the elite region touches the normalized spatial boundary.

[0099] The standardized residuals of each sample point are constructed using the predicted mean and predicted standard deviation. The range of values ​​of the standardized residuals is divided into multiple stratified intervals. The theoretical coverage probability and actual observation frequency of each stratified interval are calculated respectively, and the absolute values ​​of the difference between the two are summed to obtain a risk signal used to measure the degree of model uncertainty calibration.

[0100] For each dimension, based on the difference between adjacent samples arranged in ascending order along the coordinates of that dimension according to the predicted mean, the average gradient of that dimension is estimated to obtain a direction signal used to indicate the extension direction of the potential superior region.

[0101] Specifically, in this embodiment, the boundary signal is the primary basis for the design space expansion of this application. Its goal is to determine whether the current design domain truncates the high-performance region. First, an elite sample set with better objective function values ​​is extracted from all samples, and then an axis-aligned bounding box is constructed based on the distribution of these samples in the normalized space.

[0102] Assume there is a total The normalized coordinate matrix of the evaluated samples is: The objective function value is Sort by objective function values ​​in ascending order, then take the top [values]. The sample constitutes the elite set :

[0103]

[0104] Find the minimum and maximum values ​​of the elite set in each dimension to obtain the bounding box of the elite region:

[0105]

[0106] Considering that some dimensions may degenerate into extremely narrow intervals, the span is defined as:

[0107]

[0108] in This is the minimum width threshold. A scaling factor is further introduced. By applying a safety margin, we obtain the safety boundary:

[0109]

[0110] If a certain dimension satisfies:

[0111]

[0112]

[0113] This indicates that the elite region is approaching the design domain boundary, and the current design space may be too narrow. In practical implementation, the improved region predicted by the surrogate model at the candidate points can be merged with the observed elite bounding box to reduce the truncation bias caused by relying solely on existing optimal samples. Once the boundary signal is triggered, the controller will prioritize calculating the new target interval based on the safety boundary, and then adjust the expansion intensity in conjunction with the risk quantity.

[0114] Risk signals are used to measure the reliability of uncertainty estimates in surrogate models. This application uses hierarchical calibration error (UCE) as the core indicator, combined with the holdout set determination coefficient. Determine if the current model can be directly used for fit-driven local updates. Assume the surrogate model is... The predicted mean of each sample is The prediction standard deviation is The actual value is The standardized residual z is defined as:

[0115]

[0116] in To prevent division by zero regularization of small quantities. If the uncertainty estimate is reasonable, then It should approximately follow a standard normal distribution. The interval is uniformly divided into Equal width layered area ), for the first For each interval, the theoretical coverage probability (given by the standard normal distribution) and the actual coverage frequency are respectively:

[0117]

[0118] in Let be the cumulative distribution function of the standard normal distribution. UCE is defined as the sum of the absolute values ​​of the differences between the theoretical coverage probability and the actual coverage frequency at each stratum:

[0119]

[0120] A smaller UCE indicates a closer match between the predicted interval and the actual error; a larger UCE suggests that the model may be overconfident or overly conservative. In addition to UCE, this application also uses the OOF (Out-of-Flight) coefficient of determination. As an indicator of structural reliability. When If the model is considered to have a high degree of fit confidence, it can enter the fit-driven branch; otherwise, more conservative boundary and risk rules are preferred.

[0121] To characterize the local variation trend of the current sample set in the directions of each design variable, an average gradient auxiliary quantity is constructed as a direction signal based on the surrogate model prediction value. Let the first... In the round of iterations, the normalized sample matrix is The surrogate model's prediction values ​​on these samples are For the first Given a design variable, sort the samples in ascending order according to the coordinate of that dimension, and denote the sorting index as... The average partial derivative of this dimension can then be estimated by the difference between adjacent samples:

[0122]

[0123] in Indicates the first Adjacent in dimension and with a distance greater than a threshold The set of sample pairs. From this, we obtain the average gradient vector:

[0124]

[0125] Considering that gradient estimation near high residual samples may be unstable, this application also constructs a residual weighted gradient auxiliary quantity based on the residual magnitude. Let the... The residuals of each sample are Then its weight is taken as:

[0126]

[0127] A larger weight indicates a more reliable fit near that sample. Directional signals play two roles in hierarchical decision-making: first, providing gradient direction for local adjustments in the fitting-driven branch; and second, assisting in determining whether the optimal region points outside the boundary in the risk-driven branch, thereby helping the controller determine the appropriate expansion direction and magnitude.

[0128] In one embodiment, the calculation of the boundary signal includes: taking the top k samples with the best objective function values ​​as the elite set, calculating the minimum and maximum values ​​of the elite set in each dimension; when the minimum value in a certain dimension is less than or equal to the lower edge threshold, or the maximum value is greater than or equal to 1 minus the upper edge threshold, it is determined to be an edge; and calculating the safety boundary based on the span of the elite set in each dimension and the preset safety margin.

[0129] The calculation of the uncertainty calibration error includes: constructing a standardized residual using the predicted mean and predicted standard deviation, dividing the value range of the standardized residual into multiple stratified intervals, calculating the theoretical coverage probability and actual coverage frequency of each stratified interval, summing the absolute values ​​of the differences between the two to obtain the stratified calibration error, which serves as the risk signal.

[0130] Specifically, in this embodiment, the boundary signal and risk signal are calculated as follows: When calculating the boundary signal, all evaluated samples are first arranged in ascending order of the objective function value. The top k samples with the best objective are selected as the elite set. The minimum and maximum values ​​of this elite set in the normalized space are calculated dimension by dimension to form an elite axis-aligned bounding box. Lower and upper edge thresholds are set. When the minimum value of a certain dimension is less than or equal to the lower edge threshold, or the maximum value is greater than or equal to 1 minus the upper edge threshold, it is determined that the elite region of that dimension has reached the design domain boundary. Subsequently, the difference between the maximum and minimum values ​​of the elite interval is used as the base span, and a minimum width constraint is superimposed to obtain the effective span. Then, it is expanded outwards to both sides according to a preset safety margin. After trimming, the safety boundary of that dimension is obtained and used for subsequent geometrically driven design space expansion.

[0131] The risk signal uses the Uncertainty Calibration Error (UCE) as its core indicator. Standardized residuals are constructed using the predicted mean, predicted standard deviation, and actual sample values ​​output by the integrated surrogate model. The residual range [-3, 3] is uniformly divided into several equally wide stratified intervals. The theoretical coverage probability of each interval is calculated based on a standard normal distribution. Simultaneously, the actual sample frequency of residual occurrences within each stratified interval is statistically analyzed. The absolute values ​​of the difference between the theoretical probability and the actual frequency for each interval are sequentially accumulated to obtain the stratified calibration error (UCE). A larger UCE value indicates a worse match between the model's uncertainty estimate and the actual error, resulting in higher risk; conversely, a smaller UCE value indicates more reliable uncertainty calibration. Combined with the holdout set determination coefficient R², this constitutes the risk signal, used to determine whether to initiate risk-driven global expansion, thereby improving the reliability and stability of design space adjustments.

[0132] In one embodiment, the step of updating the design space boundary by performing hierarchical decision-making according to a preset priority includes: if the boundary signal indicates that the elite region touches the boundary of the current normalized design space, then geometric expansion is performed based on the safety boundary; otherwise, if the holdout set determination coefficient of the surrogate model satisfies a first condition, then fitting-driven adjustment is performed based on the direction signal; otherwise, if the uncertainty calibration error exceeds a set risk threshold, then risk-driven global expansion is performed.

[0133] The boundary signal is checked. If the elite region in any dimension touches the normalized space boundary, the safety boundary is calculated based on the span of the elite set in each dimension and the preset safety margin. After modulating the expansion amplitude using the uncertainty calibration error, geometric expansion is performed, and the current round of hierarchical decision-making ends.

[0134] If none of the boundary signals are triggered, it is determined whether the retention set determination coefficient meets the preset fitting reliability threshold. If it does, the boundary of each dimension is fine-tuned by a preset step size near the current optimal sample position according to the gradient direction indicated by the direction signal, and the current round of hierarchical decision-making ends.

[0135] If the holdout set determination coefficient does not meet the fitting reliability threshold, then it is determined whether the uncertainty calibration error exceeds the preset risk trigger threshold. If it does, then the global expansion ratio is linearly determined according to the degree to which the uncertainty calibration error exceeds the risk trigger threshold, and risk-driven global expansion is performed.

[0136] Specifically, in this embodiment, the specific execution process of step S4, hierarchical decision-making, is as follows:

[0137] First, check the boundary signal. If any dimensional elite region touches the normalized space boundary, calculate the safety boundary based on the elite span and the preset safety margin, and then perform geometric expansion after using the uncertainty calibration error UCE modulation expansion amplitude to end this round of decision-making.

[0138] If no boundary signals are triggered, it is determined whether the holdout set determination coefficient R²_oof meets the preset fitting reliability threshold. If it does, it indicates that the model fit is reliable. Based on the gradient direction indicated by the direction signal, the boundaries of each dimension are fine-tuned with a preset step size near the current optimal sample position, and the current round of decision-making ends.

[0139] If R²_oof is not satisfied, it is determined whether the UCE exceeds the preset risk trigger threshold. If it does, it indicates that the model uncertainty mismatch is severe. The global expansion ratio is linearly determined according to the degree to which the UCE exceeds the threshold, and risk-driven global spatial expansion is performed to enhance the model calibration and coverage capabilities.

[0140] The specific process is as follows: Figure 2 As shown, after calculating the boundary geometry signal, risk signal, and direction, hierarchical decision-making is performed according to a fixed priority.

[0141] First layer: Boundary geometry update. If the elite region touches an edge in any dimension, the controller directly enters the geometry update branch. At this point, candidate target intervals are first calculated based on the safety boundary, and then the expansion degree is modulated using UCE:

[0142]

[0143]

[0144] in This is the freezing threshold. If the UCE is large, A smaller value indicates a more conservative expansion strategy when the model is not reliable enough; if there is a clear need for expansion in the edge direction, then further expansion is added in the corresponding boundary direction.

[0145]

[0146] This allows for a more relaxed search range on that side. This branch is the main expansion path in the current implementation because the elite sample edges provide direct geometric evidence that the current design space might truncate the preferred region.

[0147] Second layer: Fitting-driven branch. If the boundary signal is not triggered, the structural reliability of the surrogate model is further checked. When the surrogate model is considered to have a good fitting ability, it is then combined with the local gradient direction provided by the direction signal and made small adjustments based on the current optimal sample position to ensure that the optimal region is completely covered by the current boundary.

[0148] The third layer: risk-driven global expansion. If the model has not yet reached a reliable fit level, and If so, a risk-driven global expansion is executed. The expansion ratio is linearly determined by the degree to which the UCE exceeds the threshold.

[0149]

[0150] in and These represent the lower and upper bounds of the expansion ratio, respectively. A higher UCE indicates a more severe mismatch in the current model's uncertainty, requiring a larger exploration range to restore the model's calibration capability.

[0151] Fourth level: No action. If none of the above conditions are met, the current design domain remains unchanged.

[0152] In one embodiment, the step of mapping the updated normalized spatial boundary back to physical space to obtain the expanded design space includes:

[0153] In the normalized space, a minimum width constraint is applied to the updated target interval to prevent interval collapse in any dimension, and a step size constraint is applied to limit the maximum change of the boundary relative to the original boundary in a single update, thus obtaining a valid new normalized interval.

[0154] Based on the lower and upper bounds of the original physical domain in this round, the lower and upper bounds of the normalized new interval are mapped back to the physical space through inverse linear transformation to obtain the expanded design space.

[0155] Specifically, in this embodiment, the updated target interval is first constrained within the normalized space: a minimum width constraint is applied, requiring the span of each dimension interval to be no less than a preset threshold to prevent the search domain from collapsing and degenerating; a step size constraint is applied, limiting the maximum change in a single boundary update relative to the original normalized boundary of this round, ensuring the stability of the space update. Subsequently, based on the lower and upper bounds of the original physical domain of this round, the constrained normalized new interval is mapped back to the physical space through an inverse linear transformation, resulting in an expanded design space with practical engineering significance, which is used for subsequent sampling and optimization.

[0156] Regardless of whether geometric updates, fit-driven fine-tuning, or risk-driven expansion are used, all decisions are first made in the normalized space and then uniformly mapped back to the physical space.

[0157] In Z-space, if the extended interval of the design space is... Therefore, the geometric candidate interval in this round can be regarded as the expanded design space:

[0158]

[0159] This geometric domain may slightly cross the boundary [0,1] in Z-space, therefore a unified legalization process is required:

[0160]

[0161] in To allow for small-scale outward expansion, slight overboundary structures resulting from geometric updates are preserved, preventing complete truncation of intervals.

[0162] Next, in Z space... Apply minimum width and step size constraints to prevent domain collapse or jumps. The minimum width constraint ensures that any dimension satisfies the following after the update:

[0163]

[0164] The step size constraint restricts this update relative to the initial domain. Maximum change:

[0165]

[0166] in Control the maximum range of movement that can be made during a single-domain update to ensure numerical stability of the update.

[0167] After completing the geometric domain synthesis in Z-space, the intervals are mapped back to physical space. This is because there is a one-to-one correspondence between the physical domain boundary and the normalized domain.

[0168]

[0169] The inverse transformation yields the updated physical domain boundary:

[0170]

[0171] To ensure that no unreasonable distortion occurs in the physical scale, additional scaling and minimum physical width constraints are applied during the mapping process, ensuring that the new physical domain remains consistent with the normalized spatial geometry update. The result obtained through the above steps... This is the updated valid search domain.

[0172] In one embodiment, the first condition is that the coefficient of determination R²_oof of the holdout set is ≥0.80; the fitting-driven adjustment includes adjusting the boundary of the position corresponding to the current optimal sample point along each dimension according to the gradient direction indicated by the direction signal.

[0173] Specifically, in this embodiment, the second-level branch fitting-driven adjustment of the hierarchical decision-making process is implemented as follows:

[0174] When the boundary signal is not triggered, the controller first checks the structural reliability of the surrogate model. In this embodiment, the first condition is set as the holdout set determination coefficient R²_oof ≥ 0.80. This threshold selection has a clear statistical meaning: when R²_oof reaches 0.80 or higher, it indicates that the surrogate model can explain more than 80% of the variation in the objective function, the predicted values ​​have a good linear correlation with the true values, and the model's fitting quality and generalization ability have reached a reliable level. Under this condition, the model's characterization of the local shape of the objective function is sufficiently accurate, and the design domain boundary can be fine-tuned based on the directional information it provides.

[0175] The specific adjustment method is as follows: based on the gradient direction indicated by the direction signal, the position corresponding to the current optimal sample point is adjusted along each dimension using a preset small step size. For the j-th design variable, if the direction signal g... j A positive value indicates that the objective function has a decreasing trend along the direction of increase in that dimension. If the current best sample is located in the middle of the dimension or the gradient direction points into the domain, the boundary of that dimension remains unchanged. After adjustment, a step size constraint is also applied to the new normalization interval, limiting the adjustment amplitude of a single dimension to no more than 10% of the current interval width, to ensure the numerical stability and engineering rationality of the adjustment action. This fitting-driven adjustment branch, under the condition that the model is reliable but does not touch the boundary, ensures that the optimal region is completely covered by the current boundary through fine-tuning, avoiding the omission of potential better points due to small offsets.

[0176] In one embodiment, the method is applied to ship hull optimization design, wherein the objective function is a ship resistance performance index, and the design space is the space spanned by the deformation parameters of the hull surface.

[0177] Specifically, in this embodiment, the method of this application is applied to a ship hull optimization design scenario, where the objective function is the ship resistance performance index, and the design space is the space spanned by the deformation parameters of the hull surface. The wave-making resistance optimization of an S60 ship model is described in detail below as an example.

[0178] In this embodiment, the S60 ship model is used as the optimization object, with a length between perpendiculars of 3.048m, a waterline length of 3.101m, a maximum beam of 0.406m, a draft of 0.163m, and a block coefficient of 0.6. The Froude number Fn = 0.305 is selected, and the optimization objective is to minimize the wave-making resistance coefficient Cw. The deformation of the hull surface is achieved using radial basis function interpolation. Eight shape point variables Y1 to Y8 are selected as design variables to control the shape of the bow waterline, bow bilge, stern waterline, and stern bilge, respectively. All shape points deform along the beam direction. The initial value ranges of each variable are shown in Table 1, collectively spanning an 8-dimensional design space. During the optimization process, displacement volume and longitudinal position of the center of buoyancy are used as constraints to ensure that the optimized hull form improves resistance performance while maintaining a constant displacement. The main hull parameters of the S60 are shown in the following table:

[0179] Table 1 Main parameters of S60 ship model

[0180]

[0181] The hull surface is deformed using the Radial Basis Function (RBF) method. Shape points Y1-Y8 are selected to control the surface deformation. Y1 and Y2 control the bow waterline shape, Y3 and Y4 control the bow and bilge shape, Y5 and Y6 control the stern waterline shape, and Y7 and Y8 control the stern and bilge shape. Y1-Y8 are all deformed along the beam direction. Since only the bow and stern sections of the hull are optimized, the hull contour is subject to fixed constraints to ensure that the contour shape does not change during the optimization process. The hull contour constraints are as follows: Figure 3 As shown in Table 2, the surface values ​​and range of variation of the ship model are given in meters (m).

[0182] Table 2 Variable Space Range

[0183] When embedding the method of this application into the ship hull optimization process, initial sample points are generated in the aforementioned 8-dimensional design space using the Sobol sequence. CFD simulation is performed using the Shipflow potential flow solver to obtain the wave-making drag coefficients of each sample. After normalization, an integrated surrogate model composed of XGBoost, RFF-Ridge, and SVR is trained. Subsequently, three signals are calculated: the distribution of elite samples in dimensions Y1 to Y8 is used to determine whether edge contact occurs; the uncertainty calibration quality of the surrogate model is evaluated using UCE; and the local influence trend of each deformation parameter on drag performance is estimated using directional signals. According to the hierarchical decision rule, geometric expansion is performed on dimensions where edge contact occurs, fitting fine-tuning is performed on reliable directions, and global expansion is performed on high-risk states, ultimately obtaining the expanded variable value range. Within the expanded design space, particle swarm optimization is used to obtain an optimized ship hull scheme with a lower wave-making drag coefficient, effectively avoiding the optimal solution truncation problem caused by improper initial variable range settings.

[0184] Table 3 shows the optimal ship form parameters obtained by the PSO and BDSE+PSO methods. It can be seen that the form value points Y4 and Y8 of the two optimal ship types are quite close, while the results for the other form value points differ significantly. In the optimized scheme obtained by PSO, the values ​​of the variables are all upper and lower limits, thus verifying that BDSE can effectively identify the optimal solution distribution at the boundary. For variables Y2, Y3, and Y5-Y7, the final results obtained by the BDSE+PSO method all exceed the original boundaries in Table 2, verifying that the BDSE+PSO method can effectively expand the design space and is more conducive to obtaining a reasonable range of design variable values.

[0185] Table 3 Comparison of Optimization Parameters

[0186]

[0187] Figure 4 as well as Figure 5 This is a comparison of the transverse and longitudinal hull lines between the optimal vessel obtained using the PSO and BDSE+PSO methods and the parent vessel. The transverse hull lines show that the optimal vessel is wider at the bow and stern than the parent vessel, narrowing towards the midships. Overall, the optimized vessel exhibits a slender forehull and a fuller aft hull shape, which, to a certain extent, especially at higher speeds, helps reduce wave-making drag.

[0188] Table 4 Comparison of Performance Parameters

[0189]

[0190] Table 4 compares some hull parameters and total resistance of the optimized ship obtained by the PSO and HSRM methods with the parent ship under the condition of Frudd number Fn=0.305. This application uses Star-ccm+ software to calculate the total resistance Rt and obtains a comparison diagram of the hull surface pressure and waveform. As shown in Table 4, the optimized ship obtained by the PSO and BDSE+PSO methods has the same displacement as the parent ship, but the wetted surface area is slightly increased. The total resistance of the optimized ship obtained by the PSO method is reduced by 2.380% compared to the parent ship; the total resistance of the optimized ship obtained by the BDSE+PSO method is reduced by 2.800%, showing better optimization results. Figure 6 and Figure 7 It can be seen that the PSO and HSRM methods improve wave-making by changing the shape of the hull (especially the bow) and generating favorable vortex structures in the flow field, thereby reducing wave-making drag and total drag, and improving wave-making drag performance and total drag performance.

[0191] It should be noted that, in this application, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0192] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0193] It should be particularly noted that, through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, or of course, by hardware. Based on this understanding, the above technical solutions, in essence or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0194] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A boundary-aware, space-adaptive expansion method for ship morphology optimization design, characterized in that, Includes the following steps: Step S1: Generate sample points within the current ship type physical design domain, calculate the objective function value corresponding to each sample point, and map the coordinates of all sample points to the normalized space; Step S2: Construct and train an integrated surrogate model, which includes at least two different types of regression models. Through resampling training and weighted fusion based on out-of-bag prediction evaluation, the predicted mean and predicted standard deviation of any point in the normalized space are obtained. Step S3: Select elite samples based on the objective function values ​​of the sample points, calculate the boundary signals of the elite samples in each dimension of the normalized space; calculate the uncertainty calibration error of the model based on the predicted mean and the predicted standard deviation, as a risk signal; and calculate the gradient direction of the model in each dimension based on the predicted mean, as a direction signal. Step S4: Make hierarchical decisions according to preset priorities to update the design space boundary. If the boundary signal indicates that the elite region touches the boundary of the current normalized design space, then perform geometric expansion based on the safety boundary. Otherwise, if the holdout set determination coefficient of the surrogate model satisfies the first condition, then a fitting-driven adjustment is performed based on the direction signal; if the uncertainty calibration error exceeds a set risk threshold, then a risk-driven global expansion is performed. Step S5: Map the updated normalized spatial boundary back to the physical space to obtain the expanded design space.

2. The method according to claim 1, characterized in that, The steps of generating sample points within the current ship type physical design domain, calculating the objective function value corresponding to each sample point, and mapping the coordinates of all sample points to the normalized space include: A set of space-filling sample points is generated in the physical domain of the current design space using a quasi-random sequence; For the sample points, the corresponding true objective function values ​​are obtained through numerical simulation; Based on the lower and upper bounds of the current physical domain, the coordinates of each sample point are mapped to [0,1] through a linear transformation. d The normalized space, where d is the dimension of the design variables.

3. The method according to claim 1, characterized in that, The step of constructing and training an ensemble surrogate model, which includes at least two different types of regression models, and obtaining the predicted mean and predicted standard deviation of any point in the normalized space through resampling training and weighted fusion based on out-of-bag prediction evaluation, includes: Each regression model is trained multiple times using Bootstrap resampling. Each training iteration yields a sub-model, and the unsampled samples constitute the out-of-bag validation set for that training iteration. The out-of-bag determination coefficients are calculated using the out-of-bag predicted values ​​of each regression model, and the fusion weights of each regression model are determined based on the out-of-bag determination coefficients. The predicted values ​​of all sub-models of each regression model are weighted and averaged according to the fusion weights to obtain the predicted mean, and the predicted standard deviation is obtained after calibration based on the dispersion of the predicted values ​​of each sub-model.

4. The method according to claim 1, characterized in that, The integrated proxy model includes an XGBoost model, a stochastic Fourier feature ridge regression model, and a support vector regression model; the resampling training is Bootstrap resampling, which obtains multiple sub-models for each model, calculates the fusion weight of each model using the out-of-bag prediction determination coefficient of each sub-model, and weights the prediction mean based on the fusion weight; the prediction standard deviation is obtained based on the dispersion of the prediction values ​​of each sub-model or after calibration.

5. The method according to claim 1, characterized in that, The process involves selecting elite samples based on the objective function values ​​of the sample points, calculating the boundary signals of the elite samples in each dimension of the normalized space, and calculating the uncertainty calibration error of the model based on the predicted mean and predicted standard deviation, which serves as a risk signal. The step of calculating the gradient direction of the model in each dimension based on the predicted mean, as a direction signal, includes: After sorting the samples in ascending order of the objective function value, the top preset proportion of samples are taken to form an elite set. The minimum and maximum values ​​of the elite set in each dimension are calculated, and it is determined whether the minimum value is lower than the lower edge threshold or whether the maximum value is higher than the upper edge threshold, so as to generate a boundary signal to indicate whether the elite region touches the normalized spatial boundary. The standardized residuals of each sample point are constructed using the predicted mean and predicted standard deviation. The range of values ​​of the standardized residuals is divided into multiple stratified intervals. The theoretical coverage probability and actual observation frequency of each stratified interval are calculated respectively, and the absolute values ​​of the difference between the two are summed to obtain a risk signal used to measure the degree of model uncertainty calibration. For each dimension, based on the difference between adjacent samples arranged in ascending order along the coordinates of that dimension according to the predicted mean, the average gradient of that dimension is estimated to obtain a direction signal used to indicate the extension direction of the potential superior region.

6. The method according to claim 5, characterized in that, The calculation of the boundary signal includes: taking the top k samples with the best objective function values ​​as the elite set, calculating the minimum and maximum values ​​of the elite set in each dimension; when the minimum value in a certain dimension is less than or equal to the lower edge threshold, or the maximum value is greater than or equal to 1 minus the upper edge threshold, it is determined to be an edge; and calculating the safety boundary based on the span of the elite set in each dimension and the preset safety margin. The calculation of the uncertainty calibration error includes: constructing a standardized residual using the predicted mean and predicted standard deviation, dividing the value range of the standardized residual into multiple stratified intervals, calculating the theoretical coverage probability and actual coverage frequency of each stratified interval, summing the absolute values ​​of the differences between the two to obtain the stratified calibration error, which serves as the risk signal.

7. The method according to claim 1, characterized in that, The design space boundary is updated by making hierarchical decisions according to preset priorities. If the boundary signal indicates that the elite region touches the boundary of the current normalized design space, geometric expansion is performed based on the safety boundary. Otherwise, if the holdout set determination coefficient of the surrogate model satisfies the first condition, then a fitting-driven adjustment is performed based on the direction signal; If the uncertainty calibration error exceeds a set risk threshold, a risk-driven global expansion step is performed, including: The boundary signal is checked. If the elite region in any dimension touches the normalized space boundary, the safety boundary is calculated based on the span of the elite set in each dimension and the preset safety margin. After modulating the expansion amplitude using the uncertainty calibration error, geometric expansion is performed, and the current round of hierarchical decision-making ends. If none of the boundary signals are triggered, it is determined whether the retention set determination coefficient meets the preset fitting reliability threshold. If it does, the boundary of each dimension is fine-tuned by a preset step size near the current optimal sample position according to the gradient direction indicated by the direction signal, and the current round of hierarchical decision-making ends. If the holdout set determination coefficient does not meet the fitting reliability threshold, then it is determined whether the uncertainty calibration error exceeds the preset risk trigger threshold. If it does, then the global expansion ratio is linearly determined according to the degree to which the uncertainty calibration error exceeds the risk trigger threshold, and risk-driven global expansion is performed.

8. The method according to claim 1, characterized in that, The step of mapping the updated normalized spatial boundary back to physical space to obtain the expanded design space includes: In the normalized space, a minimum width constraint is applied to the updated target interval to prevent interval collapse in any dimension, and a step size constraint is applied to limit the maximum change of the boundary relative to the original boundary in a single update, thus obtaining a valid new normalized interval. Based on the lower and upper bounds of the original physical domain in this round, the lower and upper bounds of the normalized new interval are mapped back to the physical space through inverse linear transformation to obtain the expanded design space.

9. The method according to claim 1, characterized in that, The first condition is that the coefficient of determination R²_oof of the holdout set is ≥0.80; the fitting-driven adjustment includes adjusting the boundary of the position corresponding to the current optimal sample point along each dimension according to the gradient direction indicated by the direction signal.

10. The method according to claim 1, characterized in that, The method is applied to ship hull optimization design, wherein the objective function is the ship resistance performance index, and the design space is the space spanned by the deformation parameters of the hull surface.

Images