A rotor airfoil design method and system based on an improved bayesian optimization algorithm

Abstract

The application belongs to the field of rotor airfoil design, and discloses a rotor airfoil design method and system based on an improved Bayesian optimization algorithm, which comprises the following steps: constructing a proxy model with initial airfoil parameters and corresponding performance indicators, iteratively selecting new candidate points through a collection function and updating the proxy model until the maximum number of iterations is reached; calculating the global expected regret according to the current observed data set; if the global expected regret is greater than the threshold, resampling the candidate evaluation point set, using the current optimal proxy model, clearing the iteration number, and returning to the step of iteratively selecting new candidate points through a collection function and updating the proxy model; if the global expected regret is less than or equal to the threshold, outputting the optimal airfoil parameters or the Pareto optimal airfoil parameters based on the current observed data set. The application can ensure that high-quality solutions are obtained while significantly improving the utilization efficiency of computing resources.

Description

Technical Field

[0001] This invention belongs to the field of rotor airfoil design, specifically relating to a rotor airfoil design method and system based on an improved Bayesian optimization algorithm. Background Technology

[0002] Rotorcraft, including helicopters, multi-rotor aircraft, and tiltrotor aircraft, play an irreplaceable role in urban air transport, mountain supply transportation, and emergency rescue applications. The rotor airfoil, as the cross-sectional geometry of the rotor blades, is one of the key factors determining the rotor's aerodynamic performance. The geometry of the airfoil directly affects the rotor's lift, drag, and moment characteristics, thus decisively influencing the aircraft's hovering efficiency, cruise performance, noise level, and flight safety.

[0003] Especially for tiltrotor aircraft, the rotor needs to switch between helicopter mode (hovering / low-speed flight) and fixed-wing mode (high-speed cruise), and the rotor must maintain good aerodynamic performance in both of these distinct flight modes. Hovering requires the airfoil to have a high maximum lift coefficient to provide sufficient vertical thrust, while cruise requires the airfoil to have low drag and a high lift-to-drag ratio under high advance ratio conditions to ensure flight efficiency. The requirements for airfoil geometry in these two modes are often contradictory, making the design of tiltrotor rotor airfoils more challenging than that of traditional helicopter rotors.

[0004] Rotor airfoil optimization design typically employs parametric methods (such as function-like / shape function transformation methods) to describe the geometry of the upper and lower surfaces of the airfoil as multiple design variables. Aerodynamic simulation methods (such as the vortex lattice method, the element method, or computational fluid dynamics (CFD) methods based on the Reynolds-averaged Navier-Stokes equations) are then used to evaluate the rotor aerodynamic performance corresponding to each set of design variables. Due to the high computational complexity of aerodynamic simulations, the computation time for a single aerodynamic simulation evaluation can range from several seconds (vortex lattice method) to tens of minutes or even hours (high-precision CFD methods), and the gradient information of the objective function with respect to the design variables is usually difficult to obtain explicitly. Therefore, rotor airfoil optimization design is a typical high-cost black-box optimization problem.

[0005] For high-cost black-box optimization problems such as rotor airfoils, traditional gradient-based optimization methods (such as quasi-Newton methods and conjugate gradient methods) are difficult to apply directly because they rely on the gradient information of the objective function. Even if the gradient is approximated by the adjoint method or finite difference method, the computational cost increases sharply with the increase of the dimension of the design variables, and it is easy to get trapped in the local optimum closest to the initial point and fail to achieve global search.

[0006] While evolutionary algorithms (such as genetic algorithms, particle swarm optimization, and differential evolution) possess global search capabilities and do not rely on gradient information, these population-based optimization methods typically require a large number of objective function evaluations to converge. Taking a typical genetic algorithm as an example, an optimization process with a population size of 50-200 individuals and 100-500 generations requires at least 5,000-100,000 objective function evaluations. Even using the least computationally expensive vortex lattice method (each evaluation takes approximately a few seconds), completing a single airfoil optimization cycle still requires several hours to several days of computation time. If high-precision CFD methods are used, the total computational cost will reach several weeks or even months, which is unacceptable in practical applications with tight engineering design iteration cycles. Although some researchers have attempted to use deep reinforcement learning methods for airfoil design in recent years, these methods require significant simulation resources to build an experience base during the training phase, and their applicability in multi-condition integrated optimization of rotor airfoils still needs further verification.

[0007] Bayesian optimization, as a highly efficient global optimization method for high-cost black-box functions, has gained widespread attention in the field of engineering optimization due to its excellent sample efficiency. The basic framework of Bayesian optimization includes two core components: a surrogate model (usually Gaussian process regression) used to establish a probabilistic prediction model for the black-box objective function; and a collection function (such as expected improvement (EI), improvement probability (PI), upper confidence bound (UCB), etc.) used to select the next evaluation point based on the predicted mean and uncertainty of the surrogate model. By iteratively updating the surrogate model and maximizing the collection function, Bayesian optimization can approximate the global optimum with far fewer evaluations than evolutionary algorithms.

[0008] In the field of aerospace aerodynamic design, Bayesian optimization has been applied to problems such as airfoil optimization and wing planform optimization. For example, the paper "Bayesian Optimization of a Low-Boom Supersonic Wing Planform" applies Bayesian optimization to the planform design of a low-boom supersonic wing, verifying its effectiveness in high-dimensional aerospace design spaces. The paper "Portfolio-Based Bayesian Optimization for Airfoil Design" proposes a Bayesian airfoil optimization method based on a combination strategy, improving the performance limitations of a single acquisition function by combining multiple acquisition functions. These works demonstrate the promising application prospects of Bayesian optimization in aerospace aerodynamic design. However, existing Bayesian optimization methods still have the following key technical limitations when applied to high-dimensional, multi-peak aerodynamic optimization problems such as rotor airfoils.

[0009] First, existing acquisition functions focus on individual candidate points for the next sampling step, and their improvement is measured at the point-level, based on the current optimal point. They lack regional-level evaluation units and stage-level pattern control logic. Taking the most widely used Expected Improvement (EI) as an example, EI considers both the probability and magnitude of improvement, but its exploration intensity is implicitly determined by the posterior variance. In multimodal, high-dimensional problems, it may repeatedly sample near a local extremum and fall into a local optimum. The Improvement Probability (PI) is overly biased towards the probability of immediate improvement, lacking global exploration capability. The Confidence Upper Bound (UCB) balances exploration and utilization using weighted mean and variance, but its weighting coefficients are sensitive and difficult to adjust adaptively. In other words, none of the above acquisition functions can answer the macro-level decision-making question: "Under a limited evaluation budget, when should we stop exploring other regions and shift to local development of the current optimal region?"

[0010] Second, while existing information-theoretic Bayesian optimization methods offer more refined sampling criteria from the perspective of information gain, they still remain at the point-level decision-making level. The entropy search method proposed in the paper "Entropy search for information-efficient global optimization" and the predictive entropy search method proposed in the paper "Predictive Entropy Search for Efficient Global Optimization of Black-box Functions" select sampling points by maximizing the information gain about the globally optimal location. The paper "Generalizing Bayesian Optimization with Decision-theoretic Entropies" further proposes a unified framework based on decision-theoretic entropy, which can degenerate into various sampling functions such as expected improvement in index (EI), knowledge gradient (KG), or entropy search under different loss function settings. However, the entropy measurement and information gain calculation of the above methods are based on the expected loss of point-level actions, which cannot assess the overall performance differences between different candidate regions. Therefore, they cannot provide a region-level quantitative criterion for the phase switching between global exploration and local development.

[0011] Third, to improve the scalability and convergence quality of Bayesian optimization in high-dimensional problems, researchers have attempted to introduce local search mechanisms. The TuRBO (Trust Region Bayesian Optimization) algorithm proposed in the paper "Scalable Global Optimization via Local Bayesian Optimization" achieves superior performance to standard Bayesian optimization in high-dimensional problems by maintaining multiple local trust regions in the design space and employing an implicit multi-armed gambling machine strategy to allocate the evaluation budget. The paper "A framework of global exploration and local exploitation using surrogates for expensive optimization" proposes a surrogate model framework combining global exploration and local exploitation, using a global Gaussian process to guide global exploration and a local radial basis function network to guide local exploitation. However, the trust region expansion and contraction rules of the TuRBO algorithm are based on a heuristic strategy of continuous success / failure counting, which lacks a unified indicator to theoretically quantify the difference in expected returns between "continuing global exploration" and "switching to local development". Other proxy model frameworks require maintaining two proxy models, one global and one local, at the same time, which increases the complexity of model management, and the switching between global and local stages still depends on empirical parameter settings.

[0012] Fourth, in the engineering practice of rotor airfoil optimization, there is a common understanding in the industry that to achieve better local convergence quality, it is necessary to superimpose gradient-based local optimizers (such as trust region methods) or introduce complex multi-stage hybrid strategies in addition to Bayesian optimization. While these hybrid methods can indeed improve convergence quality in certain scenarios, they generally suffer from the following problems: they require the design of numerous empirical parameters and complex stage switching rules, making parameter tuning difficult; they lack unified and interpretable theoretical metrics to support the stage division between global exploration and local development; and the additional local optimizers increase the algorithm's implementation complexity and computational overhead. In rotor airfoil optimization, the complex parameter tuning requirements of these hybrid strategies and the lack of theoretically guided stage divisions not only increase the uncertainty of the design process but also make it difficult to transfer and apply optimization methods between different operating conditions (such as hovering and cruising).

[0013] In summary, in rotor airfoil aerodynamic optimization design, evolutionary algorithms suffer from excessive computational costs due to the large number of evaluations required. Standard Bayesian optimization, lacking a region-level exploration-exploitation switching criterion, is prone to getting stuck in local optima or experiencing slow convergence. Information-theoretic Bayesian optimization methods remain at the point-level decision-making level and cannot provide region-level switching indices. Existing local trust region methods and global-local hybrid frameworks offer improvements but lack a unified theoretical basis for switching and are complex to implement. In other words, current technologies lack a simple, principled method that can adaptively switch between global exploration and local exploitation based on calculable region-level quantitative indices without introducing additional complex components, thereby maximizing rotor airfoil aerodynamic performance within a limited aerodynamic simulation evaluation budget. This is particularly problematic in applications with multiple flight modes (hovering, cruise), such as tiltrotor aircraft, where the objective function landscape for airfoil performance varies significantly under different operating conditions. These technical deficiencies further restrict the efficiency and final design quality of rotor airfoil optimization design. Summary of the Invention

[0014] The purpose of this invention is to provide a rotor airfoil design method and system based on an improved Bayesian optimization algorithm, which significantly improves the utilization efficiency of computing resources while ensuring the acquisition of high-quality solutions.

[0015] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0016] Firstly, a rotor airfoil design method based on an improved Bayesian optimization algorithm is provided, comprising the following steps:

[0017] A surrogate model is constructed using the initial airfoil parameters and corresponding performance indices. New candidate points are selected iteratively through a data acquisition function, and the surrogate model is updated until the maximum number of iterations is reached.

[0018] Calculate the global expected regret based on the currently observed dataset;

[0019] If the global expected regret is greater than the threshold, then resample the candidate evaluation point set, use the current best surrogate model, reset the iteration count to zero, and return to the iteration to select new candidate points through the collection function and update the surrogate model;

[0020] If the global expected regret is less than or equal to the threshold, then the optimal airfoil parameters or Pareto optimal airfoil parameters are output based on the currently observed dataset.

[0021] Several alternative methods are provided below, but they are not intended as additional limitations on the overall solution above. They are merely further additions or optimizations. Provided there are no technical or logical contradictions, each alternative method can be combined individually with respect to the overall solution above, or multiple alternative methods can be combined with each other.

[0022] Preferably, the airfoil parameters are the CST parameterization coefficients of the upper and lower surfaces of the airfoil.

[0023] Preferably, the performance index is a weighted combination of airfoil efficiency improvement and thrust relative error.

[0024] Preferably, the global expected regret is defined as the potential loss when moving from the current convex region to other search regions, where the current convex region is determined by the currently observed dataset, and other search regions are regions in the design space other than the current convex region.

[0025] Preferably, the calculation of the global expected regret includes:

[0026] Construct a convex region based on the best observation point in the currently observed dataset;

[0027] A set of support points is sampled within the design space, and function values ​​are estimated at the support points using the Monte Carlo method.

[0028] Based on the function value estimation of the support points located within the convex region, the expected performance within the region is calculated, including the mean and standard deviation of the expected performance within the region.

[0029] The global expected regret is obtained by multiplying the improvement probability and the improvement magnitude, where the improvement magnitude is the difference between the minimum estimated function value of the support point outside the convex region and the mean expected performance within the region, and the improvement probability is the cumulative distribution function value of the standard normal distribution of the ratio of the improvement magnitude to the standard deviation of the expected performance within the region.

[0030] Preferably, the construction of the convex region based on the best observation point in the currently observed dataset includes:

[0031] Centered on the optimal observation point, based on the radius Define a convex region, where the radius is... Determined adaptively based on the design space boundary:

[0032]

[0033] In the formula, The upper limit of the absolute radius, This is a relative proportionality coefficient. To design the dimensions of the space, For dimensional indexing, Represents the current optimal solution's i-th The value of dimension, Indicates the first The lower bound of the dimensional design space. Indicates the first The upper bound of the dimensional design space.

[0034] Preferably, the function value estimation at the support point using the Monte Carlo method includes estimating the function value using a nearest neighbor-based weighted average method, specifically:

[0035] Take the data from the currently observed dataset that is closest to the support point. One observation point;

[0036] Calculate the distance between each observation point and the support point, and take the reciprocal of the distance as the weight of the observation point;

[0037] Based on the weight of the observation points, The performance indicators of each observation point are weighted and averaged, and the weighted average is used as the function value estimate of the support point.

[0038] Preferably, the expected performance within the calculation region includes the mean of the expected performance within the region and the standard deviation of the expected performance within the region, including:

[0039]

[0040]

[0041] In the formula, The mean expected performance within the region. The standard deviation of expected performance within the region. It is a convex region. For observation points within a convex region, Indicates the number of Monte Carlo samples. Indicates the first During the Monte Carlo sampling, the observation points were... Function value estimation, For the support point set, For variance, For observation point Performance index values, Expressing expectations, This represents the intersection of the set of support points and the convex region.

[0042] Preferably, the global expected regret formula is expressed as follows:

[0043]

[0044] In the formula, This indicates a general expectation and regret. Indicates the first During the Monte Carlo sampling, the support points were... Function value estimation, intermediate parameters , and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively. This indicates the area where the support points are concentrated, excluding the convex region.

[0045] Secondly, a rotor airfoil design system based on an improved Bayesian optimization algorithm is provided, including:

[0046] The surrogate model initialization and update module is used to construct a surrogate model with initial airfoil parameters and corresponding performance indicators. Iteratively, new candidate points are selected through the acquisition function and the surrogate model is updated until the maximum number of iterations is reached.

[0047] The expected regret calculation module is used to calculate the global expected regret based on the currently observed dataset; if the global expected regret is greater than the threshold, it proceeds to the continued exploration module; otherwise, it proceeds to the fine-tuning module.

[0048] The Continuing Exploration module is used to resample the candidate evaluation point set, using the current optimal surrogate model, resetting the iteration count to zero, and returning to the iteration to select new candidate points and update the surrogate model through the acquisition function until the maximum number of iterations is reached, after which the expected regret calculation module is entered.

[0049] The fine-grained development module is used to output the optimal airfoil parameters or Pareto optimal airfoil parameters based on the currently observed dataset.

[0050] This invention provides a rotor airfoil design method and system based on an improved Bayesian optimization algorithm. Addressing the issues of high cost of single-simulation evaluation and the lack of regional-level exploration-development switching criteria in existing Bayesian optimization acquisition strategies, this invention proposes an adaptive acquisition strategy based on expected regret, building upon a Gaussian process surrogate model and acquisition function framework. This strategy constructs a convex region around the current posterior optimal solution, utilizes Monte Carlo sampling and nearest-neighbor weighted estimation to build a regional-level expected regret index, and adaptively switches between global exploration and local development modes based on this index. In a tiltrotor aircraft rotor airfoil optimization example, this method achieves a 5.93% aerodynamic efficiency improvement under cruise conditions and a 51.66% aerodynamic efficiency improvement under hover conditions, with total computation time only 25%–30% of traditional methods. Attached Figure Description

[0051] Figure 1 This is a flowchart of the rotor airfoil design method based on the improved Bayesian optimization algorithm of the present invention;

[0052] Figure 2 This is a schematic diagram of the optimization iteration process of the Rastrigin test function in Experiment 1 of this invention;

[0053] Figure 3 This is a schematic diagram of the baseline rotor model structure in Experiment 2 of this invention;

[0054] Figure 4 This is a schematic diagram of the rotor airfoil optimization iteration process of the tiltrotor aircraft under cruise conditions in Experiment 2 of this invention;

[0055] Figure 5 This is a schematic diagram of the rotor airfoil optimization iteration process of the tiltrotor aircraft under hovering conditions in Experiment 2 of this invention;

[0056] Figure 6 This is a comparison diagram of the rotor airfoil geometry before and after optimization under cruise conditions in Experiment 2 of this invention;

[0057] Figure 7 This is a comparison diagram of the rotor airfoil geometry before and after optimization under the hovering condition in Experiment 2 of this invention. Detailed Implementation

[0058] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0059] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to limit the invention.

[0060] In rotor airfoil aerodynamic optimization design, the design variables are the parametric geometric coefficients of the airfoil (such as the polynomial coefficients of the upper and lower surfaces generated by the CST (Class-Shape Transformation) method), and the objective function is the rotor efficiency or comprehensive aerodynamic performance index obtained through aerodynamic simulation (such as the vortex lattice method or the CFD method based on the Reynolds-averaged Navier-Stokes equations). This type of aerodynamic simulation evaluation process is a typical high-cost black-box function, with a single evaluation taking anywhere from a few seconds to tens of minutes, and function gradient information is difficult to obtain. Bayesian optimization, due to its good sample efficiency, has been widely applied in the field of airfoil and wing aerodynamic optimization design. However, existing Bayesian optimization methods still face the following technical problems when applied to rotor airfoil design:

[0061] (1) Lack of quantitative, regional-level criteria for guiding the switching between global exploration and local development. Currently, the common practice in rotor airfoil optimization engineering is to fix the maximum number of iterations or simulation budget, determined by design experience; or to make subjective judgments based on the observation of the current airfoil performance convergence trend. This approach leads to airfoil optimization efficiency and final design quality being highly dependent on the engineer's personal experience, lacking repeatable, theory-driven regional-level decision-making basis. Especially in application scenarios such as tiltrotor aircraft with multiple flight modes (hovering / cruising), the objective function landscape of airfoil performance corresponding to different operating conditions varies significantly, making the experience-based stage division method even more difficult to adapt. The regional expected regret index proposed in this invention formalizes this problem into the expected loss problem of switching between regions for the first time, so that whether to continue global exploration can be quantitatively represented by a computable scalar.

[0062] (2) Existing acquisition strategies struggle to adaptively balance exploration and utilization, easily falling into local optima or experiencing slow convergence under limited budgets. To address this, this invention constructs an expected regret mechanism, directly linking the continuation of global exploration to the anticipated gains (or losses) of leaving the current region: when the expected regret is high, it indicates significant potential for improvement in other regions, and the algorithm maintains or enhances global exploration; when the expected regret is below a threshold, it indicates the current region is highly likely to contain a global optimum, and the algorithm automatically switches to local exploration. This forms an adaptive acquisition strategy driven by expected regret, representing a novel solution to the classic problem of balancing exploration and utilization.

[0063] (3) Existing methods struggle to simultaneously achieve convergence quality and computational efficiency for multimodal complex functions. To address this, this invention introduces a scheme combining region construction, expected regret estimation, and mode switching without altering the surrogate model structure, thus changing the allocation of the evaluation budget between the global and local stages. Experiments show that, for the 8-dimensional Rastrigin function, with the same number of iterations, the optimal function value of the method presented in this invention is significantly lower than that of standard Bayesian optimization based on expected improvement EI and improved probability PI. Simultaneously, the total computation time of the method presented in this invention is only about 25% to 30% of that of the comparative methods. This combination of better solution and shorter computation time makes the aforementioned efficiency improvement directly valuable in the practical engineering scenario of rotor airfoil optimization, given the high computational cost of aerodynamic simulation. It can significantly shorten the design iteration cycle while ensuring the aerodynamic performance of the airfoil.

[0064] (4) In engineering optimization with limited simulation / experiment budgets, there is a lack of a general Bayesian optimization framework that combines algorithmic versatility, simple control, and robust convergence. There is a technical bias in the engineering community: it is believed that to achieve better local convergence quality, complex components such as local gradient optimizers and trust region modules must be added; otherwise, Bayesian optimization alone cannot achieve both global and refined results. In response, the expected regret-driven strategy proposed in this invention is structurally very simple: it does not introduce a new surrogate model or explicitly introduce a local gradient optimizer; it achieves automatic global-local mode switching simply through region construction, index calculation, and threshold decision; experiments show that its local convergence quality is superior to some hybrid schemes that add complex local optimizers. This, to some extent, breaks through the aforementioned technical bias, indicating that the previously held premise of adding a local optimizer is not a necessary condition.

[0065] Specifically, such as Figure 1 As shown, this embodiment provides a rotor airfoil design method based on an improved Bayesian optimization algorithm, including the following steps:

[0066] (1) Construct a surrogate model with the initial airfoil parameters and corresponding performance indicators, iterate through the acquisition function to select new candidate points and update the surrogate model until the maximum number of iterations is reached.

[0067] The Bayesian optimization problem can be generally described as: ,in Design space for the problem, For design variables, general dimensions , Design variables The objective function value is determined by the objective function. In the initial stage of the optimization process, several parameter points are randomly selected within the design space, and their corresponding objective function values ​​are calculated. This serves as the initial sample set, with each sample containing airfoil parameters and their corresponding comprehensive performance index values. Subsequently, an initial surrogate model is constructed based on this sample set to predict the objective function values ​​and uncertainties of new parameter points. Guided by the acquisition function, the next most promising evaluation point is selected, its true objective value is calculated, and it is added to the existing sample set to update the surrogate model. By repeatedly updating the surrogate model and selecting parameter points, Bayesian optimization gradually narrows the search range and eventually converges to the optimal parameter solution. However, in practical calculations, limited by the dimensionality of the problem and computational resources, standard Bayesian optimization methods often struggle to converge stably and quickly to the global optimum.

[0068] Therefore, this embodiment terminates the iterative process at reaching the maximum number of iterations, rather than ultimately converging to the optimal parameter solution. This integrates the overlapping iterative process as part of the overall method. After reaching the maximum number of iterations, an adaptive acquisition strategy based on expected regret is introduced into standard Bayesian optimization. By quantifying the decision risk of region transformation, a theoretical basis for balancing exploration and development is provided. The core of expected regret lies in assessing the potential loss when shifting from the current optimal region to other search regions. When the expected regret is below a preset threshold, the core objective of Bayesian optimization is adjusted from global exploration to refined development of the current optimal region. This strategy can significantly improve the utilization efficiency of computational resources while ensuring the acquisition of high-quality solutions.

[0069] In the specific application of rotor airfoil aerodynamic optimization design, the input and output definitions in this embodiment are as follows:

[0070] Input: (a) Design variable vector ,in and These are the CST parameterization coefficients for the upper and lower surfaces of the airfoil, respectively. (a) The Bayesian order; (b) The range of values ​​for each design variable is determined by airfoil geometric constraints and aerodynamic design experience; (c) The objective function That is, given airfoil parameters The rotor's overall performance index obtained through aerodynamic simulation calculations (including a weighted combination of efficiency improvement and thrust constraints) is a deterministic scalar.

[0071] Output: The optimal combination of CST parameters that achieves the best overall performance and its corresponding optimal airfoil geometry, which can be directly used for rotor blade section design. The objective function evaluation process is as follows: given design variables... Based on the CST parameters, the geometric curves of the upper and lower surfaces of the airfoil are reconstructed. The airfoil section is then applied to the rotor blade model. Aerodynamic performance calculation methods (such as the vortex lattice method or the element method) are used to evaluate the rotor efficiency and thrust, and the comprehensive performance index value is returned. This evaluation process is a high-cost black-box function, and the gradient of the objective function is unavailable.

[0072] (1.1) Proxy model.

[0073] In one embodiment, a Gaussian process is used as a surrogate model within the Bayesian optimization framework. A Gaussian process is a stochastic process whose arbitrary finite-dimensional distribution follows a multivariate normal distribution and can be uniquely determined by a mean function and a covariance function.

[0074] Mean function Defined as:

[0075]

[0076] Indicates at the input point random variables Expected value Let be a random variable representing the objective function value under the posterior distribution of the Gaussian process. Since the Gaussian process surrogate model has uncertainty in its prediction of unobserved points, for candidate sampling points... Its objective function value is modeled as a random variable that follows a normal distribution. ,in and Let be the posterior prediction mean and posterior prediction variance of the Gaussian process, respectively. Represents expectation. Covariance function. Defined as:

[0077]

[0078] when At that time, it describes the function at the input point. variance at location This measures the uncertainty at that point. When At that time, it characterized the function at two different input points. and The value at the location and Covariance between This characterizes the spatial correlation of function values. Therefore, the Gaussian process can be formally represented as:

[0079]

[0080] This represents a Gaussian process. In the Gaussian process model, the covariance function is also called the kernel function.

[0081] In Bayesian optimization, the first step is to use a pre-defined mean function. With covariance function Define the prior distribution of the Gaussian process. This is done after obtaining the observed dataset. Then, by adjusting the hyperparameters in the covariance function This updates the posterior distribution of the Gaussian process, thereby achieving the desired outcome for the unknown objective function. The fitting process is called Gaussian regression. For the first The observation point is the first one in the design space. Each parameter point corresponds to a CST parameterization coefficient on the upper / lower surface of the airfoil. For the first Each observation output corresponds to an input. The true objective function value corresponds to the comprehensive performance index value of the airfoil, i.e. , This represents the total number of observation points.

[0082] For hyperparameters The estimation can be performed using the Maximum A Posterior Estimate (MAP) method, which aims to find the most probable parameter values ​​given the observed data.

[0083]

[0084] in, Hyperparameters obtained using the MAP method The estimated value, This indicates that after obtaining all observations... under the condition of parameters The posterior distribution of . According to Bayes' theorem, it can be transformed into:

[0085]

[0086] Due to edge likelihood With hyperparameters Since it is irrelevant, we only need to maximize the molecular part. Among them, Let be the likelihood function, representing the likelihood given hyperparameters. Data observed under conditions The probability of; The prior distribution of hyperparameters is used to introduce domain knowledge. If we assume it to be a uniform distribution, that is, for all hyperparameters... If equal probabilities are assigned, the maximum a posteriori estimate degenerates into the maximum likelihood estimate (MLE):

[0087]

[0088] Hyperparameters obtained using the MLE method The estimated value. In a Gaussian process, from the observation point The multidimensional random vector formed by the corresponding random variables It follows a multivariate normal distribution. Its mean vector is... Covariance Matrix for:

[0089]

[0090]

[0091] Observations It can be viewed as a noisy sample of this distribution, whose log-likelihood function... It has the following parsing form:

[0092]

[0093] in It is the variance of the observation noise. It is the identity matrix. Maximizing this log-likelihood function is equivalent to minimizing the following objective function. :

[0094]

[0095] This function relates to hyperparameters Since it is differentiable, gradient-based optimization algorithms (such as quasi-Newton methods) can be used to efficiently solve for its optimal solution in implementation. .

[0096] In Bayesian optimization, a well-trained Gaussian process surrogate model can simultaneously optimize the objective function. The values ​​and posterior distribution of the unknown point are used for prediction. Its posterior predicted distribution is:

[0097]

[0098] This distribution represents the situation before the knowledge is known. Observation data Under the condition of unknown points True function value The probability distributions of different values ​​are taken, and their mean and variance are used as the predicted value and uncertainty measure of that point, respectively.

[0099] (1.2) Acquisition function.

[0100] Bayesian optimization methods rely on the posterior distribution of the objective function provided by the surrogate model to guide the selection of the next sampling point. A common approach is to design a sampling function. This function performs sampling on each candidate sampling point. An evaluation is conducted to determine whether sampling is worthwhile. Generally, an effective sampling function needs to balance exploration and utilization: lower scores should be assigned to areas where sampling points have already been fully explored; higher attention should be paid to areas with high prediction uncertainty to improve the model's global accuracy; at the same time, higher scores should be assigned to areas with larger prediction means for maximization problems, and higher weights should be assigned to areas with smaller prediction means for minimization problems, thereby effectively guiding the search towards the possible optimal region.

[0101] There are currently various data acquisition functions available. The following is a brief introduction to typical examples related to this invention.

[0102] A. Expected Improvement (EI): Expected Improvement is a commonly used sampling function to balance exploration and exploitation in Bayesian optimization. Its core idea is to evaluate the expected improvement that new sampling points might bring, based on existing points.

[0103] Assuming the current process has already been carried out After sampling, the known optimal function value is:

[0104]

[0105] For any candidate point Under the Gaussian process surrogate model, the objective function value at this point is modeled as a random variable. It follows a Gaussian process posterior distribution ,in and These are the posterior prediction mean and posterior prediction variance of the Gaussian process, respectively.

[0106] Define the improvement amount at this point for:

[0107]

[0108] Expected improvements This is the expected amount of improvement:

[0109]

[0110] Within the framework of Gaussian processes, Analytical solutions with closed-form:

[0111]

[0112] in , Let be the probability density function of the standard normal distribution. Let be the cumulative distribution function of the standard normal distribution. and It is a normal distribution The mean and standard deviation.

[0113] B. Confidence Bound: The confidence boundary is a classic data collection strategy used to balance exploration and exploitation. Its core idea is that when modeling the objective function using a surrogate model, in addition to focusing on the predicted mean (the best estimate of the function value) provided by the model, the uncertainty of the prediction itself (variance or standard deviation) must also be fully considered. The confidence boundary strategy combines the mean and uncertainty through a simple linear weighting formula to form a unified collection score.

[0114] The upper confidence interval boundary is defined as:

[0115]

[0116] The lower confidence interval boundary is defined as:

[0117]

[0118] in This is a hyperparameter greater than 0, used to balance exploration and exploitation. When the data is large, the acquisition function tends to explore regions with high uncertainty. In hours, the data collection function tends to utilize known high-mean regions.

[0119] C. Probability of Improvement (PI): PI is a sampling function based on improved probability. Its core idea is: after determining the current optimal function value in the existing point set, calculate the value of each candidate point using a Gaussian process model. The algorithm calculates the probability that the function value exceeds the optimal value. Then, it selects the point with the highest probability of improvement as the target for the next evaluation. The motivation behind this strategy is to directly find the point most likely to bring immediate benefit in the next evaluation, but its purely probability-driven nature also makes it prone to getting bogged down in local searches, fine-tuning around existing optimal values.

[0120] The PI acquisition function is defined as:

[0121]

[0122] in The current optimal function value is given by the existing point set. This is a very small noise term to prevent division by zero.

[0123] (2) Calculate the global expected regret based on the currently observed dataset. If the global expected regret is greater than a preset threshold, it is determined that there is still significant improvement potential outside the current optimal region, and global exploration needs to continue. At this time, the following operations are performed: First, redetermine the current optimal sampling point based on the currently updated observation dataset. And its corresponding convex region, and regenerate the Monte Carlo support point set within the design space to update the estimation basis of the expected regret (this step is called...). Figure 1The candidate evaluation point set is resampled as shown in the figure, and the test point set is updated. Then, the iteration count is reset to zero, and the standard Bayesian optimization iteration loop in step (1) is returned based on the inheritance of the previous surrogate model. That is, the calculation of the maximum value of the acquisition function, the calculation of the objective function value, the addition of the new evaluation points to the observation dataset, and the updating of the surrogate model are continued to start the next round of global exploration. If the global expected regret is less than or equal to the threshold, it indicates that the potential improvement space outside the convex region is very limited. The global exploration mode is automatically switched to the local development mode, and the optimal airfoil parameter or Pareto optimal airfoil parameter is output based on the currently observed dataset. Corresponding to the objective function of the comprehensive performance index in this embodiment, this embodiment finally outputs an optimal airfoil parameter.

[0124] To ensure consistency of notation in subsequent formula derivations, the main notations used in this invention are first uniformly agreed upon: For design space The general design variable vector in; For the observed dataset The first in One sampling point; The sampling point with the best objective function value in the currently observed dataset (the current optimal solution); The first in Monte Carlo sampling One support point.

[0125] Within the Bayesian optimization framework, this embodiment proposes an adaptive acquisition strategy based on expected regret. This method provides a theoretical basis for balancing exploration and development by quantifying the decision risk of region transformation. The core of expected regret lies in evaluating the potential loss when shifting from the current optimal region to other search regions, where the current optimal region is determined by the currently observed dataset, and other search regions are regions in the design space other than the current optimal region. Specifically, let... The global expected regret is defined as:

[0126]

[0127] in This represents the estimated optimal point within the current convex region. Indicates sampling points outside the convex region. Describes the estimated optimal point under the posterior distribution of a Gaussian process. The random variable corresponding to the objective function value, Represents the sampling points of the posterior distribution of a Gaussian process. The random variable corresponding to the objective function value.

[0128] The specific calculation process of the global expected regret is as follows: First, determine the currently observed dataset. The optimal sampling point for the objective function value (For the minimization problem), and construct a convex region centered on that point. Its radius Determined adaptively based on the search space boundary:

[0129]

[0130] in Indicates the first The lower bound of the dimensional design variable search space. Indicates the first The upper bound of the dimensional design variable search space, i.e., the th Dimensional design variables The range of values ​​is , Indicates the current optimal solution The The value of dimensionality. In rotor airfoil optimization, and Corresponding to the first The lower and upper limits of the allowed values ​​for each CST geometric parameter. The upper limit of the absolute radius, This is a relative proportionality coefficient. Therefore, the convex region... Defined as:

[0131]

[0132] in For the entire search space (design space).

[0133] Subsequently, a set of support points is sampled within the design space, and function values ​​are estimated at the support points using the Monte Carlo method to calculate the expected value of the performance difference between the inside and outside of the region. In the calculation of the expected value, a weighted average method based on nearest neighbors is used for function value estimation; specifically, the nearest neighbor to the support point in the currently observed dataset is selected. Calculate the distance between each observation point and the support point, and take the reciprocal of the distance as the weight of the observation point; use the weight of the observation point to... The performance indicators of each observation point are weighted and averaged, and this weighted average is used as the function value estimate for the support point set. any point in Its function value is estimated by the following formula:

[0134]

[0135] in As a support point Function value estimation, For observation point The comprehensive performance index value, This indicates the concentration of currently observed data and support points. Recent Each observation point, weight Based on the reciprocal of the distance, Small positive numbers are used to avoid numerical instability. The expected performance within the region is estimated using Monte Carlo sampling:

[0136]

[0137]

[0138] in, The mean expected performance within the region. The standard deviation of expected performance within the region. Indicates the number of Monte Carlo samples, where Indicates the first During the Monte Carlo sampling, the observation points were... (The formula calculation takes the intersection of the support point and the convex region, therefore the observation point) Simultaneously serving as a support point Estimation of the function value of ), For variance, For observation point The overall performance index value. The final calculation of expected regret incorporates the product of the probability of improvement and the magnitude of improvement:

[0139]

[0140] in This indicates a general expectation and regret. Indicates the first During the Monte Carlo sampling, the support points were... Function value estimation, intermediate parameters , and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively. This indicates the area where the support points are concentrated, excluding the convex region.

[0141] The mathematical structure of the above formula can be decomposed into a combination of the probability of improvement and the magnitude of improvement. Wherein, To improve the probability, the cumulative distribution function value of the standard normal distribution representing the ratio of the improvement magnitude to the standard deviation of the expected performance within the region is given. To improve the magnitude, the difference between the estimated minimum function value of the support point outside the convex region and the mean expected performance within the region is expressed as... This is an uncertainty correction term. The theoretical advantage of this method lies in providing a quantitative criterion for the trade-off between exploration and development. When A large value indicates the existence of unexplored regions with significant improvement potential, and the algorithm should continue exploring; when... When the value is small, it means that the current region is likely to contain the global optimum, and the process should proceed to the development phase. Specifically, let the preset threshold be... The algorithm is based on the global expectation regret With threshold The comparison results are used to make a mode switching decision: when When this occurs, it indicates that there are still unexplored regions with significant improvement potential outside the current optimal region. Sampling points outside the convex region may yield objective function values ​​better than the current optimal value, and the algorithm should continue to maintain the global exploration mode; when When this occurs, it indicates that the potential improvement space outside the convex region is very limited, and the current optimal region is very likely to contain the global optimal solution. The algorithm automatically switches from global exploration mode to local development mode, focusing the subsequent evaluation budget on the fine-grained search of the current optimal region.

[0142] Experiment 1: Validation of the effectiveness of the improved Bayesian optimization method:

[0143] To verify the effectiveness of the improved Bayesian optimization method proposed in this invention, numerical experiments were conducted using the classic 8-dimensional Rastrigin test function. This function, due to its numerous local extrema, serves as a benchmark problem for evaluating the performance of global optimization algorithms. In the experiments, the improved Bayesian optimization method proposed in this invention uses the expected improvement (EI) as the basic acquisition function to generate candidate evaluation points in both the global exploration and local development phases. It should be noted that the key aspect of this invention lies in the adaptive global-local strategy switching mechanism based on expected regret. This mechanism is independent of the choice of specific acquisition function; that is, the method of this invention is compatible with various mainstream acquisition functions such as expected improvement (EI), improvement probability (PI), and confidence upper bound (UCB).

[0144] This experiment compares the improved Bayesian optimization method described above with two standard Bayesian optimization methods that employ typical acquisition functions: one using a standard method that improves the expected efficiency (EI), and the other using a standard method that improves the probability (PI). The comparison of these three methods aims to highlight the performance improvement brought about by the adaptive policy switching mechanism of this invention. Figure 2 The iterative convergence curves of the three methods during the optimization process are shown, clearly illustrating the trend of the quality of the optimal solution as the number of iterations increases. Table 1 details the computation time required to complete the same number of iterations, providing a quantitative basis for evaluating the efficiency of the algorithms.

[0145] Table 1. Optimization results and optimization time of the Rastrigin test function

[0146]

[0147] according to Figure 2 As shown in Table 1, the test cases using the improved Bayesian optimization algorithm significantly outperform the standard Bayesian optimization method, which uses the expected improvement EI or improved probability PI as the acquisition function, in terms of solution quality. This improved Bayesian optimization algorithm achieves adaptive search by monitoring the global expected regret: when the expected regret value of the stage optimization result is lower than a preset threshold, the algorithm automatically shifts the search strategy from global exploration to refined development of the current optimal region. This dynamic switching mechanism maintains the breadth of global exploration while accelerating the convergence process through local reinforcement search, enabling the current solution to quickly approach the theoretical optimum. In terms of computational efficiency, the improved algorithm exhibits a significant advantage. Under the same number of iterations (500 steps), the computation time of the improved Bayesian optimization is only 25%~30% of that of the standard method. This efficiency improvement is mainly due to the adaptive mechanism based on expected regret, which avoids redundant exploration in low-potential regions, and the concentrated search in the local development stage, which reduces unnecessary function evaluation.

[0148] In the 8-dimensional Rastrigin test, this invention achieved an optimal value significantly better than EI / PI and greatly reduced computation time. This proves that the invention not only did not increase the overall burden by introducing new metrics, but also improved the global-local budget allocation. The results of this embodiment fully verify that the improved Bayesian optimization method can not only obtain a better final solution, but also significantly reduce computational resource consumption. Especially in practical engineering applications with limited computational budgets, this method greatly improves the practicality and feasibility of the optimization process while ensuring the quality of the solution, providing an effective solution for complex engineering optimization problems.

[0149] Experiment 2: Validation of the effectiveness of the improved Bayesian optimization method in tiltrotor aircraft rotor airfoil optimization:

[0150] This embodiment constructs a baseline rotor model based on Xrotor's reverse engineering capabilities. The rotor has a diameter of 2.132 meters, and the initial blade section adopts the NACA4412 airfoil (see...). Figure 3 The cruise speed is set at 600 RPM and the hover speed at 1000 RPM, corresponding to the typical flight states of the tiltrotor aircraft in fixed-wing cruise mode and helicopter hover mode, respectively.

[0151] In the optimization task, the baseline airfoil is first parametrically modeled using a shape transformation-like method. Its upper and lower surfaces are represented by independent polynomial functions, resulting in a total of 8 CST geometric parameters, which serve as bounded intervals. The internal design variables. These are used to characterize the aerodynamic characteristics of the airfoil under actual flight conditions.

[0152] Selected at radial position A representative blade section was selected, and the free-flow Mach number and Reynolds number corresponding to this section were calculated based on the rotor design speed and inflow conditions. Subsequently, the rotor aerodynamic performance was evaluated using a rapid rotor performance calculation method, and the corresponding results were returned. The optimization problem solved in this embodiment can be expressed as the following mathematical model: [Solution Object] ;maximize ; Satisfy the constraints , and .in and These are the CST parameters for the upper and lower surfaces of the airfoil, respectively. and These represent the current rotor speed and the maximum rotor speed, respectively. This indicates the rotor efficiency corresponding to the current airfoil. and These represent the rotor thrust values ​​under the current and baseline airfoils, respectively. and These represent the current airfoil thickness and the baseline airfoil thickness, respectively. Indicates the actual radial position. Indicates the rotor radius.

[0153] To improve efficiency while keeping thrust close to the baseline value, this embodiment constructs a performance index by combining the efficiency improvement and the relative thrust error, and uses it as the objective function, the expression of which is as follows:

[0154]

[0155] in and These are the weighting coefficients. This indicates the rotor efficiency corresponding to the baseline airfoil.

[0156] The formula for calculating rotor efficiency during cruise mode is:

[0157]

[0158] in For free flow velocity, Indicates torque, This refers to the rotor angular velocity. However, when the rotor is in a hovering state with zero free-flow velocity ( When running, the above efficiency formula is no longer applicable and needs to be replaced with the following expression:

[0159]

[0160] in The tensile coefficient, The torque coefficient, Indicates air density, Indicates the rotor radius. This indicates the rotor efficiency when hovering.

[0161] The optimization results and time consumption of the improved Bayesian optimization method of this invention under cruise and hovering conditions are shown in Table 2. In cruise mode, the optimization process took 2659 seconds, achieving a 5.93% improvement in aerodynamic efficiency; while in hovering mode, a significant efficiency gain of 51.66% was achieved in just 2355 seconds. The thrust parameters under both conditions were stably controlled near the baseline value, meeting engineering constraints. Figure 4 and Figure 5 The iterative trajectory of the optimization process is illustrated. In the initial stage, the algorithm quickly locates potential regions where the expected regret is below a threshold through global exploration, and then switches to a local refinement stage through an adaptive strategy. This mechanism effectively ensures search efficiency, enabling the algorithm to obtain a relatively optimal solution that meets engineering requirements within 250 iterations, demonstrating the comprehensive advantages of the improved method in terms of convergence speed and solution quality.

[0162] Table 2. Optimization results and optimization time for tiltrotor aircraft rotor airfoils

[0163]

[0164] Comparison of geometry before and after optimization, for example Figure 6 and Figure 7 As shown. Under cruise conditions, the optimized airfoil shifts the maximum thickness location rearward and optimizes the surface curvature distribution. This improvement effectively delays boundary layer separation on the airfoil surface, reduces pressure drag, and thus improves aerodynamic efficiency. Under hover conditions, the optimization mainly involves increasing the leading edge radius and adjusting the upper surface camber, allowing the airflow to better adhere to the airfoil surface, delaying flow separation, improving stall characteristics at high angles of attack, and achieving a significant improvement in aerodynamic efficiency.

[0165] The above engineering examples verify the effectiveness of the improved Bayesian optimization method of the present invention in rotor airfoil aerodynamic optimization design: compared with the traditional Bayesian optimization method, the adaptive stage switching mechanism of the present invention can allocate the limited aerodynamic simulation evaluation budget more efficiently, and significantly shorten the design cycle while ensuring the airfoil design quality.

[0166] In summary, this invention proposes a regional expected regret index, which evaluates the expected loss of leaving the current optimal region and turning to other regions at the regional scale. It measures the expected loss of retaining the current region versus turning to other regions at the regional level, representing a fundamental extension of traditional point-level acquisition functions. It provides a unified and computable criterion for switching between global exploration and local development phases. Furthermore, it constructs an expected regret-driven adaptive acquisition strategy, enabling the algorithm to automatically select between global exploration and local development modes based on the uncertainty of the current surrogate model and the magnitude of the expected regret, without requiring complex empirical rules. It forms a Bayesian optimization framework applicable to various expensive black-box optimization problems, improving local convergence characteristics and reducing implementation complexity without introducing additional complex local optimizers. This invention maintains compatibility with Gaussian process models and mainstream acquisition functions, making it easily portable to various engineering black-box optimization scenarios. Key parameters such as the expected regret threshold are easy to understand and can be flexibly set according to task characteristics.

[0167] Applying the improved Bayesian optimization method described above to rotor airfoil aerodynamic optimization design yields superior airfoil aerodynamic performance with fewer aerodynamic simulation evaluations. In engineering problems where airfoil CST parameters are the design variables and rotor efficiency is the optimization objective, the method of this invention effectively avoids redundant simulation evaluations in low-potential airfoil configuration regions through adaptive global-local strategy switching. This maximizes airfoil aerodynamic performance within a limited computational budget, shortens the rotor airfoil design cycle, and reduces engineering design costs.

[0168] In another embodiment, a rotor airfoil design system based on an improved Bayesian optimization algorithm is provided, comprising:

[0169] The surrogate model initialization and update module is used to construct a surrogate model with initial airfoil parameters and corresponding performance indicators. Iteratively, new candidate points are selected through the acquisition function and the surrogate model is updated until the maximum number of iterations is reached.

[0170] The expected regret calculation module is used to calculate the global expected regret based on the currently observed dataset; if the global expected regret is greater than the threshold, it proceeds to the continued exploration module; otherwise, it proceeds to the fine-tuning module.

[0171] The Continuing Exploration module is used to resample the candidate evaluation point set, using the current optimal surrogate model, resetting the iteration count to zero, and returning to the iteration to select new candidate points and update the surrogate model through the acquisition function until the maximum number of iterations is reached, after which the expected regret calculation module is entered.

[0172] The fine-grained development module is used to output the optimal airfoil parameters or Pareto optimal airfoil parameters based on the currently observed dataset.

[0173] In another embodiment, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the steps of a rotor airfoil design method based on an improved Bayesian optimization algorithm.

[0174] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0175] In another embodiment, a computer device is provided, including a processor and a memory storing a plurality of computer instructions, which, when executed by the processor, implement steps of a rotor airfoil design method based on an improved Bayesian optimization algorithm.

[0176] The memory and processor are electrically connected directly or indirectly to enable data transmission or interaction. For example, these components can be electrically connected to each other via one or more communication buses or signal lines. The memory stores a computer program that can run on the processor, which implements the method of the present invention by running the computer program stored in the memory.

[0177] The memory may be, but is not limited to, Random Access Memory (RAM), Read Only Memory (ROM), Programmable Read-Only Memory (PROM), Erasable Programmable Read-Only Memory (EPROM), Electrically Erasable Programmable Read-Only Memory (EEPROM), etc. The memory stores the program, and the processor executes the program upon receiving an execution instruction.

[0178] The processor may be an integrated circuit chip with data processing capabilities. The aforementioned processor can be a general-purpose processor, including a Central Processing Unit (CPU), a Network Processor (NP), etc. It can implement or execute the methods, steps, and logic block diagrams disclosed in the embodiments of this invention. The general-purpose processor can be a microprocessor or any conventional processor.

[0179] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0180] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.

Claims

1. A rotor airfoil design method based on an improved Bayesian optimization algorithm, characterized in that, Includes the following steps: A surrogate model is constructed using the initial airfoil parameters and corresponding performance indicators. New candidate points are selected iteratively through a data acquisition function, and the surrogate model is updated until the maximum number of iterations is reached. Calculate the global expected regret based on the currently observed dataset; The global expected regret is defined as the potential loss when moving from the current convex region to other search regions, where the current convex region is determined by the currently observed dataset, and other search regions are regions in the design space other than the current convex region. If the global expected regret is greater than the threshold, then resample the candidate evaluation point set, use the current best surrogate model, reset the iteration count to zero, and return to the iteration to select new candidate points through the collection function and update the surrogate model; If the global expected regret is less than or equal to the threshold, then based on the currently observed dataset, the optimal airfoil parameters or Pareto optimal airfoil parameters are output. The calculation of the global expected regret includes: Constructing a convex region based on the optimal observation point in the currently observed dataset includes: centering on the optimal observation point and calculating the radius... Define a convex region, where the radius is... Determined adaptively based on the design space boundary: ； In the formula, The upper limit of the absolute radius, This is a relative proportionality coefficient. To design the dimensions of the space, For dimensional indexing, Represents the current optimal solution's i-th The value of dimension, Indicates the first The lower bound of the dimensional design space. Indicates the first The upper bound of the dimensional design space; A set of support points is sampled within the design space, and function values ​​are estimated at the support points using the Monte Carlo method. Based on the function value estimation of the support points located within the convex region, the expected performance within the region is calculated, including the mean and standard deviation of the expected performance within the region. The global expected regret is obtained by multiplying the improvement probability and the improvement magnitude, where the improvement magnitude is the difference between the minimum estimated function value of the support point outside the convex region and the mean expected performance within the region, and the improvement probability is the cumulative distribution function value of the standard normal distribution of the ratio of the improvement magnitude to the standard deviation of the expected performance within the region.

2. The rotor airfoil design method based on the improved Bayesian optimization algorithm according to claim 1, characterized in that, The airfoil parameters are the CST parameterization coefficients of the upper and lower surfaces of the airfoil.

3. The rotor airfoil design method based on the improved Bayesian optimization algorithm according to claim 1, characterized in that, The performance index is a weighted combination of airfoil efficiency improvement and thrust relative error.

4. The rotor airfoil design method based on the improved Bayesian optimization algorithm according to claim 1, characterized in that, The method of estimating function values ​​at support points using the Monte Carlo method includes estimating function values ​​using a weighted average method based on nearest neighbors, specifically: Take the data from the currently observed dataset that is closest to the support point. One observation point; Calculate the distance between each observation point and the support point, and take the reciprocal of the distance as the weight of the observation point; Based on the weight of the observation points, The performance indicators of each observation point are weighted and averaged, and the weighted average is used as the function value estimate of the support point.

5. The rotor airfoil design method based on the improved Bayesian optimization algorithm according to claim 1, characterized in that, The expected performance within the calculation region includes the mean of the expected performance within the region and the standard deviation of the expected performance within the region, including: ； ； In the formula, The mean expected performance within the region. The standard deviation of expected performance within the region. It is a convex region. For observation points within a convex region, Indicates the number of Monte Carlo samples. Indicates the first During the Monte Carlo sampling, the observation points were... Function value estimation, For the support point set, For variance, For observation point Performance index values, Expressing expectations, This represents the intersection of the set of support points and the convex region.

6. The rotor airfoil design method based on the improved Bayesian optimization algorithm according to claim 1, characterized in that, The global expected regret formula is expressed as follows: ； In the formula, This indicates a general expectation and regret. Indicates the first During the Monte Carlo sampling, the support points Function value estimation, intermediate parameters , and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively. This indicates the area where the support points are concentrated, excluding the convex region.

7. A rotor airfoil design system based on an improved Bayesian optimization algorithm, characterized in that, include: The surrogate model initialization and update module is used to construct a surrogate model with initial airfoil parameters and corresponding performance indicators. Iteratively, new candidate points are selected through the acquisition function and the surrogate model is updated until the maximum number of iterations is reached. The expected regret calculation module is used to calculate the global expected regret based on the currently observed dataset; If the global expected regret exceeds the threshold, proceed to the continued exploration module; Otherwise, proceed to the detailed development module; The global expected regret is defined as the potential loss when shifting from the current convex region to other search regions, where the current convex region is determined by the currently observed dataset, and other search regions are regions in the design space other than the current convex region; wherein, calculating the global expected regret includes: Constructing a convex region based on the optimal observation point in the currently observed dataset includes: centering on the optimal observation point and calculating the radius... Define a convex region, where the radius is... Determined adaptively based on the design space boundary: ； In the formula, The upper limit of the absolute radius, This is a relative proportionality coefficient. To design the dimensions of the space, For dimensional indexing, Represents the current optimal solution's i-th The value of dimension, Indicates the first The lower bound of the dimensional design space. Indicates the first The upper bound of the dimensional design space; A set of support points is sampled within the design space, and function values ​​are estimated at the support points using the Monte Carlo method. Based on the function value estimation of the support points located within the convex region, the expected performance within the region is calculated, including the mean and standard deviation of the expected performance within the region. The global expected regret is obtained by multiplying the improvement probability and the improvement magnitude, where the improvement magnitude is the difference between the minimum estimated function value of the support point outside the convex region and the mean expected performance within the region, and the improvement probability is the cumulative distribution function value of the standard normal distribution of the ratio of the improvement magnitude to the standard deviation of the expected performance within the region. The exploration module is used to resample the candidate evaluation point set, use the current best surrogate model, reset the iteration count to zero, and return to the iteration to select new candidate points and update the surrogate model through the acquisition function until the maximum number of iterations is reached, then enter the expected regret calculation module. The fine-grained development module is used to output the optimal airfoil parameters or Pareto optimal airfoil parameters based on the currently observed dataset.

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