An aero-engine turbine disc load inverse prediction method and system based on non-steady active learning

Abstract

The application discloses a kind of based on non-steady active learning's aero-engine turbine disc load inverse prediction method and system, the method includes: in with load distribution as input parameter space sampling and obtaining corresponding response value, constructs data set;First GP is constructed based on data set;The importance of sample point is modeled, and second GP is constructed;Based on the uncertainty of first GP output and the prediction difficulty of second GP output, a collection function is constructed, active sampling is executed and first GP is iteratively updated, and the final GP is obtained;The feasible solution region is identified using the final GP, and then density clustering is performed to determine the local refinement interval, implement interval refinement and local sampling, and output the solution set as the load corresponding to the observed response value.The system includes an initial sample generation module, a Gaussian process proxy modeling module, a sample importance modeling and adaptive sampling module, and a multi-solution search and refinement module.The method and system can improve the prediction accuracy of turbine disc load.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aero-engines, and particularly relates to a turbo-disc load inverse prediction method and system for aero-engines based on non-steady-state active learning. BACKGROUND

[0002] The turbo-disc is a key component of an aero-engine, and its structural integrity and service reliability have an important influence on the overall performance and flight safety of the aero-engine. Under the coupling action of high-speed rotation and high-temperature environment, the turbo-disc needs to withstand significant centrifugal force and thermal stress, and its operating state is directly related to the service life and failure risk of the aero-engine. Therefore, accurately grasping the load condition and structural state during the service process is of great significance for tasks such as life assessment, health monitoring and failure prediction.

[0003] However, due to the harsh environmental conditions of the aero-engine during the service process, it is still difficult to directly measure the real load and internal state of the turbo-disc. Under this background, a class of structural inversion methods based on limited observation data (such as strain, displacement or temperature) to inversely calculate internal load or state parameters has been gradually developed, providing a feasible technical path for state recognition in unmeasurable areas under complex working conditions. This kind of inversion technology has been widely used in structural load identification, state estimation and health monitoring engineering scenarios.

[0004] The core of structural inversion is to construct a mapping model between the system input (such as load, boundary condition) and the output response (such as strain, displacement, temperature), and inversely calculate the most possible input condition through the model. Compared with the traditional forward problem, the inversion problem is ill-posed, often showing non-uniqueness of the solution, high nonlinearity of the system and high-dimensional coupling relationship between input variables. These characteristics make the inversion result easily affected by observation errors and modeling errors, further exacerbating the ambiguity and uncertainty of the solution space.

[0005] In actual problems, different combinations of input parameters may produce similar or even identical structural responses, making it difficult to determine a unique input solution through single observation data. To alleviate this problem, some methods attempt to introduce physical constraints, boundary priors or structural knowledge to narrow down the feasible solution space; some researches turn to the idea of multi-solution identification, focusing on how to effectively extract all physically reasonable solutions that meet the observation conditions, and by establishing a solution set space, improve the stability and reliability of the inversion result.

[0006] Currently, many methods have been proposed for structural inversion, including optimization-based parameter identification methods, Bayesian inference methods, and approximate inversion methods based on surrogate models. Among them, the surrogate model method constructs an approximate mapping between input and response (such as Gaussian process regression, Kriging, neural network, polynomial chaos expansion, etc.), which significantly reduces the simulation cost while ensuring a certain accuracy, and can reduce the computational cost by 1-2 orders of magnitude, showing good computational efficiency in engineering applications.

[0007] In order to further improve the sample efficiency of surrogate modeling, active learning methods have been widely used in structural modeling and inversion tasks in recent years. As an adaptive sampling strategy, active learning can prioritize data points that are most valuable to model improvement during iteration, thereby improving modeling accuracy with limited samples. However, existing active learning strategies and inversion methods still have the following shortcomings when dealing with strong nonlinearity and non-stationary responses: (1) In the process of building surrogate models, traditional sampling strategies have low sampling efficiency in high nonlinearity or non-stationary regions, resulting in low prediction accuracy of surrogate models in key regions when facing complex nonlinear systems: (1-1) It is difficult to effectively identify local high nonlinearity or high error regions, such as single sampling index or linear weighting, which cannot fully exploit the complex characteristics of samples; (1-2) The distribution of sampling points lacks global balance control, and is prone to sampling redundancy, such as most active learning methods only rely on model prediction uncertainty (such as variance) for sampling, ignoring the difference in sample contribution to model training; (2) It is difficult to identify multiple reasonable domains: existing methods focus on single optimal solution, lack of systematic identification ability for multiple solution problems, reducing the comprehensiveness and robustness of the results: such as single solution inversion as the goal, or using global optimization method to identify multiple approximate solutions, but lacking organization and control of solution set structure.

[0008] Based on the above problems, it is necessary to propose an active modeling and inversion method with regional adaptability identification ability and information-driven sampling ability to achieve accurate, efficient, and multi-solution identification of structural state. SUMMARY

[0009] The purpose of the present application is to provide a non-stationary active learning-based aero-engine turbine disc load prediction method and system, which can improve the prediction accuracy of turbine disc load.

[0010] To achieve the above invention purposes, the technical scheme adopted by the present application is as follows:

[0011] A non-stationary active learning-based aero-engine turbine disc load inverse prediction method, the method comprising the following steps:

[0012] (1) sampling to obtain initial sample points in a given input parameter space, and evaluating the sample points to obtain corresponding response values, constructing a training data set, the input parameter space is the load distribution, in addition, densely sampling in the input space range as candidate sample points;

[0013] (2) constructing a first Gaussian process proxy model based on the training data set, denoted as the first GP, for predicting the response value and uncertainty of the sample points;

[0014] (3) modeling the importance of the sample points, constructing a second Gaussian process model, denoted as the second GP, for outputting the predicted difficulty corresponding to the sample points; based on the uncertainty output by the first GP and the predicted difficulty output by the second GP, constructing a collection function, performing active sampling, taking the candidate sample point with the maximum collection function value as a new sample, and updating the first GP based on the new sample to obtain the final Gaussian process proxy model;

[0015] (4) under the constraint of observed response values, using the final Gaussian process proxy model to identify the feasible solution region in the input parameter space; performing density clustering on the feasible solution region to determine the local refinement interval, implementing interval refinement and local sampling in the local refinement interval, and outputting the solution region satisfying the preset error constraint condition as the solution set to obtain the load corresponding to the observed response value.

[0016] Further, in step (1), the response value is selected from one or more of strain, temperature or displacement, and the load distribution is selected from centrifugal load distribution, axial load distribution and outer ring load distribution; in step (1), the initial sample points are obtained based on Latin hypercube sampling in a given input parameter space, and the sample points are evaluated by a real system to obtain corresponding response values.

[0017] Further, in step (2), the uncertainty of the response value is the predicted standard deviation.

[0018] Further, in step (3), specifically:

[0019] (3-1) evaluating each sample point in the training data set using the leave-one-out method, calculating the corresponding negative log likelihood as a sample importance indicator, and forming a new training set with the sample points and the negative log likelihood, constructing a second Gaussian process proxy model based on the training set to output the importance prediction mean value corresponding to the candidate sample points as the predicted difficulty;

[0020] (3-2) densely sampling in the input space range as candidate sample points, normalizing the prediction standard deviation of the candidate sample points by the first Gaussian process proxy model and the importance prediction mean of the candidate sample points by the second Gaussian process proxy model, and taking the product of the two as a collection function, performing active sampling in the candidate sample points to obtain new samples, and iteratively training the first Gaussian process proxy model based on the new samples to obtain the final Gaussian process proxy model.

[0021] Further, in step (3-1), specifically: temporarily remove a certain sample point in the training data set to form a subset, and retrain the first Gaussian process proxy model based on the subset, output the response value and its uncertainty of the removed sample point as the prediction result; calculate the corresponding negative log-likelihood according to the prediction result, and take the negative log-likelihood as the sample importance indicator; then take each sample point and its corresponding negative log-likelihood as the training set to construct the second Gaussian process proxy model to output the importance prediction mean corresponding to the candidate sample point;

[0022] Further, in step (3-2), specifically: the candidate sample points are obtained by densely sampling based on Latin hypercube sampling in the input parameter space range; the prediction standard deviation of the candidate sample points by the first Gaussian process proxy model and the prediction importance mean of the candidate sample points by the second Gaussian process proxy model are normalized, and the product of the two is taken as a collection function to perform active sampling in the candidate sample points; after obtaining new samples, update the training data set and retrain the first Gaussian process proxy model, and the above process is performed in a loop until a preset iteration number is reached or a target accuracy is met, and finally a high-precision Gaussian process proxy model is obtained.

[0023] Further, in step (4), specifically:

[0024] (4-1) densely sample based on Latin hypercube sampling in the input parameter space to generate a candidate point set, and take it as the input of the final Gaussian process model to predict the mean and standard deviation of the candidate points; combine the observed response value, and use the tolerance of the given error to identify the approximate solution set that meets the conditions from the candidate points as the feasible solution region by using the double screening standards of absolute error and relative error.

[0025] (4-2) perform density clustering on the feasible solution region, extract the center point of each clustering cluster, and combine the corresponding response value standard deviation predicted by the final Gaussian process proxy model to determine the local refinement space;

[0026] (4-3) perform local sampling in each local refinement interval to iteratively update the approximate solution set;

[0027] (4-4) After all iterations are completed, all approximate solutions meeting the error constraint in each iteration are re-clustered, independent solution clusters in the global range are uniformly identified, and the solution with the minimum error is extracted from each independent solution cluster as the representative candidate solution of the cluster, and the final solution set is output, and the load corresponding to the observed response value is obtained.

[0028] The application also provides an aero-engine turbine disc load inverse prediction system based on non-steady active learning, which comprises:

[0029] An initial sample generation module is configured to construct a training data set based on a given input parameter space and corresponding response values, wherein the input parameter space is a load distribution, and the response values are selected from one or more of strain, temperature or displacement;

[0030] A Gaussian process proxy modeling module is configured to construct a first Gaussian process proxy model based on the training data set to predict the response values and standard deviations of sample points in the input parameter space;

[0031] A sample importance modeling and adaptive sampling module is configured to model the importance of the input parameters, construct a second Gaussian process model to output a predicted difficulty, and construct an acquisition function based on the standard deviations output by the first Gaussian process proxy model and the predicted difficulty output by the second Gaussian process proxy model, perform active sampling in candidate sample points, select the candidate sample point with the maximum acquisition function value as a new sample, and iteratively train the first Gaussian process proxy model based on the new sample to obtain a final Gaussian process proxy model;

[0032] A multi-solution search and refinement module is configured to identify a feasible solution region in the input parameter space using the final Gaussian process proxy model under the constraint of the observed response values, perform density clustering on the feasible solution region to determine a local refinement interval, implement interval refinement and local sampling in the local refinement interval, and output a solution region meeting a preset error constraint condition as a final solution set to obtain a load corresponding to the observed response value.

[0033] The aero-engine turbine disc load inverse prediction method and system based on non-steady active learning provided by the application guide data sampling (step 3) through an active learning strategy, and identify all physical solutions meeting the observed response (step 4) by combining a multi-solution search framework, thereby constructing an input-output mapping model through proxy modeling and active learning, and achieving efficient and reliable load inversion in the scene of high nonlinearity and non-stationarity of input-response.

[0034] The application further provides an aero-engine turbine disc load inverse prediction device based on non-steady active learning, comprising a memory and one or more processors, wherein the memory stores an executable program, and the one or more processors can realize the aero-engine turbine disc load inverse prediction method when executing the program.

[0035] The application further provides a computer readable storage medium, which stores a program, and the program is used to realize the aero-engine turbine disc load inverse prediction method when executed by a processor.

[0036] Compared with the prior art, the application has the following excellent effects:

[0037] The method and system provided by the application can improve the efficiency and precision of the construction of a proxy model: in the process of constructing the final proxy model, the application introduces a prediction difficulty index, takes sample likelihood as a sample importance index based on cross-validation, and constructs a prediction difficulty field in the input space, which is used to accurately guide subsequent sampling, thereby improving the fitting precision of the model in a non-stationary region, enhancing sample utilization rate, and reducing redundant sample collection; and the application fuses prediction uncertainty and prediction difficulty by a normalized product method to construct a more diverse composite sampling index, so that sampling is concentrated in a key region with large information amount and modeling difficulty, and the modeling quality under the condition of limited samples is improved.

[0038] In the inverse prediction process of the input parameters by using the final proxy model, the method and system provided by the application introduce multi-solution identification and local refinement: a density clustering is used to identify multiple feasible solution domains, and interval refinement and local sampling are implemented in each solution domain to construct a hierarchical inversion structure; thereby the identification and management ability of the inversion multi-solution is significantly enhanced, and the completeness and reliability of the inversion result are improved.

[0039] The method and system provided by the application combine active learning modeling and inversion multi-solution search method to improve the efficiency, precision of proxy model construction, and accuracy and comprehensiveness of solution identification.

[0040] The method and system provided by the application are convenient to deploy in different industrial systems, adapt to inversion problems of multiple dimensions and scales, and have strong engineering adaptability. BRIEF DESCRIPTION OF DRAWINGS

[0041] Figure 1 A flowchart of an aero-engine turbine disc load inverse prediction method provided for embodiment 1;

[0042] Figure 2 A one-dimensional test function is used to verify the adaptive sampling ability of the method provided in embodiment 1 in a non-stationary function in embodiment 2;

[0043] Figure 3In Example 2, a one-dimensional test function is used to verify the ability of the method provided in Example 1 to perform inverse prediction by refining the solution interval and clustering the solution set of a non-stationary function;

[0044] Figure 4 Example 2 uses a two-dimensional test function to verify the adaptive sampling capability of the method provided in Example 1 for non-stationary functions;

[0045] Figure 5 In Example 2, a two-dimensional test function is used to verify the ability of the method provided in Example 1 to perform inverse prediction by refining the solution interval and clustering the solution set of a non-stationary function;

[0046] Figure 6 In Example 2, a multidimensional test function is used to verify the adaptive sampling capability of the method provided in Example 1 for non-stationary functions;

[0047] Figure 7 This is a simplified before-and-after comparison of the turbine disk component of the aero-engine in Example 3;

[0048] Figure 8 These are the load distribution and strain measurement points of the turbine disk of the aero-engine in Example 3;

[0049] Figure 9 The adaptive sampling capability of the turbine disk of an aero-engine is applied to the method provided in Example 1 in Example 3;

[0050] Figure 10 This is a comparison of the identified loads with the actual values ​​under different load components obtained by inverse prediction of the turbine disk of an aero-engine using the method provided in Example 1 in Example 3. Detailed Implementation

[0051] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0052] Example 1

[0053] like Figure 1 As shown, the inverse prediction method for turbine disk load of aero-engines provided in this embodiment specifically includes the following steps:

[0054] S1. Initial sample points are obtained based on Latin hypercube sampling within a given input parameter space, and the sample points are evaluated using a real system to obtain the corresponding response values, thus constructing a training dataset. The input parameter space represents the load distribution, and the response value is strain. Specifically:

[0055] S1-1. Initial samples are uniformly generated in the input parameter space using Latin hypercube sampling (LHS);

[0056] S1-2, real system evaluation is performed on these samples, and the corresponding response values are obtained through finite element simulation or experimental evaluation to form input-output training pairs as the training data set ;

[0057] In this embodiment, the input parameter space is the load distribution, and the response value is the strain.

[0058] S2, a first Gaussian process surrogate model is constructed based on the training data set to predict the response value and uncertainty of the sample point under the current input parameter space. Specifically:

[0059] S2-1, given the input space , the Gaussian process is defined as: for any finite sample point set , the function value obeys the multivariate Gaussian distribution:

[0060] ;

[0061] Among them is the mean function, which is usually set to a constant or zero mean, and the covariance function determines the smoothness and generalization characteristics of the function; commonly used covariance functions include square exponential kernel (SE):

[0062] ;

[0063] The square exponential kernel includes two key parameters, the characteristic standard deviation and the length scale parameter , which together constitute the hyperparameters .

[0064] S2-2, when constructing the first Gaussian process regression model, consider the noisy observations:

[0065] ;

[0066] Among them represents independent Gaussian noise with zero mean and variance .

[0067] Given the training set , the response at the new location obeys the prior distribution, and the joint distribution of the sample combined with this new data point finally forms a new joint Gaussian distribution that satisfies:

[0068] ;

[0069] Among them is the covariance matrix of the training points, To train-test point covariance vector, .

[0070] The prediction posterior distribution of the first Gaussian process proxy model GPR model at the new input is:

[0071] ;

[0072] where the prediction mean and variance are:

[0073] ;

[0074] ;

[0075] S3, model the importance of the input parameters, build a second Gaussian process model for the prediction difficulty index corresponding to the output candidate sample point; based on the uncertainty output by the first Gaussian process proxy model and the prediction difficulty output by the second Gaussian process proxy model, build a collection function, perform active sampling, and iteratively update the first Gaussian process proxy model based on the new sample until the final Gaussian process proxy model is obtained; specifically:

[0076] S3-1, evaluate the negative log likelihood (NLL) of each training sample by leave-one-out method as the "prediction difficulty" indicator; build a new Gaussian process model (Importance-GP), also called the second Gaussian process model, to model the importance of the sample in the entire input space; use the Importance-GP model to extrapolate the prediction difficulty distribution of the training sample in the whole space;

[0077] In order to calculate the likelihood of the sample, the following operations are performed for the th sample point :

[0078] S3-1-1, temporarily remove from the training set , get subset ;

[0079] S3-1-2, retrain the GP model based on , get the prediction distribution at ;

[0080] S3-1-3, define the importance of the sample point as the negative log likelihood:

[0081] ;

[0082] S3-1-4, for from 1 to , repeat the above steps to obtain all sample points and their importance data set;

[0083] S3-1-5, constructing a second Gaussian process model (Importance-GP) to fit the "hard-to-predict" distribution as the training set, see formula:

[0084] ;

[0085] The resulting Importance-GP model for any candidate sample point gives the predictive mean , i.e. the extrapolated estimate of the negative log-likelihood for this sample point; the negative log-likelihood is used to quantify the "hard-to-predict" indicator - the larger the value, the more unique information this sample point carries, and the more significant its contribution to the model performance.

[0086] S3-2, calculate the predictive standard deviation of the first Gaussian process proxy model and the predictive likelihood of the second Gaussian process, and the product of the two after normalization is the acquisition function, guiding the new sample to sample in the area with the most information, and selecting the new sampling point according to the maximum value of the acquisition function for real evaluation; specifically:

[0087] To better balance the trade-off between exploration and exploitation, the posterior standard deviation of the first Gaussian process model is introduced as a measure of model uncertainty; at the same time, the predictive mean from the importance Gaussian process model is used to represent the expected prediction difficulty at the candidate sample point . Normalize and integrate these two indicators to build a collaborative acquisition function:

[0088] ;

[0089] This acquisition function not only encourages attention to areas with high prediction difficulty, but also ensures sufficient exploration of areas with high uncertainty. Its multiplication formula ensures that when the prediction difficulty or uncertainty is close to zero, the acquisition score naturally disappears, thus avoiding redundant sampling in easily understood and highly trusted areas. Conversely, when both components are high, the function will prioritize points with the greatest potential information gain, thus achieving a dynamic balance between exploration and exploitation. This design supports an adaptive sampling strategy that guides data collection towards areas with the most information.

[0090] ​S3-3, calculate the acquisition function value at the candidate sample point according to the above formula, select a new sample point from the candidate sample points according to the maximum value of the acquisition function for real evaluation, and add the new sample point to the current training set; constantly update the training data and the first Gaussian process proxy model; perform multiple iterations in the candidate sample points to obtain the final Gaussian process proxy model.

[0091] In order to improve the modeling efficiency and inversion accuracy, the active sampling criterion based on the prediction difficulty and uncertainty weighting is introduced in S3.

[0092] S-4, under the constraint of the observation response value, the final Gaussian process proxy model obtained in step S3 is used to identify the input parameter space corresponding to the observation response value as a feasible solution region; the density clustering is performed on the feasible solution region to determine the local refinement interval, the interval refinement and local sampling are implemented in the local refinement interval, and the solution region meeting the preset error constraint condition is output as the final solution set, so that the load corresponding to the observation response value is obtained; specifically:

[0093] S4-1, after the training of the Gaussian process proxy model is completed, the search and identification of the solution are entered: under the condition that the observation response is known, the prediction ability and uncertainty expression ability of the final Gaussian process proxy model are fully utilized, and all the multiple physical reasonable input solution regions in the input space that can cause the target response are efficiently searched to support the multi-solution inversion requirement of complex structures. Based on the observed response , all the input solution regions meeting the response constraint condition are identified in the input parameter space . The specific process is as follows:

[0094] A set of uniformly distributed candidate points is generated in the parameter space by Latin hypercube sampling , and the final Gaussian process proxy model obtained in S3 is used to obtain the response value to construct an initial sample set. The output and the standard deviation of the candidate points are predicted by the final Gaussian process proxy model, the target observation value is combined, the approximate solution set meeting the condition is identified from the candidate point set by using the absolute error and the relative error through the given error tolerance:

[0095] ;

[0096] Wherein and are the absolute and relative error thresholds respectively. The samples in the set are considered as the candidate solutions with potential physical feasibility under the current iteration.

[0097] S4-2. To identify multiple potential solution regions, the OPTICS density clustering algorithm is introduced. Compared with traditional density clustering methods such as DBSCAN, OPTICS can adaptively handle cluster structures with different densities without requiring a preset number of clusters, making it particularly suitable for non-uniformly distributed solution spaces. By adjusting the parameters of minimum sample point count and maximum neighborhood scale, OPTICS can adaptively identify multiple potential solution clusters with uneven density based on the local density differences in the sample distribution, without specifying the number of clusters.

[0098] right Perform OPTICS density clustering for each cluster. Extract its center point And combined with the uncertainty of this point in GP Determine the local search interval:

[0099] ;

[0100] In the formula For the first Round The center point of the cluster Its standard deviation corresponds to the 95% confidence interval. This mechanism ensures that the search interval gradually shrinks as the confidence of the surrogate model increases, causing the sampling to gradually concentrate around the reliable solution domain.

[0101] S4-3. In each iteration, in all refinement intervals Local sample sampling is performed separately, and the GP model is used to predict grid points to update the sample set. Compared with the traditional global sampling strategy, this refinement mechanism significantly improves sample utilization and effectively avoids redundant computation in low-value areas.

[0102] S4-4. To enhance the stability and comprehensiveness of local search solutions, this invention also introduces a final candidate solution re-clustering mechanism. After all iterations are completed, all approximate solutions that satisfy the error constraints in each iteration are re-clustered to uniformly identify independent solution clusters globally. For each final solution cluster... Extract the solution with the smallest error as a representative candidate solution for this class, and output the final solution set:

[0103] .

[0104] Through the closed-loop iterative optimization mechanism, the application realizes high-precision and multi-solution structure state inversion starting from a small amount of observation, and is especially suitable for a system inverse problem scene with strong nonlinearity and complex solution space. In the embodiment, the OPTICS algorithm is used to realize unsupervised clustering of the preliminary solution set of inversion, and algorithms such as DBSCAN and KMeans can also be used instead. The advantages of the OPTICS algorithm are as follows: The OPTICS algorithm is suitable for complex multi-solution structures with non-uniform density, and improves the recognition ability of different density solution sets.

[0105] To verify the effectiveness and engineering adaptability of the method in the proxy modeling and inversion problem of a non-stationary system, simulation experiments are carried out in a typical function test scene and a turbine disk case of an aero-engine, and the performance and accuracy of proxy modeling, inversion precision and multi-solution recognition ability are evaluated.

[0106] Embodiment 2: Simulation experiment in a typical function test scene

[0107] I. Experimental setup: Through typical test functions, the performance of the proposed active learning modeling strategy and adaptive inversion method in non-stationary problems is evaluated.

[0108] For the active learning modeling part, the method provided in embodiment 1 is compared with two typical benchmark methods:

[0109] (1) Standard Gaussian process (standard GP): The sampling strategy of the Bayesian optimization method based on the standard Gaussian process only relies on the prediction standard deviation for uncertainty-driven sampling.

[0110] (2) Pure K-fold cross-validation: An active sampling method based on neural networks, which uses K-fold cross-validation to train a fully connected neural network with two hidden layers (8 and 4 neurons respectively, with a tanh activation function) using the Adam optimizer. The uncertainty of the prediction is estimated as the standard deviation of the prediction results of K=8 sub-models, which is used to drive the selection of new sampling points.

[0111] The acquisition function in the proposed method is constructed based on the importance score, which combines the prediction standard deviation and the posterior likelihood, so as to more comprehensively measure the sampling value. To ensure fair comparison, the same initial sample design, number of iterations, evaluation indicators and test function configuration are used for evaluation.

[0112] One-dimensional (1D) and two-dimensional (2D) benchmark functions with typical nonstationary characteristics were selected. These functions were used to evaluate the proposed method's ability to adaptively sample in locally nonlinear regions. To further evaluate the generality and robustness of the proposed method, this embodiment also tested some other widely used optimization benchmark functions, including the Griewank function (2D), Langermann function (2D), Shubert function (2D), Ackley function (4D), the improved Sphere function (6D), and the Hartmann function (6D), which cover different features and dimensions.

[0113] To comprehensively evaluate the fitting and predictive performance of the surrogate model, this embodiment uses the coefficient of determination (COP). The coefficient of determination (COP) and mean squared error (MSE) are used as evaluation metrics. The measure of how well a regression model fits the data is that the model can explain the proportion of variation in the dependent variable (target variable). For the first The true function value at each test point For the predicted values ​​of the surrogate model, For all test points The average value. When When the value is close to 1, it indicates that the model's predictions are highly consistent with the actual values, and the model has a good fitting ability. This indicates that the surrogate model achieved a perfect fit across all test points. Mean Squared Error (MSE) measures the average deviation between predicted and true values; a smaller MSE value indicates lower prediction error and higher accuracy.

[0114] ;

[0115] ;

[0116] The above settings allow for a systematic evaluation of key performance indicators such as adaptability, convergence speed, accuracy, and stability of various sampling strategies when handling non-stationary modeling tasks.

[0117] For the inverse modeling task, a high-precision alternative model is first constructed using the proposed active sampling method. Then, by specifying different target response values, the accuracy of predicting the inverse input and the method's ability to capture multiple solutions are evaluated, thus demonstrating its effectiveness for complex inverse problems.

[0118] II. Test Function

[0119] 1. One-dimensional non-stationary test function

[0120] Select a one-dimensional test function To verify the algorithm's adaptive sampling capability for non-stationary functions.

[0121] All experiments started with the same initial sample set: the initial sample set adopted a one-time Latin hypercube sampling scheme of 10 points to achieve a uniform distribution of samples within the domain interval, which was used to train the initial Gaussian process model; subsequently, the algorithm iteratively selected 40 new sample points to enhance the model performance.

[0122] To evaluate the impact of different initial sample configurations on the performance of the three sampling algorithms, this paper employs a design of 10 independent replicates, each starting with a different initial sample: Figure 2 (Performance comparison of adaptive sampling methods in 10 independent LHS experiments: (a) During 40 iterations) (b) Evolution; final (a) Distribution; (b) Evolution of MSE during 40 iterations; (c) Final MSE distribution) shows the evolution of prediction performance and final statistics during 40 sampling iterations. Figure 2 (a) and Figure 2 In (c), the solid line represents the average of 10 runs, and the shaded area represents ±1 standard deviation. Figure 2 (b) and Figure 2 (d) uses a box plot to represent the distribution of the final performance metrics, where the box represents the interquartile range, the horizontal line represents the median, and the points represent the average performance of each method.

[0123] Taking a one-dimensional non-stationary function as an example, the adaptive sampling method provided in Example 1 significantly outperforms the comparative methods in terms of modeling accuracy and stability. Specifically, as Figure 2 In (a) and (c) of the data, the proposed method consistently outperforms the standard Gaussian process and pure K-fold cross-validation in most stages of the sampling process. Figure 2 (b) Predictive Determination Coefficient The average error reached 0.979, significantly higher than that of the standard Gaussian process (0.909) and pure K-fold cross-validation (0.663). Simultaneously, it exhibited the lowest mean squared error (MSE), the most concentrated error distribution, and no significant outliers, demonstrating stronger prediction accuracy and robustness. In contrast, the standard Gaussian process had a slow convergence speed, and the pure K-fold cross-validation method showed large error fluctuations and insufficient stability. These results validate that our proposed method possesses advantages such as high accuracy, fast convergence, and strong stability in one-dimensional non-stationary modeling tasks, providing reliable support for subsequent inversion analysis.

[0124] To evaluate the basic capability of the proposed inversion framework and its effectiveness in identifying the multi-solution structure, the proposed active learning strategy was first applied to construct a high-fidelity surrogate model, and the proposed adaptive interval refinement method was used in the inversion process. The parameter space was set as [-5, 5], the maximum number of iterations was 3, the initial sampling size was 5000, and the refinement factor was 2. To evaluate the performance in dense and sparse solution scenarios, two target output values and were considered.

[0125] Figure 3 The inversion results of (a) in Figure 3 and (b) in Figure 3 are shown. The black dashed line in the figure is the true function curve , the blue solid line represents the final Gaussian process surrogate model prediction curve obtained in Example 1, the light blue shaded area represents the 95% confidence interval; the red dashed line represents the target value, the orange triangle represents the predicted solution, and the green hollow circle represents the true solution; the method successfully identifies multiple inverse solutions, and all solutions are within the predicted confidence interval, highlighting the value of uncertainty quantification in evaluating the reliability of solutions.

[0126] The relative errors between the predicted solutions and their corresponding true solutions are calculated and summarized in Tables 1 and 2. For the case, 6 solutions are identified, and the maximum relative error is less than 3.9%. The error of most solutions is less than 1%, and the error of solutions 1 and 2 is less than 0.07%, reflecting that these regions have higher alternative fidelity. Larger errors (e.g., solutions 3 and 4) correspond to regions where the function changes rapidly and the GP uncertainty is high due to sparse training data. For the case, two symmetric solutions are accurately identified, and the relative errors are both less than 2%, verifying the effectiveness of the method in the sparse solution scenario.

[0127] In summary, both visual and numerical results confirm the accuracy and stability of the proposed inversion method in handling one-dimensional multi-solution problems. The predicted solution distribution closely matches the true solution set, and the prediction error is still well controlled, laying a solid foundation for extending the method to higher-dimensional and more complex inverse problems.

[0128] Table 1 Comparison of predicted solutions and true solutions for the target value case

[0129] .

[0130] Table 2 Comparison of predicted solutions and true solutions for the target value case

[0131] .

[0132] 2. Two-dimensional non-stationary test function

[0133] To further verify the adaptive sampling capability of the proposed method in non-stationary function modeling, a two-dimensional test function is used for evaluation:

[0134]

[0135] To consider both stationary and non-stationary regions simultaneously, the domain of this field is defined as follows: .

[0136] The experiments followed the same procedure as in the one-dimensional scenario: 60 initial samples were generated within the domain using LHS to construct the initial GP agent. Then, 60 new samples were added incrementally through active learning using three sampling strategies (the method of this invention, standard Gaussian process, and pure K-fold cross-validation). To ensure statistical robustness, each experiment was independently repeated 10 times, with a different initial sampling point configuration each time. Figure 4 (Comparison of different adaptive sampling methods based on 10 independent LHS runs: (a) During 60 iterations) (a) the evolution; (b) the final (c) Evolution of MSE over 60 iterations; (d) Final MSE distribution; This shows the evolution of prediction performance and final statistics over 60 sampling iterations.

[0137] from Figure 4 (a) and Figure 4 As shown in (c), the proposed method consistently outperforms the two comparative methods throughout the entire sampling process. It exhibits rapid growth in the first 10 iterations and maintains high stability thereafter.

[0138] from Figure 4 (b) and Figure 4 As shown in (d), the average coefficient of determination R² of this method reaches 0.974, which is much higher than that of the standard Gaussian process (0.911) and pure K-fold cross-validation (0.621). At the same time, it has the lowest mean square error, the most concentrated error distribution, and no obvious outliers, which verifies its advantages in modeling stability and accuracy in high-dimensional complex nonlinear scenarios.

[0139] To further evaluate the performance of the proposed inversion method in a high-dimensional parameter space, we tested it on a two-dimensional multimodal function. First, we constructed a high-fidelity surrogate model using an active learning strategy. The inversion settings were: 3 iterations, 100,000 initial samples, and a refinement factor of 2. Four different target response values ​​were selected for the inversion test. Figure 5 (a) , Figure 5 (b) in (c) in (d) in (e) in (f) in , Figure 5 (c) in (d) in (e) in (f) in and Figure 5 (d) in (e) in (f) in to investigate the performance of the method in different decryption scenarios.

[0140] From Figure 5 , in the inversion task, the method can accurately identify multiple solution regions for multiple target response values, including sparsely distributed elliptical solution domains and geometrically complex multi-connected high-density solution sets. Table 3 summarizes the number of inverse solutions identified for each target value and their corresponding relative errors. The average relative error is stably controlled at about 5%, with a minimum of 4.91%. All predicted solutions are within the confidence interval of the surrogate model, reflecting strong uncertainty control ability and robustness of multi-solution identification. The number of solutions ranges from 492 to 1079, indicating that the method can find as many effective solutions as possible at different levels of solution sparsity and complexity. In all cases, the predicted contours closely match the true level sets, reflecting the high fidelity of the surrogate model and the reliability of the inversion strategy.

[0141] Table 3 Evaluation indicators of two-dimensional inverse solution results

[0142] .

[0143] 3. Multi-dimensional test functions

[0144] To further evaluate the generalization ability of the proposed adaptive sampling strategy in different problem types, the method is applied to six widely used benchmark functions in the optimization and machine learning fields. The experiment follows the same model configuration and sampling procedure used in the previous non-stationary function test, and is compared with the standard Gaussian process and pure K-fold cross-validation method.

[0145] The selected benchmark functions include: Griewank function (2D), Langermann function (2D), Shubert function (2D), Ackley function (4D), Sphere Modified function (6D), and Hartmann function (6D).

[0146] .

[0147] The above formulas define the analytical expressions of these functions. These benchmark functions exhibit a wide range of complexity characteristics, including different dimensions and diverse nonlinear behaviors. Without modifying the sampling procedure or model parameters, the above three sampling strategies are directly applied to these functions to evaluate the generalization performance of the proposed method. Table 4 summarizes the dimensions, input domains, and sampling configurations of all test functions.

[0148] Table 4. Test function and sample settings

[0149] .

[0150] Following the protocol in Table 4, each experiment was independently repeated 10 times. (Decision coefficient) It is used as the primary performance metric to evaluate the accuracy of a model under a fixed sampling budget. Figure 7 All methods are shown in the six benchmark functions. Scored box plots. Red dots represent the average, and the midline is clearly marked on each plot. The proposed method consistently demonstrates superior or comparable performance across all six functions, measured by mean and median. For Griewank (2D), Shubert (2D), and Hartmann (6D), the proposed method significantly outperforms standard Gaussian processes and pure K-fold cross-validation. It achieves the highest mean and median values. The median value was calculated, and a compact box plot was drawn, demonstrating the model's excellent stability and generalization ability. Particularly for the Shubert function, the median value... Reaching 0.891, far exceeding the standard Gaussian distribution (0.686) and pure K-fold cross-validation (0.407), it highlights its significant performance improvement in handling complex multimodal surfaces.

[0151] For Langermann (2D), the proposed method produces the median. The proposed method is comparable to the standard Gaussian distribution and outperforms pure K-fold cross-validation. Although the standard Gaussian distribution is slightly more compact, the proposed method demonstrates greater robustness in most trials. For the Ackley (4D) sample, the proposed method achieves higher average yield. The proposed method reduces variability and demonstrates superior overall modeling performance, despite a similar median to the standard Gaussian process (GP). For the sphere correction function (6D), the proposed method again shows a slightly higher median and significantly higher mean than the standard Gaussian process. Furthermore, the standard Gaussian process exhibits some strong outliers, further highlighting the robustness of the proposed strategy. In all tests, the pure K-fold cross-validation method performed poorly in both accuracy and stability. Its results are highly variable and sensitive to initial sample configuration, indicating limited robustness of its alternative models in complex landscapes.

[0152] The proposed adaptive sampling method achieves significant improvements in modeling accuracy and convergence speed for low-dimensional non-stationary functions, while maintaining competitive or superior performance in high-dimensional problems. These results clearly demonstrate that the proposed method excels at solving problems with complex structures and significant local nonlinearities, while exhibiting excellent versatility and robustness.

[0153] Example 3: Load Identification in Aero Engine Turbine Floats

[0154] To verify the practical effect of this invention in the identification of complex engineering structures, an inversion identification experiment was conducted using the turbine disk, a key rotating component in an aero-engine, as the object. During service, the turbine disk is subjected to strong centrifugal force and thermal stress, making structural condition identification crucial for lifespan prediction and health assessment. However, due to its complex internal structure and high-temperature, high-speed operating environment, real load data is difficult to obtain directly.

[0155] 1. Finite element model construction

[0156] The turbine disk structure includes key features such as the outer edge, hub, and tenon. To simplify calculations and avoid stress singularities caused by geometric abrupt changes, the tenon area is locally simplified, retaining the main load-bearing structure, resulting in the simplified turbine disk structure as follows: Figure 7 As shown ( Figure 7 (a) in the diagram represents the unsimplified turbine disk structure. Figure 7 (b) in the figure represents the simplified turbine disk structure. In this experiment, a finite element model of a certain type of turbine disk was constructed. A hexahedral dominant meshing strategy was adopted, generating a total of 45,656 elements and 181,935 nodes to ensure a balance between computational accuracy and computational efficiency.

[0157] The turbine disk is entirely made of GH901 high-temperature alloy. This study analyzed the material under a fixed temperature, and the temperature was taken as... Set its material parameters as follows: tensile and compressive elastic modulus ,density Poisson's ratio .

[0158] Regarding boundary conditions, full constraints are applied to the inner cylindrical surface at the connection between the turbine disk center and the shaft to simulate fixed support under actual assembly conditions. Load conditions include three categories:

[0159] 1. Centrifugal Load: A uniformly distributed centrifugal load is applied to the outer edge region to simulate the centrifugal force caused by high-speed rotation. The centrifugal load amplitude is set within a certain range. .

[0160] 2. Axial load: A uniformly distributed axial load is applied to the disk surface, with a load amplitude range of... ;

[0161] 3. Outer ring load: Apply an axial uniform load on the outer ring end face, with a load amplitude range of ;

[0162] Strain measurement points are arranged at 4 key positions, respectively at the rim (Node1) and the hub transition area (Node2, Node3, Node4), covering the key areas with significant strain changes, and each strain measurement position is shown in Figure 8 (b).

[0163] Figure 8 (a) in the figure is the boundary condition of load application: A represents the centrifugal force applied on the edge of the turbine disc, B represents the axial uniform distributed pressure applied on the disc surface, C represents the axial uniform distributed pressure applied on the outer disc surface, and D represents the fixed constraint position.

[0164] 2. Comparison of performance of surrogate model construction

[0165] To verify the performance of the active learning algorithm proposed in Example 1 in the modeling of turbine discs, the proposed method is compared with the standard GP and K-fold neural network structure benchmark method, and and MSE are used as performance indicators.

[0166] All methods are based on the same initial sample set: 10 points are generated by LHS to ensure uniform coverage in the entire design domain. Then, 10 additional points are iteratively added to improve model accuracy. To consider the randomness in the sampling process, 10 independent experiments are performed, each using a different LHS design for initialization. The final model performance of each method is evaluated after 10 iterations of adaptive sampling, and statistical analysis is performed across all runs.

[0167] Figure 9 (a) in the figure and Figure 9 (b) in the figure show the statistical comparison of the final model performance using and MSE (adaptive sampling performance comparison of 10 independent LHS experiments). In the box plot, the horizontal line represents the median, and the red dot represents the mean. From the indicator, the proposed method outperforms the standard Gaussian process in 10 experiments, and significantly outperforms the pure K-fold cross-validation. The MSE results further confirm this conclusion, with the proposed method having lower overall prediction error and higher consistency. Overall, the proposed active sampling strategy has obvious advantages in the accuracy and stability of turbine disc surrogate modeling, surpassing existing benchmark methods.

[0168] 3. Structural load inversion

[0169] Before performing the load inversion, a high-precision surrogate model is constructed using the proposed active learning strategy. To evaluate the inversion accuracy, 100 new load combinations are generated within a predefined range, and the corresponding strain responses are simulated using the finite element model, which are considered as the "sensor measurements". The three load types applied in this example are designed to reflect the typical dynamic load conditions experienced by the turbine disk during operation. Specifically, the centrifugal load is modeled using a sinusoidal function to simulate the smooth periodic variation during high-speed rotation. The axial load applied on the disk surface is modeled using a triangular wave pattern, representing the load abruptness that can occur due to structural interactions. Finally, the axial load on the outer rim is defined as a double-frequency coupled excitation, simulating complex non-stationary load behavior involving multiple frequency components. These different load patterns are designed to comprehensively evaluate the robustness of the inversion algorithm under complex conditions. All loads are sampled over 1 second. These synchronized load curves are applied to the model to generate the true inverse dataset.

[0170] ;

[0171] ;

[0172] .

[0173] The inversion is performed using the interval refinement algorithm. The parameter space is defined as with a maximum number of iterations of 3, an initial sample size of 100,000, and a refinement factor of 2. The candidate solution selection is based on the absolute error ( ) and the relative error ( ).

[0174] Figure 10 The comparison of the true load curves and the identified load curves is shown in FIG. 6 Figure 10 (a) for the non-stationary centrifugal load; Figure 10 (b) for the triangular wave axial load (disk surface); Figure 10 (c) for the sinusoidal coupled axial load (outer rim) of FIG. 6. The inversion curves are closely related to the true curves in both amplitude and frequency components, confirming the effectiveness of the algorithm. To quantify the inversion accuracy, Table 5 shows the average relative error (ARE) and the linear correlation coefficient (LCC) for each load type.

[0175] To quantitatively evaluate the accuracy of the inversion results, the average relative error (ARE) and the linear correlation coefficient (LCC) of the three load components during the operating period are calculated, respectively, and the specific results are shown in Table 5.

[0176] Table 5 Numerical errors of the identified physical loads under different load components

[0177] .

[0178] As shown in Table 5, the proposed method achieved high accuracy and consistency in all three load components. For the centrifugal load (case (a)), the ARE was as low as 3.03%, and the LCC was 0.9986. This indicates that the reconstructed load was very close to the true value, with minimal bias and excellent overall trend consistency, especially for the relatively smooth changes typical in centrifugal effects. For the axial load applied to the disk surface (case (b)), the ARE reached 7.44%, and the LCC was 0.9964. The relatively high error can be due to the sharp transitions of the triangular waveform, which posed a challenge to the numerical inversion. Nevertheless, the reconstructed load was still very close to the true trend, indicating that the algorithm could handle abrupt changes with reasonable accuracy. For the rim axial load involving non-stationary dual-frequency excitation (case (c)), the inversion results showed an ARE of 3.26% and an LCC of 0.9983. The low error and high correlation confirmed that the method could accurately reconstruct complex multi-frequency load patterns, highlighting its practical value in real-world scenarios.

[0179] Overall, the system and method provided by the present application combine an active learning-based surrogate model with interval refinement search, demonstrating excellent accuracy and robustness in reconstructing multiple concurrent load types in turbine disks. Especially in non-stationary high-frequency excitation scenarios, the method exhibits reliable inversion capabilities, verifying its versatility and applicability in structural health monitoring and fault diagnosis of complex engineering systems.

Claims

1. A non-stationary active learning based inverse prediction method for aeroengine turbine disk loads, characterized in that, The method comprises: (1) obtaining initial sample points in a given input parameter space, evaluating the sample points to obtain corresponding response values, constructing a training data set, and inputting the parameter space as a load distribution; (2) constructing a first Gaussian process proxy model for predicting the response value and the uncertainty of the response value of the sample point based on the training data set, denoted as the first GP; the uncertainty of the response value is the prediction standard deviation; (3) modeling the importance of the sample points, constructing a second Gaussian process model for outputting the predicted difficulty corresponding to the sample points, denoted as the second GP, and the predicted difficulty is the importance prediction mean; constructing an acquisition function based on the uncertainty output by the first GP and the predicted difficulty output by the second GP, performing active sampling to obtain new samples, and iteratively updating the first GP based on the new samples to obtain the final Gaussian process proxy model; (4) under the constraint of the observed response value, using the final Gaussian process proxy model to identify the feasible solution region in the input parameter space; performing density clustering on the feasible solution region to determine the local refined interval, implementing interval refinement and local sampling in the local refined interval, and outputting the solution region satisfying the preset error constraint condition as the solution set to obtain the load corresponding to the observed response value.

2. The non-stationary active learning based inverse prediction method of turbine disk loads of an aero-engine according to claim 1, characterized in that, In step (1), the initial sample points are obtained in the given input parameter space based on Latin hypercube sampling, and the sample points are evaluated by the real system to obtain the corresponding response values, the response values are selected from one or more of strain, temperature or displacement, and the load distribution is selected from centrifugal load distribution, axial load distribution and outer ring load distribution.

3. The non-stationary active learning based inverse prediction method of turbine disk loads of an aero-engine of claim 1, wherein, In step (3), specifically: (3-1) evaluating each sample point in the training data set using the leave-one-out method, calculating the corresponding negative log-likelihood as the sample importance indicator, and forming a new training set with the sample points and the corresponding negative log-likelihood, constructing a second Gaussian process proxy model based on the training set, and outputting the importance prediction mean corresponding to the candidate sample point as the predicted difficulty; (3-2) densely sampling within the input space range as candidate sample points, normalizing the prediction standard deviation of the candidate sample points by the first Gaussian process proxy model and the importance prediction mean of the candidate sample points by the second Gaussian process proxy model, and taking the product of the two as the acquisition function, performing active sampling on the candidate sample points, selecting the candidate sample point with the maximum acquisition function value as the new sample, and iteratively training the first Gaussian process proxy model based on the new sample to obtain the final Gaussian process proxy model.

4. The non-stationary active learning based inverse prediction method of turbine disk loads of an aero-engine according to claim 3, characterized in that, In step (3-1), specifically: temporarily remove a certain sample point in the training data set to form a subset, and retrain the first Gaussian process proxy model based on the subset to output the response value and its uncertainty of the removed sample point as the prediction result; calculate the corresponding negative log-likelihood according to the prediction result, and use the negative log-likelihood as the sample importance indicator.

5. The inverse prediction method of an aeroengine turbine disc load according to claim 1, characterized in that, In step (4), specifically: (4-1) Dense sampling is performed in the input parameter space based on Latin hypercube sampling to generate a candidate point set, which is used as the input of the final Gaussian process model to predict the mean and standard deviation of the candidate points; In combination with the observed response value, the candidate points that meet the conditions are identified from the candidate points as the feasible solution set by using the absolute error and relative error double screening criteria with a given error tolerance; (4-2) Density clustering is performed on the feasible solution set to extract the center points of each cluster, and the corresponding response value standard deviation is predicted by combining the final Gaussian process surrogate model to determine the local refinement space; (4-3) Local sampling is performed in each local refinement interval to iteratively update the approximate solution set; (4-4) After all iterations are completed, all approximate solution sets that meet the error constraint in each iteration are re-clustered to identify the independent solution clusters in the global range, and the solution with the minimum error in each independent solution cluster is extracted as the representative candidate solution of the cluster, and the final solution set is output to obtain the load corresponding to the observed response value.

6. A non-stationary active learning based inverse prediction system for turbine disk loads of an aero-engine, characterized in that, The system comprises: An initial sample generation module configured to construct a training data set based on a given input parameter space and corresponding response value, wherein the input parameter space is a load distribution, and the response value is selected from one or more of strain, temperature, or displacement; A Gaussian process surrogate modeling module configured to construct a first Gaussian process surrogate model based on the training data set to predict the response value and standard deviation of the sample points in the input parameter space; A sample importance modeling and adaptive sampling module configured to model the importance of the input parameters, construct a second Gaussian process model to output a predicted difficulty, and construct a sampling function based on the standard deviation output by the first Gaussian process surrogate model and the predicted difficulty output by the second Gaussian process surrogate model to perform active sampling to obtain new samples, and iteratively train the first Gaussian process surrogate model based on the new samples to obtain a final Gaussian process surrogate model; A multi-solution search and refinement module configured to identify a feasible solution region in the input parameter space using the final Gaussian process surrogate model under the constraint of the observed response value, perform density clustering on the feasible solution region to determine a local refinement interval, and implement interval refinement and local sampling in the local refinement interval, and output the solution region that meets the preset error constraint condition as the final solution set to obtain the load corresponding to the observed response value.

7. An apparatus for inverse prediction of a turbine disk load of an aero-engine based on non-steady active learning, comprising a memory and one or more processors, wherein the memory stores an executable program, and the one or more processors can implement the method for inverse prediction of a turbine disk load of an aero-engine based on non-steady active learning according to any one of claims 1-5 when executing the program.

8. A computer-readable storage medium having a program stored thereon, wherein the program, when executed by a processor, is used to implement the method for inverse prediction of a turbine disk load of an aero-engine based on non-steady active learning according to any one of claims 1-5.

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