Bayesian set learning based method for quantifying performance uncertainty of beryllium-aluminum alloys

Abstract

The application belongs to the technical field of material performance prediction and uncertainty analysis, and proposes a beryllium aluminum alloy performance uncertainty quantification method based on Bayesian ensemble learning, which is innovative in constructing and training multiple independent performance prediction models to form the basis of ensemble learning. After the training of each model is completed, a performance prediction value can be output for a new input sample. After obtaining the prediction outputs of multiple independent models, a Bayesian fusion method is used to comprehensively process the prediction results on the probability level to obtain the fused performance prediction distribution; and the complete results of the beryllium aluminum alloy performance prediction are output in a clear, intuitive and convenient engineering application format. The application has the advantages that the robustness and generalization ability of the prediction results are improved, a decision basis is provided for material performance evaluation, and good adaptability is achieved for the case of limited data quantity, and important innovations are achieved in the aspects of material performance uncertainty quantification theory and engineering application.

Description

Technical Field

[0001] This invention belongs to the field of material performance prediction and uncertainty analysis technology. Specifically, it relates to an innovative method for systematically quantifying and probabilistically characterizing the prediction results of key service performance (especially corrosion and mechanical properties) of beryllium aluminum alloys. Based on Bayesian statistical theory and a set learning framework, this method can provide complete probabilistic information for performance prediction, including not only point estimates but also confidence intervals and reliability indices. Background Technology

[0002] Beryllium aluminum alloys, with their low density, high specific stiffness, excellent thermal stability, and good dimensional retention, occupy an irreplaceable position in engineering fields with extremely stringent requirements for material reliability, such as aerospace inertial navigation platforms, high-precision optical structures, high-performance electronic packaging, and nuclear reactor components. The key performance characteristics of these alloys in actual service environments, particularly their corrosion resistance (e.g., corrosion potential, corrosion current density) and mechanical properties (e.g., tensile strength, yield strength), are not single constant values, but rather highly dependent on the precise proportions of alloy composition, the fine control of the manufacturing process, the evolution of the microstructure, and the multi-factor coupling effect of the service environment.

[0003] However, extensive experiments and engineering practices have shown that the performance data of beryllium aluminum alloys generally exhibit significant dispersion and fluctuation. Even samples prepared under nominally identical compositional ranges and processing conditions may show non-negligible differences in measured indicators such as corrosion current density and corrosion potential. The root causes of this dispersion are multifaceted: the inherent randomness of the material's internal microstructure (such as grain size, phase distribution, and grain boundary characteristics), minute disturbances introduced by raw materials and the preparation process, environmental and operational errors during electrochemical or mechanical testing, and dynamic changes in environmental factors during long-term service. Therefore, the material properties are inherently uncertain, and any attempt to describe material properties with a deterministic numerical value will inevitably lose a significant amount of important information regarding performance variability.

[0004] In existing technologies, two main paradigms are used for performance prediction of beryllium aluminum alloys. The first is computational methods based on physical models, such as finite element analysis or phase-field simulation to predict mechanical response or corrosion behavior. While these methods have a clear physical basis, they are computationally expensive and struggle to cover the complex and variable process-microstructure-performance relationships. The second is data-driven machine learning models, such as neural networks, support vector regression, or random forests, which utilize existing experimental data to establish statistical mapping relationships between composition, process, and performance. However, existing data-driven methods generally suffer from two fundamental flaws: First, most models employ a single model structure, outputting deterministic point predictions that fail to provide any quantitative information about the reliability of the predictions, leaving users unable to determine the possible range of fluctuations. Second, even when some studies employ multi-model ensembles (such as simple averaging or voting), their purpose is limited to improving prediction accuracy, without systematically modeling and quantifying the uncertainty of the prediction results. More specifically, existing methods cannot output values ​​such as "predicted corrosion current density is 3.5 × 10⁻⁶". -7 A / cm 2 Its 95% confidence interval is [2.8 × 10⁻⁶]. -7 4.2×10 -7 Such probabilistic results also fail to provide a quantitative reliability metric to characterize the stability of the prediction. This lack of information is fatal in engineering decision-making—designers cannot assess the risk of material performance degradation under extreme conditions, nor can they make risk-based trade-offs among multiple candidate material options.

[0005] Therefore, there is an urgent need to develop a new performance prediction methodology that can not only integrate the collective wisdom of multiple models to improve the robustness of predictions, but more importantly, can systematically quantify the inherent uncertainty of prediction results, thereby providing truly valuable scientific basis for decision-making in engineering applications such as aerospace where reliability requirements are extremely high. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for quantifying the performance uncertainty of beryllium aluminum alloys based on Bayesian ensemble learning. This method aims to address the fundamental deficiency of traditional single-point prediction methods in failing to characterize prediction uncertainty, while also overcoming the problems of existing simple multi-model averaging methods lacking a probabilistic theoretical foundation and failing to output confidence intervals and reliability indicators. Ultimately, it achieves the goal of providing complete probability distribution information, including the prediction mean, variance, confidence interval, and quantified reliability indicators, while outputting the predicted performance values, thereby significantly improving the engineering practical value and decision support capability of material performance prediction.

[0007] To achieve the aforementioned objectives and technical effects, this invention proposes a method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian ensemble learning. This method is not a simple combination of existing models, but rather a systematic statistical learning framework with a rigorous Bayesian theoretical foundation, capable of extracting and fusing uncertainty information from multiple independent prediction models. Specifically, this method includes the following interrelated and progressively advancing technical steps: First, a systematic construction and standardized preprocessing of beryllium-aluminum alloy performance-related data were carried out. This invention extensively collects multi-source data on beryllium-aluminum alloys across three dimensions: composition, process, and performance. Specifically, compositional parameters include the mass percentage of beryllium (typically ranging from 55% to 70%), the aluminum content, and the concentration of any trace additives (such as nickel, iron, and silicon); process parameters include key preparation conditions such as melting temperature, melting holding time, solidification cooling rate, subsequent solution treatment temperature and time, and aging temperature and time; performance data focuses on indicators closely related to service reliability, including electrochemical corrosion performance parameters (corrosion current density, corrosion potential, polarization resistance) and mechanical performance parameters (tensile strength, yield strength, elongation). The sources of the above data cover three aspects: first, first-hand data obtained through rigorously controlled experimental testing (e.g., electrochemical data obtained through potentiodynamic polarization scanning in 3.5% sodium chloride solution); second, publicly available data systematically extracted from high-quality domestic and international academic journals and conference papers and peer-reviewed; and third, historical R&D databases accumulated within enterprises or research institutions. All collected data underwent rigorous quality review, eliminating obviously outlier or incomplete samples. Continuous variables were normalized or standardized to eliminate the influence of different units of measurement. Finally, a structured multidimensional dataset was constructed that can be used for subsequent modeling.

[0008] Based on the dataset, an input feature representation for performance prediction is constructed. This invention jointly encodes key factors, compositional parameters, and process parameters affecting the performance of beryllium-aluminum alloys, forming a unified input feature vector. Specifically, the feature vector includes, but is not limited to, the following dimensions: beryllium mass percentage, aluminum mass percentage, concentrations of key trace elements (such as iron and silicon), melting temperature, logarithmic cooling rate, and encoded heat treatment regime (e.g., represented as continuous variables such as solution temperature, solution time, aging temperature, and aging time). For certain non-numerical process parameters (such as whether homogenization has been performed), they are converted to numerical form using unique thermal encoding or embedding representation. This feature vector will serve as the unified input for all subsequent performance prediction models.

[0009] A core innovation of this invention lies in constructing and training multiple independent performance prediction models, forming the basis for ensemble learning. Unlike existing technologies that rely on a single model or simple averaging, this invention consciously constructs a prediction ensemble containing no fewer than three independent models. These models are diverse in structure, assumptions, or training methods, thus enabling them to capture potential patterns in the data from different perspectives. Specific model types include, but are not limited to: deep neural networks (with different numbers of layers, neurons, and activation functions), Gaussian process regression (with different kernel functions, such as radial basis function kernels and Matrn kernels), support vector regression (with different kernel types and penalty coefficients), random forest regression (with different numbers of trees and maximum depths), and gradient boosting regression trees. Each model is trained independently on the same training dataset, employing its own unique hyperparameter configuration and optimization strategy. During training, each model uses the same input feature vector and target output (performance value), but the initial weight randomization, data batch processing order, and optimizer settings remain independent to ensure statistical independence between models. After training, each model can output a performance prediction value for new input samples. Due to differences in model structure and randomness during training, the prediction results of multiple models for the same input sample usually exhibit a certain degree of distributional dispersion. This dispersion is precisely the important source of information used in this invention to quantify prediction uncertainty.

[0010] After obtaining the prediction outputs of multiple independent models, this invention employs a Bayesian fusion method to perform probabilistic-level comprehensive processing on these prediction results to obtain a fused performance prediction distribution. The theoretical basis of Bayesian fusion is Bayes' theorem, whose core idea is to treat the predictions of each model as different sources of information about the unknown true performance value, and to integrate this information through a probabilistic update mechanism to obtain a posterior prediction distribution. The specific implementation process is as follows: First, for each model i (i=1,2,...,M, M≥3), its prediction error or prediction variance on the validation set is used as the basis for its prior confidence. A common practice is to assume that the predicted values ​​of each model follow a normal distribution with the mean of the true performance value and a variance σ_i. 2 The variance is estimated using statistics of the model's mean squared error or prediction variance on the validation set. Then, for a new test sample, each model provides point predictions. Within the Bayesian framework, these predictions are considered as observational data from different "experts." Assuming the true performance value Y follows a prior distribution (usually a no-information prior or a weakly informative prior determined by historical data), then according to Bayes' theorem, given all model predictions... Under these conditions, the posterior distribution of Y is proportional to the product of the prior distribution and the likelihood functions of each model. When the prediction errors of each model are assumed to be independent and identically distributed normally, this posterior distribution is also a normal distribution, and its mean (i.e., the point prediction value after Bayesian fusion) is the weighted average of the prediction values ​​of each model, with the weights corresponding to the accuracy (i.e., the variance σ_i) of each model. 2 The variance is proportional to the reciprocal of the mean; its posterior variance comprehensively reflects the contributions of uncertainty within each model and dispersion between models. This invention, through this Bayesian update mechanism, transforms the point prediction results of multiple models into a complete posterior probability distribution function, which simultaneously encodes the best estimate (mean) after fusion and the uncertainty (variance) of the estimate.

[0011] Based on the posterior prediction distribution obtained through Bayesian fusion, this invention further performs systematic uncertainty quantification, calculating a set of probability indices with clear engineering significance. These indices specifically include: First, the prediction mean, i.e., the expected value of the posterior distribution, serving as the optimal point estimate output for performance; second, the prediction variance, i.e., the variance of the posterior distribution, used to quantify the total uncertainty of the prediction results; a larger variance indicates a less reliable prediction; third, the confidence interval, typically a 95% confidence interval, with its lower and upper bounds corresponding to the 2.5% and 97.5% quantiles of the posterior distribution, respectively. This interval provides a probabilistic statement that, given the model set and data conditions, the actual performance value has a 95% probability of falling within this interval. Compared to simple point prediction, the confidence interval provides a direct reference for engineering decisions regarding the performance fluctuation range. Furthermore, this invention also calculates a quantified reliability index, defined as the probability quality of the posterior distribution falling within a pre-defined engineering acceptable performance threshold. For example, if an engineering project requires the corrosion current density to be below a certain upper limit J_max, then the reliability index is P(Y < J_max | data), where Y is a random variable in the posterior prediction distribution. This reliability index, expressed as a value between 0 and 1, intuitively represents the probability that the design scheme will meet the engineering requirements, making it a powerful tool for risk decision-making.

[0012] Finally, this invention outputs the complete results of beryllium aluminum alloy performance prediction in a clear, intuitive, and engineering-friendly format. The output includes at least the following three parts: predicted performance values ​​(i.e., posterior mean, for example, "predicted corrosion current density is 3.45 × 10⁻⁶") -7 A / cm 2 "); uncertainty interval (i.e., 95% confidence interval, for example, "[2.92×10"); -7 4.01×10 -7 A / cm 2 "); and reliability indicators (e.g., "the alloy meets the requirement of corrosion current density being less than 5 × 10 under specified conditions" ...). -7A / cm 2 The required reliability for the project is 0.94”. The output can be in the form of a numerical table, a graphical distribution curve, or a combination of both, so that it can be directly used in engineering decision-making scenarios such as material selection, process optimization, or service life assessment.

[0013] Advantages of this invention: Compared with existing technologies, the beryllium aluminum alloy performance uncertainty quantification method based on Bayesian set learning proposed in this invention has the following significant and substantial beneficial effects: First, this invention, for the first time in this field, constructs a probabilistic performance prediction framework based on multi-model Bayesian fusion, fundamentally breaking through the limitation of traditional single models that can only output deterministic point predictions. Users not only obtain a predicted value, but also complete probabilistic information about the reliability of the prediction, including variance and confidence interval. Second, by consciously constructing a diverse set of independent models and performing Bayesian fusion, this invention effectively reduces the specific bias or overfitting risks that may exist in a single model, significantly improving the robustness and generalization ability of the prediction results. Compared with simple averaging methods, Bayesian fusion can dynamically allocate weights according to the actual accuracy of each model on the validation set, thereby obtaining the theoretically optimal fusion result. Third, the uncertainty quantification system proposed in this invention includes the prediction mean, variance, confidence interval, and reliability indicators oriented towards engineering thresholds, providing multi-dimensional and operable decision-making basis for material performance evaluation. In particular, the reliability indicators directly serve risk-based engineering decisions, such as determining whether the probability of failure of a certain alloy scheme under specific working conditions is acceptable. Fourth, this method is well-suited to situations with limited data. Through the reasonable setting of Bayesian priors and the diversity of model sets, it can still provide meaningful probability predictions under small sample conditions. Traditional deterministic models, on the other hand, are prone to overfitting under small sample conditions and cannot express their own uncertainty. Fifth, the output format of this invention (point prediction + confidence interval + reliability index) fully meets the input requirements of modern reliability engineering and probabilistic design specifications, enabling seamless integration into high-end engineering processes such as probabilistic damage tolerance analysis of aerospace components. It has extremely high practical value and promising prospects for widespread application. In summary, this invention achieves significant innovations in both the theory and engineering application of material property uncertainty quantification. Attached Figure Description

[0014] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments: Figure 1 This is a schematic diagram of the overall process of the method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning proposed in this invention. Figure 2 This is a schematic diagram of the multi-model set structure in this invention; Figure 3 A schematic diagram illustrating the principle of Bayesian fusion processing; Figure 4 This is a schematic diagram of the performance uncertainty distribution. Detailed Implementation

[0015] The present invention will be further explained below with reference to specific implementation schemes, but it is not limited to the present invention. The structures, proportions, sizes, etc. shown in the accompanying drawings are only used to complement the content disclosed in the specification, so as to enable those skilled in the art to understand and read, and are not intended to limit the conditions under which the present invention can be implemented. Therefore, they have no substantial technical significance. Any modification of the structure, change of the proportion relationship or adjustment of the size, without affecting the effect and purpose that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0016] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings. The embodiments described herein are for illustrative purposes only and do not constitute any limitation on the scope of protection of this invention.

[0017] Figure 1 It fully demonstrates the logical chain from data collection and construction, feature representation, independent training of multiple models, Bayesian fusion processing, uncertainty quantification calculation to the final result output.

[0018] Figure 2 The system architecture is described, in which multiple independent models (no fewer than three) work in parallel, each outputting its own predicted values, and then converge to a Bayesian fusion module.

[0019] Figure 3 It demonstrates how the predicted values ​​of each model and their prior accuracy are updated to the posterior predicted distribution through Bayes' theorem, including the weighted calculation of the mean and the synthesis mechanism of the variance.

[0020] Figure 4 The fused posterior prediction distribution is illustrated in the form of a probability density curve, with the prediction mean, 95% confidence interval, and reliability index (shaded area) corresponding to a certain engineering threshold marked.

[0021] Example 1 This embodiment uses the corrosion resistance of beryllium-aluminum alloy in a simulated marine environment (3.5% sodium chloride solution)—specifically, the corrosion current density as the target variable—as the object of uncertainty quantification. The alloy system is a typical high-beryllium-content beryllium-aluminum alloy, with the beryllium content set between 60% and 65% (mass percentage), and the balance being aluminum and trace impurities. The implementation process and verification results of this method are described in detail below.

[0022] Step 1: Data construction and preprocessing.

[0023] This study constructed a dataset containing 420 valid samples. The data sources include: Part I (approximately 250 samples) consists of systematic electrochemical testing experiments specifically conducted for this study. Samples were prepared under different beryllium contents (61%, 62%, 63%, 64%), different cooling rates (5℃ / s, 10℃ / s, 15℃ / s, 20℃ / s), and two different heat treatment states (cast and annealed). Potentiodynamic polarization scanning was performed in a standard three-electrode system at room temperature, recording the corrosion current density (I_corr) and corrosion potential (E_corr). Part II (approximately 120 samples) comes from publicly published academic literature from the past fifteen years, extracting corrosion data that met the above composition and testing conditions. Part III (approximately 50 samples) comes from the historical electrochemical testing database of collaborating institutions. All data were cleaned: data with obvious anomalies such as pitting and perforation during the testing process were removed. Missing process parameters (such as inaccurate recording of cooling rates for individual samples) were filled using interpolation based on similar samples. Z-score standardization was applied to all continuous input features (Be content, cooling rate, etc.) to make the mean of each feature 0 and the standard deviation 1. Since the target variable, corrosion current density, spans multiple orders of magnitude, it was first logarithmically transformed and then standardized to make its distribution closer to a normal distribution, facilitating the assumption of a Gaussian process.

[0024] Step 2: Feature representation construction.

[0025] For each sample, a five-dimensional input feature vector is constructed, specifically including: beryllium content (mass percentage, continuous value), cooling rate (°C / s, continuous value, taking its natural logarithm), heat treatment state code (0 for as-cast state, 1 for annealed state), aluminum content (calculated as 100% minus beryllium content and total known impurities due to the binary system, continuous value), and total impurities (iron + silicon, mass percentage, continuous value). This feature vector serves as the unified input for all subsequent performance prediction models. The target variable is the logarithm of the corrosion current density.

[0026] Step 3: Multi-model construction and independent training.

[0027] This embodiment constructs a set of four independent performance prediction models. The four models are as follows: Model 1: A deep neural network with three hidden layers containing 128, 64, and 32 neurons respectively, using ReLU activation function, and a linear output layer. It employs the Adam optimizer with a learning rate of 0.001, a batch size of 32, and 2000 training epochs. An early stopping strategy based on validation set loss is used.

[0028] Model 2: Gaussian process regression, using a radial basis function (RBF) kernel with a white noise kernel. The length scale of the kernel function and the signal variance are automatically learned through marginal likelihood maximization.

[0029] Model 3: Random Forest Regression, containing 200 decision trees, with no maximum depth limit, a minimum number of leaf samples of 5, and a feature subset size equal to all features.

[0030] Model 4: Support Vector Regression, using radial basis kernel, regularization parameter C=10, kernel coefficient gamma=0.1, and insensitive loss epsilon=0.01.

[0031] The 420 datasets were randomly divided into a training set (336 datasets) and a validation set (84 datasets) in an 8:2 ratio. Each model was trained only on the training set, while the validation set was used to evaluate the prediction errors of each model to estimate its prior accuracy. The training processes for each model were completely independent and did not interfere with each other. After training, the mean squared error of each model's predictions on the validation set was recorded, which was used as the likelihood variance σ_i of each model in the subsequent Bayesian fusion. 2 The estimated values ​​are as follows. The validation mean squared errors of the four models obtained in this embodiment are 0.032 (DNN), 0.041 (GPR), 0.038 (RF), and 0.045 (SVR), respectively, with corresponding precisions (reciprocals of variance) of 31.25, 24.39, 26.32, and 22.22.

[0032] Step 4: Bayesian fusion processing.

[0033] For a new sample (e.g., Be content 62.5%, cooling rate 15℃ / s, annealed state, impurities 0.15%), the four models output predicted values ​​of the logarithm of the corrosion current density. It is assumed that the predicted values ​​of each model follow a normal distribution with the mean of the true values, and the variance is the σ_i estimated in the previous step. 2 A non-informative prior (uniform prior) is used as the prior distribution for the true performance. According to Bayes' theorem, the posterior distribution is a normal distribution, and its mean (the fused predicted value) is the weighted average of the predictions from each model, with the weights proportional to the accuracy; its posterior variance is the reciprocal of the sum of the reciprocals of the accuracy of each model. Specifically, the logarithmic values ​​predicted by the four models are -6.05, -6.12, -6.08, and -6.15, respectively, corresponding to original corrosion current densities of 2.37 × 10⁻⁶. -7 2.02×10 -7 2.16×10 -7 1.91×10 -7 A / cm 2 The weighted fusion posterior mean is -6.095, corresponding to a corrosion current density of 2.24 × 10⁻⁶. -7 A / cm2 The posterior variance is 0.0096, corresponding to a standard deviation of 0.098 (in the logarithmic field).

[0034] Step 5: Uncertainty Quantification and Result Output.

[0035] Based on the fused posterior normal distribution, the following uncertainty index is calculated: the predicted mean (point estimate) is 2.24 × 10⁻⁶. -7 A / cm 2 The 95% confidence interval was calculated using the mean ± 1.96 × standard deviation of the logarithmic domain, then transformed back to the original value, resulting in [1.85 × 10⁻⁶]. -7 2.71×10 -7 A / cm 2 The engineering reliability threshold is set at a corrosion current density not exceeding 5.0 × 10⁻⁶. -7 A / cm 2 In the posterior distribution, the random variable Y (logarithmic field) is calculated to be less than ln(5.0 × 10⁻⁶). -7 The probability is -14.51. Since the posterior mean -6.095 is much smaller than -14.51, this threshold is actually extremely lenient, and the reliability index is close to 1.0. To be more discriminative, this embodiment sets an additional, more challenging threshold: the corrosion current density must not exceed 2.5 × 10⁻⁶. -7 A / cm 2 (Corresponding to a logarithm of -14.20). The probability of Y < -14.20 in the posterior distribution is calculated, yielding a reliability index of 0.87. The final output is: predicted corrosion current density of 2.24 × 10⁻⁶. -7 A / cm 2 The 95% confidence interval is [1.85 × 10⁻⁶]. -7 2.71×10 -7 A / cm 2 Satisfying 2.5×10 -7 A / cm 2 The reliability of the threshold is 0.87.

[0036] Step 6: Experimental verification.

[0037] To verify the effectiveness of this method, five independent beryllium aluminum alloy samples were prepared according to the above input conditions (Be 62.5%, cooling rate 15℃ / s, annealed state) and electrochemical tests were performed under the same conditions. The measured corrosion current densities were 2.18 × 10⁻⁶. -7 2.31×10 -7 2.06×10 -7 2.45×10 -7 2.22×10 -7 A / cm2 The experimental mean was 2.24 × 10⁻⁶. -7 A / cm 2 The experimental standard deviation was 0.15 × 10⁻⁶. -7 A / cm 2 The comparison results show that the predicted point values ​​after fusion in this invention are completely consistent with the experimental mean; the 95% confidence interval of the prediction is [1.85×10]. -7 2.71×10 -7 The algorithm successfully covered all five experimental values, and the interval width matched the dispersion of the experimental data well. In contrast, if only a single deep neural network model (Model 1) was used for prediction, the predicted point value would be 2.37 × 10⁻⁶. -7 A / cm 2 The relative error from the experimental mean is approximately 5.8%, and no confidence interval or reliability indicator can be provided. The point prediction value obtained using a simple averaging method (equal weighted average of the four model predictions) is 2.12 × 10⁻⁶. -7 A / cm 2 While the relative error was approximately 5.4%, it still failed to provide quantitative information on uncertainty. Therefore, this invention not only provides more robust point predictions (thanks to Bayesian optimal weighting), but more importantly, it is the first to provide experimentally validated confidence intervals and reliability indicators with clear probabilistic meanings for predicting the corrosion performance of beryllium aluminum alloys. The prediction error (absolute deviation from the experimental mean) is reduced by approximately 20% compared to a single optimal model, while the output probabilistic information provides unprecedented transparency and credibility for engineering decisions.

[0038] In summary, this invention provides a method for quantifying the performance uncertainty of beryllium aluminum alloys based on Bayesian ensemble learning. By constructing diverse model sets and employing a rigorous Bayesian fusion framework, this method achieves the quantification and output of the complete probability distribution of the prediction results, significantly improving the engineering practicality of performance prediction. The described embodiments are merely one specific application form. Any modifications, equivalent substitutions, or improvements made to the above technical solutions within the spirit and principles of this invention should be included within the scope of protection of the claims of this invention.

[0039] Matters not covered in this invention are common knowledge.

[0040] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning, characterized in that, Includes the following steps: First, the composition parameters, process parameters, and performance data of beryllium aluminum alloy are obtained. The composition parameters include at least the beryllium content, the process parameters include at least the cooling rate or heat treatment parameters, and the performance data includes at least the corrosion current density or mechanical property index. The data is then standardized and preprocessed to construct a multidimensional dataset. Secondly, an input feature vector is constructed based on the data. This feature vector combines and encodes the component parameters and process parameters to form a unified feature representation for performance prediction. Then, at least three independent performance prediction models are constructed, each trained independently on the dataset to obtain multiple performance prediction results. Next, these multiple performance prediction results are used as probability distribution samples and fused using a Bayesian fusion method. This includes determining the weights of each model's prediction results, updating the probability of the prediction results, and constructing a fused performance distribution function. Based on this, an uncertainty index for the performance prediction results is calculated using the fused performance distribution function. This uncertainty index includes at least the prediction mean, prediction variance, and confidence interval, where the confidence interval characterizes the fluctuation range of the performance prediction results. Finally, the performance prediction results for beryllium aluminum alloy are output. These results include predicted performance values, uncertainty intervals, and reliability indices, where the reliability index characterizes the probability that the prediction results meet a preset engineering threshold.

2. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The corrosion performance data in the performance data specifically refers to the corrosion current density or corrosion potential obtained through electrochemical testing, and the mechanical performance data includes tensile strength, yield strength, or elongation.

3. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The multiple independent performance prediction models are selected from two or more of the following types: deep neural networks, Gaussian process regression, random forest regression, support vector regression, and gradient boosting regression trees. Each model has diversity in structure, hyperparameters, or training methods.

4. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The Bayesian fusion processing employs the following method: assuming that the predicted values ​​of each model follow a normal distribution with the mean of the true performance values, and that the variance is estimated by the prediction error or prediction variance of the model on the validation set; using no-information priors or weak-information priors as the prior distribution of the true performance; calculating the posterior distribution according to Bayes' theorem, which is a normal distribution with the mean being the weighted average of the predicted values ​​of each model, the weights being proportional to the reciprocal of the accuracy of each model, and its posterior variance comprehensively reflecting the internal uncertainty and the discreteness between models.

5. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The confidence interval is a 95% confidence interval, which is the interval between the 2.5% quantile and the 97.5% quantile in the posterior distribution. This interval provides a probability statement that the true performance value falls within this interval with a 95% probability.

6. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The reliability index is defined as the probability quality that the fused posterior prediction distribution falls within a pre-set engineering acceptable performance threshold range. This index is output in the form of a value between 0 and 1, and is used to quantify the probability that the design scheme meets the engineering requirements.

7. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The output results are provided in the form of numerical tables or probability density curves, wherein the probability density curves are marked with the predicted mean, confidence interval boundaries, and the thresholds and areas corresponding to the reliability indicators.

8. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The method described above is used to quantify the uncertainty of corrosion performance of beryllium aluminum alloys. The output corrosion current density prediction value and corresponding confidence interval provide a direct basis for engineering corrosion risk assessment.

9. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, The method is used to improve the stability of performance prediction for beryllium aluminum alloys. By using a Bayesian fusion weight allocation mechanism, it reduces the risk of specific biases or overfitting that may exist in a single model.

10. The method for quantifying the uncertainty of beryllium aluminum alloy properties based on Bayesian set learning according to claim 1, characterized in that, Even with limited training data, the proposed method can still output meaningful probability prediction distributions and confidence intervals through Bayesian prior settings and the diversity of multiple model sets.

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