Method for predicting elastic modulus of face-centered cubic single-phase multi-principal element alloy with preferred occupation

Abstract

This invention proposes a method for predicting the elastic modulus of face-centered cubic single-phase multi-principal element alloys based on preferred occupancy. First, candidate alloy compositions are generated and FCC single-phase candidate systems are screened. Then, the occupancy fractions of each element in the 1a and 3c sublattices at the target temperature are obtained through thermodynamic phase equilibrium calculations. Based on this, a preferred occupancy supercell structure is constructed, and the bulk elastic modulus label is obtained through energy-volume relationships derived from first-principles calculations. Further, the samples are cleaned, enhanced, and feature-constructed. A regression model is trained using elemental composition, physical property parameters, and preferred occupancy thermodynamic features to achieve batch prediction of the elastic modulus of the alloy body. This invention integrates candidate composition generation, phase classification and screening, preferred occupancy structure file construction, bulk elastic modulus label calculation, data cleaning, data enhancement, feature engineering, Support Vector Regression (SVR) model training, and independent prediction into a reproducible technical process.

Description

Technical Field

[0001] This invention relates to the field of computational simulation of metallic materials and design of alloy materials, specifically a machine learning method for predicting the bulk elastic modulus of face-centered cubic single-phase multi-principal alloys based on preferred site selection. Background Technology

[0002] Multi-principal element alloys (MPEAs) use three or more principal elements as the basis for alloy design, breaking through the traditional design mode of using a single principal element as the matrix and adding a small amount of alloying elements to control performance. These alloys have a wide compositional design space and exhibit significant material design potential in terms of strength, plasticity, thermal stability, corrosion resistance, and radiation resistance. Among them, face-centered cubic single-phase MPEAs typically possess good plasticity and processing properties, making them a representative system in structural materials research and engineering applications.

[0003] Bulk modulus is a crucial intrinsic parameter characterizing a material's resistance to uniform compression, reflecting its volumetric deformation response under service loads. For multi-principal element alloys, the bulk modulus is closely related to alloy composition, phase structure, atomic-scale local environment, and heat treatment path. Traditional experimental measurement methods require the preparation and characterization of samples individually, resulting in long cycles and high costs. While first-principles calculations can provide highly accurate theoretical data, point-by-point calculations over a large composition space still require significant computational resources. Therefore, limited by the high dimensionality of the composition space and the diversity of heat treatment processes, both traditional experimental preparation and theoretical calculation methods for multi-principal element alloys face the challenges of high cost and low efficiency. Furthermore, existing modeling methods for special quasirandom structures (SQS) are overly simplistic, neglecting the influence of temperature on the configuration. Therefore, a modeling method based on sublattice occupies fractions (SOFs) is proposed to consider the competitive preference for occupancy of different constituent elements in different sublattices at different heat treatment temperatures.

[0004] Machine learning methods can learn the complex nonlinear relationships between composition, structure, physical properties, and performance from existing data, providing a new technical approach for high-throughput screening of multi-principal element alloys. Therefore, it is necessary to propose a complete method that simultaneously considers composition screening, phase structure discrimination, preferred site structure construction, tag data acquisition, data cleaning and enhancement, material feature engineering, and machine learning regression prediction to achieve efficient prediction of the bulk elastic modulus of face-centered cubic single-phase multi-principal element alloys. Summary of the Invention

[0005] This invention proposes a method for predicting the bulk elastic modulus of face-centered cubic single-phase multi-principal element alloys based on optimal occupancy. It addresses the challenges of high-dimensional composition space in multi-principal element alloys, difficulties in screening face-centered cubic single phases, acquisition of optimal occupancy structure information, and high costs associated with high-throughput prediction of unknown alloys. The method integrates candidate component generation, phase classification and screening, construction of optimal occupancy structure files, calculation of bulk elastic modulus labels, data cleaning, data augmentation, feature engineering, support vector regression (SVR) model training, and independent prediction into a reproducible technical process.

[0006] The present invention adopts the following technical solution.

[0007] A method for predicting the bulk elastic modulus of face-centered cubic (FCC) single-phase multi-principal element alloys based on preferred site selection can be applied to alloy systems that tend to form FCC single phases but do not fall into the category of body-centered cubic (BCC) or intermetallic compound phases. The method first generates candidate alloy compositions and screens FCC single-phase candidate systems. Then, thermodynamic phase equilibrium calculations are used to obtain the site fractions of each element in the 1a and 3c sublattices at the target temperature. Based on this, a preferred site selection supercell structure is constructed, and the bulk elastic modulus label is obtained through energy-volume relationships derived from first-principles calculations. Further, the samples are cleaned, enhanced, and feature-constructed. A regression model is trained using elemental composition, physical property parameters, and preferred site selection thermodynamic features to achieve batch prediction of the bulk elastic modulus of the alloys to be predicted. This method achieves rapid prediction of the bulk elastic modulus of multi-principal element alloys with low computational cost, improving the efficiency of candidate alloy screening and accelerating the composition design and performance evaluation of multi-principal element alloys.

[0008] The prediction method includes the following steps:

[0009] Step S1: Candidate alloy composition generation and initial screening based on empirical criteria; set the candidate element set, number of principal components, upper and lower limits of atomic fraction for each principal component, composition step size, and empirical phase formation criterion threshold; iterate through the candidate element combinations and generate composition allocations that satisfy a total atomic fraction of 1 using a loop backtracking method; calculate the weighted average melting point, mixing entropy, mixing enthalpy, thermodynamic parameter Omega (Ω), atomic size mismatch parameter delta (δ), and valence electron concentration VEC for each effective composition; and output the candidate alloy compositions that simultaneously satisfy the empirical criteria as the candidate alloy composition pool of the candidate dataset.

[0010] Step S2, face-centered cubic single-phase classification and screening: The candidate alloy composition obtained in step S1 is converted into a fixed-dimensional element mole fraction feature vector required by the phase classification model, and input into the trained neural network multi-label phase classification model to obtain the probability values ​​of face-centered cubic phase FCC, body-centered cubic phase BCC or intermetallic compound phase IM respectively, and the FCC single-phase candidate alloy is screened according to the probability threshold.

[0011] Step S3: Constructing a structure file based on preferred occupancy; For the FCC single-phase candidate alloys screened in Step S2, using Thermo-Calc thermodynamics and phase equilibrium software, by writing a macro command file ***.tcm for the corresponding alloy system, thermodynamic calculations and phase equilibrium data are automatically calculated to obtain the occupancy fractions of different elements in the 1a and 3c sublattices at the target heat treatment temperature. Based on the FCC_L12_AuCu3 prototype structure and the set supercell size, the integer number of atoms of each element in each sublattice is calculated, and the coordinates of each sublattice are randomized and the elements are allocated according to the integer number of atoms to generate a preferred occupancy structure file POSCAR for first-principles calculations;

[0012] Step S4, constructing the bulk elastic modulus label; the structure file POSCAR obtained in step S3 is further optimized by first-principles calculation using the VASP software package. Based on the optimized equilibrium unit cell, multiple volume-scaled states are constructed and the total energy of the system is calculated. The bulk elastic modulus is obtained by fitting the state equation using the energy-volume method, and the bulk elastic modulus is used as the target label for supervised learning in the machine learning task.

[0013] Step S5, initial data cleaning; perform cross-validation prediction on the initial dataset composed of element composition and bulk elastic modulus labels, calculate the sample residuals and robust z-values ​​using at least two regression detection models respectively, identify noise samples using the intersection judgment method, and perform label replacement or deletion operations on the noise samples to obtain the cleaned sample dataset.

[0014] Step S6, small sample data augmentation; for the cleaned sample dataset, the mole fraction of alloy composition is mapped to the combined data transformation space to construct local interpolation augmentation candidate samples, and then the augmentation samples are screened through composition constraints, deduplication constraints, nearest neighbor distance constraints, label range constraints and local consistency constraints to obtain the augmented training dataset;

[0015] Step S7, Machine learning feature construction; parse the alloy chemical formula or read the element mole fraction to generate elemental composition features; calculate the combined features based on elemental physical property parameters, binary mixing enthalpy parameters, electronegativity, and the preferred site thermodynamic database to form a feature table for predicting the bulk elastic modulus.

[0016] Step S8: Support Vector Regression Model Training; Using the feature table obtained in Step S7 as input and the bulk elastic modulus obtained in Step S4 as the target column, the Support Vector Regression model is trained, and the optimal model is determined through repeated cross-validation, two-stage hyperparameter search, and recursive feature elimination cross-validation; the training results are saved as independently callable model files, parameter files, and metadata files.

[0017] Step S9, Independent Prediction: Read the feature table of the FCC single-phase multi-principal alloy to be predicted, load the model file and metadata file saved in step S8, verify and extract the features to be predicted according to the feature column order recorded in the metadata file, and call the trained support vector regression model to output the predicted value of the bulk elastic modulus of the alloy to be predicted.

[0018] In step S1, the candidate element set is Co, Cr, Cu, Fe, Mn, Ni, and V; the number of principal elements in the alloy system is 3, 4, or 5; when the number of principal elements is 3, the atomic fraction of each principal element is 30% to 38%; when the number of principal elements is 4, the atomic fraction of each principal element is 20% to 28%; when the number of principal elements is 5, the atomic fraction of each principal element is 16% to 24%; the composition step size is 2%, and elements not involved in the current alloy composition are filled with 0.

[0019] Step S1 uses script 1 to generate candidate alloy compositions and perform preliminary screening based on empirical criteria. This corresponds to script 1 - generating n-element multi-principal element alloy compositions, and includes the following steps:

[0020] Step S11: Initialize the candidate element set, element property dictionary, and binary mixing enthalpy parameter; the candidate element set includes Co, Cr, Cu, Fe, Mn, Ni, and V, the element properties include melting point, atomic radius, and valence electron concentration, and the binary mixing enthalpy parameter is used to calculate the mixing enthalpy characteristics of the candidate alloy;

[0021] Step S12: Set the number of principal components, the upper and lower limits of the atomic fraction of each principal component, the component step size, and the empirical phase formation criterion threshold; wherein, the number of principal components is 3, 4, or 5, when the number of principal components is 3, the atomic fraction of each principal component ranges from 30% to 38%, when the number of principal components is 4, the atomic fraction of each principal component ranges from 20% to 28%, when the number of principal components is 5, the atomic fraction of each principal component ranges from 16% to 24%, and the component step size is 2%.

[0022] Step S13: Use a combination traversal method to obtain the element combination under the specified number of principal elements, and use a loop algorithm to generate the composition allocation that satisfies the total atomic fraction of 1; fill the elements that do not participate in the current alloy composition with 0, so that each candidate sample forms a composition vector with Co, Cr, Cu, Fe, Mn, Ni and V in a fixed order.

[0023] Step S14: For each candidate alloy, calculate the weighted average melting point Tm, mixing entropy Smix, mixing enthalpy Hmix, thermodynamic parameter Omega, atomic size mismatch parameter delta, and valence electron concentration VEC based on the element mole fraction.

[0024] Step S15: Based on empirical phase formation criteria, candidate components are screened, and only candidate components that simultaneously satisfy Omega≥1.1, delta≤6.6%, and VEC≥8.0 are retained to form an initial candidate alloy composition pool.

[0025] Step S16 outputs the filtered candidate alloy composition pool as a data table containing fields for Co, Cr, Cu, Fe, Mn, Ni, V, formula, Omega, delta, VEC, Smix, and Hmix, providing input for subsequent phase classification and filtering.

[0026] Face-centered cubic single-phase multi-principal element alloys refer to candidate alloys that have a tendency to form face-centered cubic single phases and do not contain alloy systems that are identified as body-centered cubic phases or intermetallic compound phases.

[0027] In step S2, the output result is (FCC: 1, BCC: 0, IM: 0), and the phase stability at the target heat treatment temperature is confirmed by thermodynamic phase equilibrium calculation software. The face-centered cubic single-phase alloy is constructed based on the 3×3×3 supercell of the FCC_L12_AuCu3 prototype structure to introduce temperature-dependent ordered preferred site information. This does not mean that the candidate alloy must have an AuCu3 stoichiometric ratio or a completely L12 long-range ordered structure. When the site fractions of different elements in the 1a and 3c sublattices tend to be consistent, this double sublattice model can characterize the site state of a near-completely disordered FCC solid solution.

[0028] Step S2 uses script 2 for face-centered cubic single-phase classification and screening, corresponding to script2 - machine learning phase classifier, which includes the following steps:

[0029] Step S21: Read the multi-principal alloy phase structure dataset, take the mole fraction of each element in the alloy as the input feature, and take FCC, BCC and IM as phase labels. If the corresponding phase exists, it is represented by 1, and if it does not exist, it is represented by 0. Multiple phases are allowed to exist in a system at the same time.

[0030] Step S22: Convert the training data or candidate alloy composition to be predicted into a fixed-dimensional element mole fraction input matrix, and use the Min-Max normalizer to numerically normalize the input features to reduce the impact of different element content scale differences on model training.

[0031] Step S23: Construct a neural network multi-label phase classification model; the model includes an input layer, a first fully connected hidden layer, a second fully connected hidden layer, and an output layer. The number of nodes in the first hidden layer is set to 128, and the number of nodes in the second hidden layer is set to 64. The hidden layer uses the ReLU activation function and combines Dropout or batch normalization units to suppress overfitting. The output layer uses the Sigmoid activation function to output the probabilities of the three phase labels FCC, BCC, and IM respectively.

[0032] Step S24: The multi-label phase classification model is trained using the binary cross-entropy loss function, Adam optimizer, weight decay and 10-fold cross-validation to obtain a stable phase classification model;

[0033] Step S25: Input the initial candidate alloy composition pool obtained in step S1 into the trained phase classification model to obtain the label probabilities of FCC, BCC and IM respectively; when the label probability of the corresponding phase is greater than the classification threshold of 0.5, it is considered that the corresponding phase exists; only when the label probability of FCC is greater than 0.5 and the label probabilities of BCC and IM are both less than 0.5, the candidate system is determined as a candidate alloy with a face-centered cubic single phase.

[0034] Step S26: The selected FCC single-phase candidate alloys are output according to fields such as Co, Cr, Cu, Fe, Mn, Ni, V and formula, providing detailed information of the candidate system for subsequent construction of preferred site structures and high-throughput prediction.

[0035] In step S3, the target heat treatment temperature is set to 973K. The FCC_L12_AuCu3 prototype structure is composed of nested 1a sublattices occupied by Au and 3c sublattices occupied by Cu. The conventional unit cell of the face-centered cubic lattice consists of lattice sites located at the eight vertices and six face centers of the unit cell. After calculation based on the periodic symmetry of the lattice sites, each unit cell contains one 1a site and three 3c sites. When a 3×3×3 supercell is used, the conventional FCC_L12_AuCu3 unit cell is periodically expanded by a factor of 3 along the three lattice directions to form a supercell with 108 lattice sites, including 27 1a sublattice sites and 81 3c sublattice sites.

[0036] Step S3 involves constructing the structure file based on optimal placeholders, including the following steps:

[0037] Step S31: For the FCC single-phase candidate alloys selected in step S2, based on the existing end-group thermodynamic database ***.TDB file, perform phase equilibrium calculations using the Thermo-Calc thermodynamic and phase equilibrium calculation software package to generate a YSITE-T.EXP output file containing sublattice occupied fractions (SOFs);

[0038] Step S32: Extract the placeholder fraction rows corresponding to the target heat treatment temperature from the YSITE-T.EXP data;

[0039] Step S33: Using the FCC_L12_AuCu3 prototype structure as the structural basis, the vertex sites in the conventional unit cell are defined as 1a sublattices, and the face-centered sites are defined as 3c sublattices. The total number of sites in the 1a and 3c sublattices is determined according to the selected supercell size. The unit cell contains one 1a sublattice site and three 3c sublattice sites. When constructing the required computational model, the cell is expanded to 3×3×3, at which point the number of 1a sublattice sites is 27 and the number of 3c sublattice sites is 81.

[0040] Step S34: Multiply the occupancy fraction of each element on the corresponding sublattice by the total number of sites in the corresponding sublattice to obtain the theoretical number of atoms occupied by each element in each sublattice; round the theoretical number of atoms occupied, and perform residual iterative correction based on the residual before and after rounding, so that the sum of the number of atoms of all elements in each sublattice is equal to the total number of sites in the corresponding sublattice.

[0041] Step S35: Fill each element into the corresponding sublattice coordinates according to the integer number of atoms determined in step S34, assign a random number generated by the rand() function of the preset random seed to each sublattice coordinate, and perform random arrangement of the coordinates according to the random number.

[0042] Step S36: Merge the coordinates and total number of atoms of the same type according to the element symbol order, fill the allocated site information into the standard coordinate file POSCAR, and add a selective dynamic marker at the end of each atomic coordinate to eliminate the rigid body translational degree of freedom; mark the origin with fractional coordinates of 0 0 0 as FFF, and fix one atom in the x-axis, y-axis and z-axis directions respectively, which moves only in that direction, and mark them as TFF, FTF and FFT respectively, and mark all other coordinates as T TT.

[0043] In step S4, structural optimization includes volume relaxation and complete relaxation; based on the optimized equilibrium unit cell, the three lattice vectors are scaled proportionally using multiple lattice linear scaling factors, which include at least 0.96, 0.98, 1.00, 1.02 and 1.04; the energy-volume equation of state is the third-order Birch-Murnaghan equation of state, and the bulk elastic modulus is obtained from the fitted zero-pressure bulk elastic modulus parameters;

[0044] The acquisition of the bulk elastic modulus label in step S4 includes the following steps:

[0045] Step S41: Perform first-principles structure optimization on the POSCAR structure file obtained in step S36 based on the VASP software package, including cell volume, cell shape and atomic position relaxation, to obtain a balanced crystal structure.

[0046] Step S42: Construct multiple volume scaling states based on the equilibrium volume V0 obtained in step S41; in a preferred embodiment, the lattice scaling factors are set to 0.96, 0.98, 1.00, 1.02, and 1.04, so that the three lattice vectors are scaled proportionally according to the corresponding scaling factors to obtain 0.96. 3 V0, 0.98 3 V0, 1.00 3 V0, 1.02 3 V0, 1.04 3 V0, and calculate the total energy of the system under different volumes to form an energy-volume dataset;

[0047] Step S43: The energy-volume data are fitted using the third-order Birch-Murnaghan equation of state to extract the zero-pressure body elastic modulus.

[0048]

[0049] Step S44: The bulk elastic modulus label obtained in step S43 is merged with the corresponding alloy composition and preferred occupancy structure information to form basic sample data for model training.

[0050] Step S5, use script 3 to perform initial sample data cleaning, corresponding to script3-data cleaning, including the following steps:

[0051] Step S51: Read the initial data table containing the element composition column and the bulk elastic modulus label column, and set the number of input feature columns, the target column position, the number of cross-validation folds, the random seed, the robust z-value threshold, and the cleaning action; the cleaning action includes deleting noise samples or replacing noise sample labels;

[0052] Step S52: Construct at least two independent regression detection models; the regression detection models include a k-nearest neighbor regression detection model and a Gaussian process regression detection model, and generate cross-validation prediction values ​​for each sample based on K-fold cross-validation with random seeds;

[0053] Step S53: Calculate the prediction residuals for each sample under the k-nearest neighbor regression detection model and the Gaussian process regression detection model, and calculate the corresponding robust z-value based on the median and absolute median of the residuals to reduce the impact of extreme residuals on the anomaly detection threshold.

[0054] Step S54: Noise samples are identified using the intersection determination method. Only when the absolute value of the residual robust z-value of the same sample under the k-nearest neighbor regression detection model and the absolute value of the residual robust z-value under the Gaussian process regression detection model are both greater than the preset threshold, the sample is determined to be a noise sample.

[0055] Step S55: When the cleaning action is deletion, the noise sample is removed from the initial dataset; when the cleaning action is replacement, the bulk elastic modulus label of the noise sample is replaced with the median or average of the predicted values ​​of the two detection models.

[0056] Step S56: Output the cleaned sample data table for subsequent small sample data augmentation and model training.

[0057] In step S6, the local interpolation enhancement includes: before performing the logarithmic ratio transformation, performing zero-value replacement and closed-loop normalization on the element mole fraction composition vector, replacing the element mole fractions less than or equal to the preset threshold with the preset threshold, and re-normalizing all element mole fractions after replacement so that the sum of all element mole fractions is 1; during interpolation in the transformation space, interpolating the transformed composition vector and bulk elastic modulus label of the paired samples according to the interpolation coefficients to obtain the candidate enhanced composition vector and candidate enhanced label; after enhancement, inversely transforming the candidate enhanced composition vector to the original composition space, and determining the optional element set of the candidate enhanced sample according to the union of the non-zero element sets in the paired samples, selecting a preset number of elements from the optional element set as the non-zero elements of the candidate enhanced sample, and setting the remaining element mole fractions to 0.

[0058] In step S6, the enhancement scheme and enhancement amount are determined by comparing the final results of the outer layer repeated hold-out validation and the inner layer cross-validation; the candidate enhancement amounts are multiple preset enhancement amounts; the evaluation learner is a support vector regression model; the evaluation index includes the coefficient of determination R. 2 The root mean square error (RMSE) and the actual proportion of true values ​​in the test set falling within the 95% prediction range are used to determine the optimal net augmentation based on the comprehensive evaluation of the three indicators. The true samples and augmented samples together constitute the dataset used for machine learning training.

[0059] Step S6 uses script 4 to perform small sample data augmentation, corresponding to script4-data augmentation, which includes the following steps:

[0060] Step S61: Read the cleaned sample dataset obtained in step S5, use Co, Cr, Cu, Fe, Mn, Ni, and V as component feature columns, use Bulk_Modulus as the target column, and set the outer layer repeat retention validation seed, outer layer test set ratio, inner layer cross-validation fold number, and candidate net enhancement amount.

[0061] Step S62, perform zero-value replacement processing on the zero component in the element mole fraction, using 10 -6 Replace the zero components and scale the remaining non-zero components proportionally so that the sum of the mole fractions of each element remains 1 after processing; perform a centered logarithmic ratio transformation or an equidistant logarithmic ratio transformation on the closed normalized component vector to map it to a transformation space that can perform Euclidean distance calculations.

[0062] Step S63: Select nearest neighbor paired samples for the real sample in the transformation space, generate candidate enhancement components according to the interpolation coefficients, and synchronously interpolate the bulk elastic modulus label to obtain candidate enhancement labels;

[0063] Step S64: Inversely transform the candidate enhancement components from the transformation space back to the original component space, and perform component normalization and bit retention processing to ensure that the sum of the mole fractions of each element in the enhancement sample remains 1. Identify the elements in the real sample and paired sample whose mole fractions are greater than the non-zero threshold, and determine the union of the non-zero element sets of the two samples as the optional element set for the candidate enhancement sample. Based on the preset range of non-zero principal components and the magnitude of the mole fractions of each element after the inverse transformation, select a corresponding number of elements from the optional element set as the non-zero elements of the candidate enhancement sample, and set the mole fractions of the remaining elements to 0.

[0064] Step S65: Perform multiple screening on the candidate enhanced samples. The multiple screening includes at least non-zero principal component number constraint, component range constraint, nearest neighbor distance filtering, local label consistency filtering, label quantile range constraint, duplicate sample removal, and component retention number constraint.

[0065] Step S66: Compare model performance under different augmentation methods and different augmentation amounts through outer layer hold-out validation and inner layer cross-validation; evaluation metrics include the coefficient of determination R. 2 The root mean square error (RMSE) and 95% prediction interval coverage were evaluated, and the final enhancement scheme was determined based on the comprehensive evaluation.

[0066]

[0067] in, All three indicators were normalized to the [0,1] interval.

[0068] R respectively 2 In a preferred approach, the weights of RMSE and Coverage are set equally, each to 1 / 3.

[0069] In step S7, based on the calculation results of the Thermo-Calc thermodynamic and phase equilibrium calculation software package, the placeholder fraction line corresponding to the target heat treatment temperature in the YSITE-T.EXP output file is read for each candidate system to calculate Sigma (deltaG), configuration entropy Sconfig and Gibbs free energy characteristic G under ordered placeholder conditions.

[0070] Step S7, use script 5 to construct machine learning input features, corresponding to script5 - feature engineering, including the following steps:

[0071] Step S71: Read the data table containing alloy chemical formulas or element mole fractions; when the input is a chemical formula, use regular expressions to identify the element symbols and their corresponding values, and extract the mole fractions of seven elements: Co, Cr, Cu, Fe, Mn, Ni and V.

[0072] Step S72: Normalize the mole fractions of the seven elements in each alloy sample within the row so that the sum of the mole fractions of the elements is 1, and form the element composition characteristics in a fixed element order.

[0073] Step S73: Based on the element's melting point, atomic radius, valence electron concentration, Pauling electronegativity, and binary mixing enthalpy parameter, calculate the empirical physical properties and thermodynamic combination characteristics, which include at least Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, and X;

[0074] Step S74: Based on Thermo-Calc thermodynamic and phase equilibrium calculation results, read the occupancy fraction of different elements on the 1a and 3c sublattices at the target heat treatment temperature, and calculate the preferred occupancy thermodynamic characteristics in combination with the end-base thermodynamic database. The preferred occupancy thermodynamic characteristics include Sigma (deltaG), Sconfig and G.

[0075] Step S75: The element composition features obtained in step S72, the empirical property combination features obtained in step S73, and the preferred occupancy thermodynamic features obtained in step S74 are merged to generate a machine learning input feature matrix for predicting the bulk elastic modulus.

[0076] In step S8, the input feature columns of the support vector regression model include elemental composition features: Co, Cr, Cu, Fe, Mn, Ni, V; physical property features: Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X; and thermodynamic features: Sigma(deltaG), Sconfig, and G.

[0077] Step S8, use script 6 to train and save the support vector regression model, corresponding to script6 - prediction model training, including the following steps:

[0078] Step S81: Read the data table containing complete input features and bulk elastic modulus labels, determine Bulk_Modulus as the target column, and determine the input feature columns including Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G;

[0079] Step S82: Divide the data into training set and test set according to the preset test set ratio; within the training set, use 10-fold repeated cross-validation to train the model.

[0080] Step S83: Construct a two-stage hyperparameter search strategy. The first stage involves a random search within a large parameter space to quickly locate the parameter region with superior performance. The second stage involves constructing a local grid around the advantageous parameter region obtained in the first stage and using grid search to further optimize the hyperparameters. In a preferred embodiment, the hyperparameter space for the support vector regression model is constructed as follows: kernel = ['linear', 'rbf', 'poly'], C = [1, 3, 10, 30, 100], epsilon = [0.01, 0.05, 0.1, 0.3], gamma (rbf, poly) = ['scale', 0.01, 0.03, 0.1, 0.3], degree (poly) = [2, 3], coef0 (poly) = [0.0, 0.5, 1.0]; the parameter combination is selected to maximize the cross-validation determination coefficient R. 2 As the main indicator, when R 2 When the differences are small, compare RMSE. 2 When both RMSE and MAE are close, further compare them.

[0081] Step S84: Construct a support vector regression model using the optimal parameters determined in step S83. Use permutation importance to determine the importance of each input feature and perform recursive feature elimination and cross-validation. The permutation importance is calculated five times, and the recursive feature elimination step size is one feature deleted each time, with a minimum of one feature retained. Cross-validation is then performed using R0. 2 The feature subset corresponding to the maximum value is used as the final input feature subset;

[0082] Step S85: Refit the support vector regression model using the filtered input features, and calculate the coefficient of determination R on the independent test set. 2 The root mean square error (RMSE), mean absolute error (MAE), and mean squared error (MSE) were evaluated. After the model performance was evaluated, the prediction pipeline, including missing value handling, feature standardization, fixed feature selection, and support vector regression models, was refitted using all sample data to obtain the final model for independent prediction.

[0083] Step S86: Save the missing value handling and normalization preprocessor, the fixed feature selector established based on the recursive feature elimination results, and the final prediction pipeline of the support vector regression model as best_model.joblib, save the optimal parameters as best_params.json, and write the training file, target column, input feature column, numerical feature column, random seed, test set ratio, cross-validation settings, model selection rules, cross-validation metrics, test set metrics, and feature selection information into model_metadata.json;

[0084] The prediction method also includes step S9, which uses script 7 to perform independent prediction of the bulk elastic modulus. The corresponding script 7 - Bulk Elastic Modulus Prediction includes the following steps:

[0085] Step S91: Load the best_model.joblib model file and model_metadata.json metadata file saved in step S8, and read the new Excel spreadsheet data of the FCC single-phase multi-principal alloy to be predicted. The input feature columns include the aforementioned Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G; and make appropriate deletions based on the feature filtering results of step S85.

[0086] Step S92: Extract the features of the sample to be predicted according to the order of the feature_columns field in model_metadata.json, and construct the input matrix.

[0087] Step S93: Input the input matrix into the prediction pipeline in best_model.joblib. The prediction pipeline automatically completes the same missing value processing, feature scaling and feature filtering as in the training phase to obtain the predicted value of the bulk elastic modulus of the alloy to be predicted. The predicted value is written into the prediction column. The data table containing the original input information and the prediction column is exported as an Excel file to realize the batch prediction output of the bulk elastic modulus of unknown FCC single-phase multi-principal element alloys.

[0088] This invention proposes a machine learning method for predicting the bulk elastic modulus of face-centered cubic single-phase multi-principal alloys based on preferred occupancy. This method can achieve rapid prediction of the bulk elastic modulus of multi-principal alloys with low computational cost, improve the efficiency of candidate alloy screening, and accelerate the composition design and performance evaluation of multi-principal alloys.

[0089] Compared with the prior art, the present invention has the following advantages:

[0090] First, this invention improves the efficiency of screening target phase structures in high-dimensional composition space by combining empirical criteria for initial screening with a neural network multi-label phase classification model to screen single-phase candidate alloys for FCC.

[0091] Second, compared to the stochastic solid solution (SQS) modeling method that treats all lattice sites as statistically equivalent and randomly assigns constituent elements, this invention uses the temperature-dependent sublattice occupancy fraction as an atomic distribution constraint, thereby reducing the oversimplification of the real local atomic environment by the stochastic model.

[0092] Third, this invention uses energy-volume equation of state fitting based on first-principles calculation data to obtain bulk elastic modulus labels, so that the training data for machine learning has a clear physical source;

[0093] Fourth, this invention uses the intersection of k-nearest neighbor regression and Gaussian process regression to identify noisy samples, and reduces the interference of abnormal samples on model training by deleting or replacing labels.

[0094] Fifth, this invention employs a data augmentation method that combines centered logarithmic ratio transformation with local interpolation for component data, preventing the generated augmented samples from deviating from the local component distribution of the original samples and improving the model's generalization ability under small sample conditions;

[0095] Sixth, this invention uses elemental composition, empirical physical properties, and preferred site thermodynamic characteristics as model inputs, thereby improving the ability to characterize the composition-structure-performance relationship. Attached Figure Description

[0096] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0097] AppendixFigure 1 This is a schematic diagram of the overall process for predicting the elastic modulus of face-centered cubic single-phase multi-principal element alloys based on optimal occupancy.

[0098] Appendix Figure 2 This is a flowchart illustrating the process of generating candidate multi-principal element alloy compositions based on empirical phase formation criteria.

[0099] Appendix Figure 3 This is a schematic diagram showing the relationship between the site fraction and temperature of the FCC_CoCuNi multi-principal element alloy according to a preferred embodiment of the present invention.

[0100] Appendix Figure 4 This is a schematic diagram of the sublattice site distribution of the FCC_L12_AuCu3 prototype;

[0101] Appendix Figure 5 This is a schematic diagram of the sublattice combination nesting of a 3×3×3 expanded cell rear-centered cubic single-phase multi-principal element alloy based on the FCC_L12_AuCu3 prototype of the present invention.

[0102] Appendix Figure 6 This is a schematic diagram of the atomic number allocation based on the occupancy fraction when constructing a POSCAR using an FCC_CoCuNi multi-principal alloy according to a preferred embodiment of the present invention.

[0103] Appendix Figure 7 This is a schematic diagram of atomic position allocation when constructing a POSCAR in the FCC_CoCuNi multi-principal element alloy of a preferred embodiment of the present invention, taking the 1a sublattice as an example;

[0104] Appendix Figure 8 This is a schematic diagram of the sublattice combination nesting of the FCC_CoCuNi multi-principal element alloy according to a preferred embodiment of the present invention;

[0105] Appendix Figure 9 This is a schematic diagram of the bulk elastic modulus fitting results based on the Birch-Murnaghan equation of state after constructing five volume-scaled states of the FCC_CoCuNi multi-principal element alloy and calculating the total energy of the system in a preferred embodiment of the present invention.

[0106] Appendix Figure 10 This is a schematic diagram of the predicted label distribution after noise sample identification and cleaning in a preferred embodiment of the present invention;

[0107] Appendix Figure 11 This is a schematic diagram comparing the evaluation metrics of the support vector regression model with the base model without augmentation under different augmentation levels in a preferred embodiment of the present invention (Figure (a) represents the coefficient of determination R). 2 (b) represents the root mean square error (RMSE), and (c) represents the coverage of the prediction interval.

[0108] Appendix Figure 12 This is a schematic diagram of the screening results for the optimal enhancement amount n=500 obtained based on the comprehensive evaluation index Composite score in a preferred embodiment of the present invention;

[0109] Appendix Figure 13 It is a feature vector construction diagram excluding elemental composition, including schematic diagrams of physical property parameters and high-entropy thermodynamic parameters;

[0110] Appendix Figure 14 This is a training result diagram of the support vector regression model in a preferred embodiment of the present invention (in the figure, (a) represents the coefficient of determination curve during the parameter adjustment stage, and (b) represents the feature selection curve).

[0111] Appendix Figure 15 This is a schematic diagram of the high-throughput prediction results of the bulk elastic modulus of 4129 candidate FCC multi-principal alloy systems in a preferred embodiment of the present invention. Detailed Implementation

[0112] The present invention will be further described below with reference to embodiments. These embodiments are for illustrative purposes only and are not intended to limit the scope of protection of the present invention. Without departing from the technical concept of the present invention, those skilled in the art can make equivalent substitutions or conventional adjustments to the element range, heat treatment temperature, supercell size, machine learning model training parameters, and prediction output format.

[0113] As shown in the figure

[0114] A method for predicting the bulk elastic modulus of face-centered cubic (FCC) single-phase multi-principal element alloys based on preferred site occupancy is proposed. This method first generates candidate alloy compositions and screens FCC single-phase candidate systems. Then, thermodynamic phase equilibrium calculations are used to obtain the site fractions of each element in the 1a and 3c sublattices at the target temperature. Based on this, a preferred site supercell structure is constructed, and the bulk elastic modulus label is obtained through energy-volume relationships derived from first-principles calculations. Further, the samples are cleaned, enhanced, and feature-constructed. A regression model is trained using elemental composition, physical property parameters, and preferred site thermodynamic features to achieve batch prediction of the bulk elastic modulus of the alloys to be predicted. This method enables rapid prediction of the bulk elastic modulus of multi-principal element alloys at a lower computational cost, improving the efficiency of candidate alloy screening and accelerating the composition design and performance evaluation of multi-principal element alloys.

[0115] The prediction method includes the following steps:

[0116] Step S1: Candidate alloy composition generation and initial screening based on empirical criteria; set the candidate element set, number of principal components, upper and lower limits of atomic fraction for each principal component, composition step size, and empirical phase formation criterion threshold; iterate through the candidate element combinations and generate composition allocations that satisfy a total atomic fraction of 1 using a loop backtracking method; calculate the weighted average melting point, mixing entropy, mixing enthalpy, thermodynamic parameter Omega (Ω), atomic size mismatch parameter delta (δ), and valence electron concentration VEC for each effective composition; and output the candidate alloy compositions that simultaneously satisfy the empirical criteria as the candidate alloy composition pool of the candidate dataset.

[0117] Step S2, face-centered cubic single-phase classification and screening: The candidate alloy composition obtained in step S1 is converted into a fixed-dimensional element mole fraction feature vector required by the phase classification model, and input into the trained neural network multi-label phase classification model to obtain the probability values ​​of face-centered cubic phase FCC, body-centered cubic phase BCC or intermetallic compound phase IM respectively, and the FCC single-phase candidate alloy is screened according to the probability threshold.

[0118] Step S3: Constructing a structure file based on preferred occupancy; For the FCC single-phase candidate alloys screened in Step S2, using Thermo-Calc thermodynamics and phase equilibrium software, by writing a macro command file ***.tcm for the corresponding alloy system, thermodynamic calculations and phase equilibrium data are automatically calculated to obtain the occupancy fractions of different elements in the 1a and 3c sublattices at the target heat treatment temperature. Based on the FCC_L12_AuCu3 prototype structure and the set supercell size, the integer number of atoms of each element in each sublattice is calculated, and the coordinates of each sublattice are randomized and the elements are allocated according to the integer number of atoms to generate a preferred occupancy structure file POSCAR for first-principles calculations;

[0119] Step S4, constructing the bulk elastic modulus label; the structure file POSCAR obtained in step S3 is further optimized by first-principles calculation using the VASP software package. Based on the optimized equilibrium unit cell, multiple volume-scaled states are constructed and the total energy of the system is calculated. The bulk elastic modulus is obtained by fitting the state equation using the energy-volume method, and the bulk elastic modulus is used as the target label for supervised learning in the machine learning task.

[0120] Step S5, initial data cleaning; perform cross-validation prediction on the initial dataset composed of element composition and bulk elastic modulus labels, calculate the sample residuals and robust z-values ​​using at least two regression detection models respectively, identify noise samples using the intersection judgment method, and perform label replacement or deletion operations on the noise samples to obtain the cleaned sample dataset.

[0121] Step S6, small sample data augmentation; for the cleaned sample dataset, the mole fraction of alloy composition is mapped to the combined data transformation space to construct local interpolation augmentation candidate samples, and then the augmentation samples are screened through composition constraints, deduplication constraints, nearest neighbor distance constraints, label range constraints and local consistency constraints to obtain the augmented training dataset;

[0122] Step S7, Machine learning feature construction; parse the alloy chemical formula or read the element mole fraction to generate elemental composition features; calculate the combined features based on elemental physical property parameters, binary mixing enthalpy parameters, electronegativity, and the preferred site thermodynamic database to form a feature table for predicting the bulk elastic modulus.

[0123] Step S8: Support Vector Regression Model Training; Using the feature table obtained in Step S7 as input and the bulk elastic modulus obtained in Step S4 as the target column, the Support Vector Regression model is trained, and the optimal model is determined through repeated cross-validation, two-stage hyperparameter search, and recursive feature elimination cross-validation; the training results are saved as independently callable model files, parameter files, and metadata files.

[0124] Step S9, Independent Prediction: Read the feature table of the FCC single-phase multi-principal alloy to be predicted, load the model file and metadata file saved in step S8, verify and extract the features to be predicted according to the feature column order recorded in the metadata file, and call the trained support vector regression model to output the predicted value of the bulk elastic modulus of the alloy to be predicted.

[0125] In step S1, the candidate element set is Co, Cr, Cu, Fe, Mn, Ni, and V; the number of principal elements in the alloy system is 3, 4, or 5; when the number of principal elements is 3, the atomic fraction of each principal element is 30% to 38%; when the number of principal elements is 4, the atomic fraction of each principal element is 20% to 28%; when the number of principal elements is 5, the atomic fraction of each principal element is 16% to 24%; the composition step size is 2%, and elements not involved in the current alloy composition are filled with 0.

[0126] Step S1 uses script 1 to generate candidate alloy compositions and perform preliminary screening based on empirical criteria. This corresponds to script 1 - generating n-element multi-principal element alloy compositions, and includes the following steps:

[0127] Step S11: Initialize the candidate element set, element property dictionary, and binary mixing enthalpy parameter; the candidate element set includes Co, Cr, Cu, Fe, Mn, Ni, and V, the element properties include melting point, atomic radius, and valence electron concentration, and the binary mixing enthalpy parameter is used to calculate the mixing enthalpy characteristics of the candidate alloy;

[0128] Step S12: Set the number of principal components, the upper and lower limits of the atomic fraction of each principal component, the component step size, and the empirical phase formation criterion threshold; wherein, the number of principal components is 3, 4, or 5, when the number of principal components is 3, the atomic fraction of each principal component ranges from 30% to 38%, when the number of principal components is 4, the atomic fraction of each principal component ranges from 20% to 28%, when the number of principal components is 5, the atomic fraction of each principal component ranges from 16% to 24%, and the component step size is 2%.

[0129] Step S13: Use a combination traversal method to obtain the element combination under the specified number of principal elements, and use a loop algorithm to generate the composition allocation that satisfies the total atomic fraction of 1; fill the elements that do not participate in the current alloy composition with 0, so that each candidate sample forms a composition vector with Co, Cr, Cu, Fe, Mn, Ni and V in a fixed order.

[0130] Step S14: For each candidate alloy, calculate the weighted average melting point Tm, mixing entropy Smix, mixing enthalpy Hmix, thermodynamic parameter Omega, atomic size mismatch parameter delta, and valence electron concentration VEC based on the element mole fraction.

[0131] Step S15: Based on empirical phase formation criteria, candidate components are screened, and only candidate components that simultaneously satisfy Omega≥1.1, delta≤6.6%, and VEC≥8.0 are retained to form an initial candidate alloy composition pool.

[0132] Step S16 outputs the filtered candidate alloy composition pool as a data table containing fields such as Co, Cr, Cu, Fe, Mn, Ni, V, formula, Omega, delta, VEC, Smix, and Hmix, providing input for subsequent phase classification and screening.

[0133] Face-centered cubic single-phase multi-principal element alloys refer to candidate alloys that have a tendency to form face-centered cubic single phases and do not contain alloy systems that are identified as body-centered cubic phases or intermetallic compound phases.

[0134] In step S2, the output result is (FCC: 1, BCC: 0, IM: 0), and the phase stability at the target heat treatment temperature is confirmed by thermodynamic phase equilibrium calculation software. The face-centered cubic single-phase alloy is constructed based on the 3×3×3 supercell of the FCC_L12_AuCu3 prototype structure to introduce temperature-dependent ordered preferred site information. This does not mean that the candidate alloy must have an AuCu3 stoichiometric ratio or a completely L12 long-range ordered structure. When the site fractions of different elements in the 1a and 3c sublattices tend to be consistent, this double sublattice model can characterize the site state of a near-completely disordered FCC solid solution.

[0135] Step S2 uses script 2 for face-centered cubic single-phase classification and screening, corresponding to script2 - machine learning phase classifier, which includes the following steps:

[0136] Step S21: Read the multi-principal alloy phase structure dataset, take the mole fraction of each element in the alloy as the input feature, and take FCC, BCC and IM as phase labels. If the corresponding phase exists, it is represented by 1, and if it does not exist, it is represented by 0. Multiple phases are allowed to exist in a system at the same time.

[0137] Step S22: Convert the training data or candidate alloy composition to be predicted into a fixed-dimensional element mole fraction input matrix, and use the Min-Max normalizer to numerically normalize the input features to reduce the impact of different element content scale differences on model training.

[0138] Step S23: Construct a neural network multi-label phase classification model; the model includes an input layer, a first fully connected hidden layer, a second fully connected hidden layer, and an output layer. The number of nodes in the first hidden layer is set to 128, and the number of nodes in the second hidden layer is set to 64. The hidden layer uses the ReLU activation function and combines Dropout or batch normalization units to suppress overfitting. The output layer uses the Sigmoid activation function to output the probabilities of the three phase labels FCC, BCC, and IM respectively.

[0139] Step S24: The multi-label phase classification model is trained using the binary cross-entropy loss function, Adam optimizer, weight decay and 10-fold cross-validation to obtain a stable phase classification model;

[0140] Step S25: Input the initial candidate alloy composition pool obtained in step S1 into the trained phase classification model to obtain the label probabilities of FCC, BCC and IM respectively; when the label probability of the corresponding phase is greater than the classification threshold of 0.5, it is considered that the corresponding phase exists; only when the label probability of FCC is greater than 0.5 and the label probabilities of BCC and IM are both less than 0.5, the candidate system is determined as a candidate alloy with a face-centered cubic single phase.

[0141] Step S26: The selected FCC single-phase candidate alloys are output according to fields such as Co, Cr, Cu, Fe, Mn, Ni, V and formula, providing detailed information of the candidate system for subsequent construction of preferred site structures and high-throughput prediction.

[0142] In step S3, the target heat treatment temperature is set to 973K. The FCC_L12_AuCu3 prototype structure is composed of nested 1a sublattices occupied by Au and 3c sublattices occupied by Cu. The conventional unit cell of the face-centered cubic lattice consists of lattice sites located at the eight vertices and six face centers of the unit cell. After calculation based on the periodic symmetry of the lattice sites, each unit cell contains one 1a site and three 3c sites. When a 3×3×3 supercell is used, the conventional FCC_L12_AuCu3 unit cell is periodically expanded by a factor of 3 along the three lattice directions to form a supercell with 108 lattice sites, including 27 1a sublattice sites and 81 3c sublattice sites.

[0143] Step S3 involves constructing the structure file based on optimal placeholders, including the following steps:

[0144] Step S31: For the FCC single-phase candidate alloys selected in step S2, based on the existing end-group thermodynamic database ***.TDB file, perform phase equilibrium calculations using the Thermo-Calc thermodynamic and phase equilibrium calculation software package to generate a YSITE-T.EXP output file containing sublattice occupied fractions (SOFs);

[0145] Step S32: Extract the placeholder fraction rows corresponding to the target heat treatment temperature from the YSITE-T.EXP data;

[0146] Step S33: Using the FCC_L12_AuCu3 prototype structure as the structural basis, the vertex sites in the conventional unit cell are defined as 1a sublattices, and the face-centered sites are defined as 3c sublattices. The total number of sites in the 1a and 3c sublattices is determined according to the selected supercell size. The unit cell contains one 1a sublattice site and three 3c sublattice sites. When constructing the required computational model, the cell is expanded to 3×3×3, at which point the number of 1a sublattice sites is 27 and the number of 3c sublattice sites is 81.

[0147] Step S34: Multiply the occupancy fraction of each element on the corresponding sublattice by the total number of sites in the corresponding sublattice to obtain the theoretical number of atoms occupied by each element in each sublattice; round the theoretical number of atoms occupied, and perform residual iterative correction based on the residual before and after rounding, so that the sum of the number of atoms of all elements in each sublattice is equal to the total number of sites in the corresponding sublattice.

[0148] Step S35: Fill each element into the corresponding sublattice coordinates according to the integer number of atoms determined in step S34, assign a random number generated by the rand() function of the preset random seed to each sublattice coordinate, and perform random arrangement of the coordinates according to the random number.

[0149] Step S36: Merge the coordinates and total number of atoms of the same type according to the element symbol order, fill the allocated site information into the standard coordinate file POSCAR, and add a selective dynamic marker at the end of each atomic coordinate to eliminate the rigid body translational degree of freedom; mark the origin with fractional coordinates of 0 0 0 as FFF, and fix one atom in the x-axis, y-axis and z-axis directions respectively, which moves only in that direction, and mark them as TFF, FTF and FFT respectively, and mark all other coordinates as T TT.

[0150] In step S4, structural optimization includes volume relaxation and complete relaxation; based on the optimized equilibrium unit cell, the three lattice vectors are scaled proportionally using multiple lattice linear scaling factors, which include at least 0.96, 0.98, 1.00, 1.02 and 1.04; the energy-volume equation of state is the third-order Birch-Murnaghan equation of state, and the bulk elastic modulus is obtained from the fitted zero-pressure bulk elastic modulus parameters;

[0151] The acquisition of the bulk elastic modulus label in step S4 includes the following steps:

[0152] Step S41: Perform first-principles structure optimization on the POSCAR structure file obtained in step S36 based on the VASP software package, including cell volume, cell shape and atomic position relaxation, to obtain a balanced crystal structure.

[0153] Step S42: Construct multiple volume scaling states based on the equilibrium volume V0 obtained in step S41; in a preferred embodiment, the lattice scaling factors are set to 0.96, 0.98, 1.00, 1.02, and 1.04, so that the three lattice vectors are scaled proportionally according to the corresponding scaling factors to obtain 0.96. 3 V0, 0.98 3 V0, 1.00 3 V0, 1.02 3 V0, 1.04 3 V0, and calculate the total energy of the system under different volumes to form an energy-volume dataset;

[0154] Step S43: The energy-volume data are fitted using the third-order Birch-Murnaghan equation of state to extract the zero-pressure body elastic modulus.

[0155]

[0156] Step S44: The bulk elastic modulus label obtained in step S43 is merged with the corresponding alloy composition and preferred occupancy structure information to form basic sample data for model training.

[0157] Step S5, use script 3 to perform initial sample data cleaning, corresponding to script3-data cleaning, including the following steps:

[0158] Step S51: Read the initial data table containing the element composition column and the bulk elastic modulus label column, and set the number of input feature columns, the target column position, the number of cross-validation folds, the random seed, the robust z-value threshold, and the cleaning action; the cleaning action includes deleting noise samples or replacing noise sample labels;

[0159] Step S52: Construct at least two independent regression detection models; the regression detection models include a k-nearest neighbor regression detection model and a Gaussian process regression detection model, and generate cross-validation prediction values ​​for each sample based on K-fold cross-validation with random seeds;

[0160] Step S53: Calculate the prediction residuals for each sample under the k-nearest neighbor regression detection model and the Gaussian process regression detection model, and calculate the corresponding robust z-value based on the median and absolute median of the residuals to reduce the impact of extreme residuals on the anomaly detection threshold.

[0161] Step S54: Noise samples are identified using the intersection determination method. Only when the absolute value of the residual robust z-value of the same sample under the k-nearest neighbor regression detection model and the absolute value of the residual robust z-value under the Gaussian process regression detection model are both greater than the preset threshold, the sample is determined to be a noise sample.

[0162] Step S55: When the cleaning action is deletion, the noise sample is removed from the initial dataset; when the cleaning action is replacement, the bulk elastic modulus label of the noise sample is replaced with the median or average of the predicted values ​​of the two detection models.

[0163] Step S56: Output the cleaned sample data table for subsequent small sample data augmentation and model training.

[0164] In step S6, the local interpolation enhancement includes: before performing the logarithmic ratio transformation, performing zero-value replacement and closed-loop normalization on the element mole fraction composition vector, replacing the element mole fractions less than or equal to the preset threshold with the preset threshold, and re-normalizing all element mole fractions after replacement so that the sum of all element mole fractions is 1; during interpolation in the transformation space, interpolating the transformed composition vector and bulk elastic modulus label of the paired samples according to the interpolation coefficients to obtain the candidate enhanced composition vector and candidate enhanced label; after enhancement, inversely transforming the candidate enhanced composition vector to the original composition space, and determining the optional element set of the candidate enhanced sample according to the union of the non-zero element sets in the paired samples, selecting a preset number of elements from the optional element set as the non-zero elements of the candidate enhanced sample, and setting the remaining element mole fractions to 0.

[0165] In step S6, the enhancement scheme and enhancement amount are determined by comparing the final results of the outer layer repeated hold-out validation and the inner layer cross-validation; the candidate enhancement amounts are multiple preset enhancement amounts; the evaluation learner is a support vector regression model; the evaluation index includes the coefficient of determination R. 2 The root mean square error (RMSE) and the actual proportion of true values ​​in the test set falling within the 95% prediction range are used to determine the optimal net augmentation based on the comprehensive evaluation of the three indicators. The true samples and augmented samples together constitute the dataset used for machine learning training.

[0166] Step S6 uses script 4 to perform small sample data augmentation, corresponding to script4-data augmentation, which includes the following steps:

[0167] Step S61: Read the cleaned sample dataset obtained in step S5, use Co, Cr, Cu, Fe, Mn, Ni, and V as component feature columns, use Bulk_Modulus as the target column, and set the outer layer repeat retention validation seed, outer layer test set ratio, inner layer cross-validation fold number, and candidate net enhancement amount.

[0168] Step S62, perform zero-value replacement processing on the zero component in the element mole fraction, using 10 -6 Replace the zero components and scale the remaining non-zero components proportionally so that the sum of the mole fractions of each element remains 1 after processing; perform a centered logarithmic ratio transformation or an equidistant logarithmic ratio transformation on the closed normalized component vector to map it to a transformation space that can perform Euclidean distance calculations.

[0169] Step S63: Select nearest neighbor paired samples for the real sample in the transformation space, generate candidate enhancement components according to the interpolation coefficients, and synchronously interpolate the bulk elastic modulus label to obtain candidate enhancement labels;

[0170] Step S64: Inversely transform the candidate enhancement components from the transformation space back to the original component space, and perform component normalization and bit retention processing to ensure that the sum of the mole fractions of each element in the enhancement sample remains 1. Identify the elements in the real sample and paired sample whose mole fractions are greater than the non-zero threshold, and determine the union of the non-zero element sets of the two samples as the optional element set for the candidate enhancement sample. Based on the preset range of non-zero principal components and the magnitude of the mole fractions of each element after the inverse transformation, select a corresponding number of elements from the optional element set as the non-zero elements of the candidate enhancement sample, and set the mole fractions of the remaining elements to 0.

[0171] Step S65: Perform multiple screening on the candidate enhanced samples. The multiple screening includes at least non-zero principal component number constraint, component range constraint, nearest neighbor distance filtering, local label consistency filtering, label quantile range constraint, duplicate sample removal, and component retention number constraint.

[0172] Step S66: Compare model performance under different augmentation methods and different augmentation amounts through outer layer hold-out validation and inner layer cross-validation; evaluation metrics include the coefficient of determination R. 2 The root mean square error (RMSE) and 95% prediction interval coverage were evaluated, and the final enhancement scheme was determined based on the comprehensive evaluation.

[0173]

[0174] in, All three indicators were normalized to the [0,1] interval.

[0175] R respectively 2 In a preferred approach, the weights of RMSE and Coverage are set equally, each to 1 / 3.

[0176] In step S7, based on the calculation results of the Thermo-Calc thermodynamic and phase equilibrium calculation software package, the placeholder fraction line corresponding to the target heat treatment temperature in the YSITE-T.EXP output file is read for each candidate system to calculate Sigma (deltaG), configuration entropy Sconfig and Gibbs free energy characteristic G under ordered placeholder conditions.

[0177] Step S7, use script 5 to construct machine learning input features, corresponding to script5 - feature engineering, including the following steps:

[0178] Step S71: Read the data table containing alloy chemical formulas or element mole fractions; when the input is a chemical formula, use regular expressions to identify the element symbols and their corresponding values, and extract the mole fractions of seven elements: Co, Cr, Cu, Fe, Mn, Ni and V.

[0179] Step S72: Normalize the mole fractions of the seven elements in each alloy sample within the row so that the sum of the mole fractions of the elements is 1, and form the element composition characteristics in a fixed element order.

[0180] Step S73: Based on the element's melting point, atomic radius, valence electron concentration, Pauling electronegativity, and binary mixing enthalpy parameter, calculate the empirical physical properties and thermodynamic combination characteristics, which include at least Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, and X;

[0181] Step S74: Based on Thermo-Calc thermodynamic and phase equilibrium calculation results, read the occupancy fraction of different elements on the 1a and 3c sublattices at the target heat treatment temperature, and calculate the preferred occupancy thermodynamic characteristics in combination with the end-base thermodynamic database. The preferred occupancy thermodynamic characteristics include Sigma (deltaG), Sconfig and G.

[0182] Step S75: The element composition features obtained in step S72, the empirical property combination features obtained in step S73, and the preferred occupancy thermodynamic features obtained in step S74 are merged to generate a machine learning input feature matrix for predicting the bulk elastic modulus.

[0183] In step S8, the input feature columns of the support vector regression model include elemental composition features: Co, Cr, Cu, Fe, Mn, Ni, V; physical property features: Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X; and thermodynamic features: Sigma(deltaG), Sconfig, and G.

[0184] Step S8, use script 6 to train and save the support vector regression model, corresponding to script6 - prediction model training, including the following steps:

[0185] Step S81: Read the data table containing complete input features and bulk elastic modulus labels, determine Bulk_Modulus as the target column, and determine the input feature columns including Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G;

[0186] Step S82: Divide the data into training set and test set according to the preset test set ratio; within the training set, use 10-fold repeated cross-validation to train the model.

[0187] Step S83: Construct a two-stage hyperparameter search strategy. The first stage involves a random search within a large parameter space to quickly locate the parameter region with superior performance. The second stage involves constructing a local grid around the advantageous parameter region obtained in the first stage and using grid search to further optimize the hyperparameters. In a preferred embodiment, the hyperparameter space for the support vector regression model is constructed as follows: kernel = ['linear', 'rbf', 'poly'], C = [1, 3, 10, 30, 100], epsilon = [0.01, 0.05, 0.1, 0.3], gamma (rbf, poly) = ['scale', 0.01, 0.03, 0.1, 0.3], degree (poly) = [2, 3], coef0 (poly) = [0.0, 0.5, 1.0]; the parameter combination is selected to maximize the cross-validation determination coefficient R. 2 As the main indicator, when R 2 When the differences are small, compare RMSE. 2 When both RMSE and MAE are close, further compare them.

[0188] Step S84: Construct a support vector regression model using the optimal parameters determined in step S83. Use permutation importance to determine the importance of each input feature and perform recursive feature elimination cross-validation. The permutation importance is calculated repeatedly 5 times. The step size of recursive feature elimination is to delete 1 feature each time, retain at least 1 feature, and use the feature subset corresponding to the maximum cross-validation R² as the final input feature subset.

[0189] Step S85: Refit the support vector regression model using the filtered input features, and calculate the coefficient of determination R on the independent test set. 2 The root mean square error (RMSE), mean absolute error (MAE), and mean squared error (MSE) were evaluated. After the model performance was evaluated, the prediction pipeline, including missing value handling, feature standardization, fixed feature selection, and support vector regression models, was refitted using all sample data to obtain the final model for independent prediction.

[0190] Step S86: Save the missing value handling and normalization preprocessor, the fixed feature selector established based on the recursive feature elimination results, and the final prediction pipeline of the support vector regression model as best_model.joblib, save the optimal parameters as best_params.json, and write the training file, target column, input feature column, numerical feature column, random seed, test set ratio, cross-validation settings, model selection rules, cross-validation metrics, test set metrics, and feature selection information into model_metadata.json;

[0191] The prediction method also includes step S9, which uses script 7 to perform independent prediction of the bulk elastic modulus. The corresponding script 7 - Bulk Elastic Modulus Prediction includes the following steps:

[0192] Step S91: Load the best_model.joblib model file and model_metadata.json metadata file saved in step S8, and read the new Excel spreadsheet data of the FCC single-phase multi-principal alloy to be predicted. The input feature columns include the aforementioned Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G; and make appropriate deletions based on the feature filtering results of step S85.

[0193] Step S92: Extract the features of the sample to be predicted according to the order recorded in the feature_columns field of model_metadata.json, and construct the input matrix.

[0194] Step S93: Input the input matrix into the prediction pipeline in best_model.joblib. The prediction pipeline automatically completes the same missing value processing, feature scaling and feature filtering as in the training phase to obtain the predicted value of the bulk elastic modulus of the alloy to be predicted. The predicted value is written into the prediction column. The data table containing the original input information and the prediction column is exported as an Excel file to realize the batch prediction output of the bulk elastic modulus of unknown FCC single-phase multi-principal element alloys.

[0195] Example 1: Candidate Component Generation and FCC Single-Phase Screening

[0196] In this embodiment, by running script 1, a candidate element set E={Co, Cr, Cu, Fe, Mn, Ni, V} is first established. Ternary, quaternary, and pentagonal near-equimolar multi-principal element alloys are generated according to the different numbers of principal components. For the pentagonal system, the atomic fraction of each element is set to 16% to 24%; for the quaternary system, the atomic fraction of each element is set to 20% to 28%; and for the ternary system, the atomic fraction of each element is set to 30% to 38%. The composition step size for each system is set to 2%.

[0197] During the generation process, candidate element combinations are traversed, and compositions satisfying a total mole fraction of 1 are assigned. For each candidate composition, its melting point weighted average, mixing entropy, mixing enthalpy, thermodynamic parameter Ω, atomic size mismatch parameter δ, and valence electron concentration VEC are calculated. The mixing enthalpy is calculated by weighted summation based on the binary mixing enthalpy table and element mole fractions, the mixing entropy is calculated using the ideal mixing entropy formula, and Ω is determined by the melting point weighted average, mixing entropy, and mixing enthalpy.

[0198] When a candidate composition simultaneously satisfies Ω ≥ 1.1, δ ≤ 6.6, and VEC ≥ 8, it is retained as a candidate composition with the potential to form a face-centered cubic solid solution. This yields the initial pool of candidate alloy compositions, and the screening process is as follows: Figure 2 As shown, each row in the composition pool includes seven fixed headers: Co, Cr, Cu, Fe, Mn, Ni, and V. Elements not involved in the current alloy composition are represented by 0.

[0199] Subsequently, a phase classification model was constructed using script 2. The phase classification model used experimentally validated multi-principal alloy phase structure data as training data. The input was element mole fraction features, and the output was three labels: FCC, BCC, and IM. The model employed a two-layer fully connected neural network, with a first hidden layer of 128 nodes and a second hidden layer of 64 nodes. The hidden layers used the ReLU activation function and Dropout was applied to suppress overfitting. The output layer used the Sigmoid function to output the probabilities of the three phase labels.

[0200] During training, a binary cross-entropy loss function and the Adam optimizer are used to learn the nonlinear relationship between composition and phase structure through multi-label classification. After predicting the candidate composition pool, if the FCC label probability of a candidate alloy is higher than the threshold of 0.5, and the BCC and IM labels are not determined to exist, then the candidate alloy is selected as a face-centered cubic single-phase candidate system.

[0201] Example 2: Construction of a structural model based on optimal occupancy and calculation of bulk elastic modulus labels

[0202] In this embodiment, a preferred site-occupying structure model is performed on the face-centered cubic single-phase candidate system selected in Example 1. First, for the target alloy system, using Thermo-Calc thermodynamics and phase equilibrium software, an existing thermodynamic database TDB file and the corresponding alloy system macro command file ***.tcm are input to perform automated thermodynamic calculations and phase equilibrium data calculations, obtaining the site fractions of different elements in the 1a and 3c sublattices at the target heat treatment temperature. Figure 3The image shows the occupancy fraction of the FCC_CoCuNi multi-principal element alloy. The heat treatment temperature for calculating the bulk elastic modulus was selected as 973 K to obtain the occupancy fraction of each element at the corresponding temperature, which is used to represent the occupancy probability of different elements in the 1a and 3c sublattices of the FCC_L12_AuCu3 prototype structure.

[0203] according to Figure 4 The FCC_L12_AuCu3 prototype structure shown, after periodic adjustments, actually contains one 1a sublattice site and three 3c sublattice sites per unit cell, meaning the POSCAR file contains one Au atom and three Cu atoms. When constructing a 3×3×3 supercell, the supercell contains 27 1a sublattice sites and 81 3c sublattice sites, for a total of 10⁸ atoms. Figure 5 Classification diagram of the 3×3×3 supercell sublattice of the FCC_L12_AuCu3 structure. Figure 6 As shown, the theoretical number of atoms of each element in each sublattice is obtained by multiplying the occupancy probability of each element in each sublattice by the total number of sites in the corresponding sublattice. If the calculation result is not an integer, it is adjusted by rounding and residual correction to keep the total number of atoms in the corresponding sublattice unchanged.

[0204] Figure 7 As shown, after determining the integer number of atoms of each element on the 1a and 3c sublattices, the coordinates of each sublattice site are randomly arranged and assigned to the randomized coordinates according to the element symbol and the corresponding number of atoms, thereby generating a structure file that meets the requirements of preferred occupancy statistics. To facilitate subsequent first-principles structure optimization, selective dynamic markers can be set in the structure file, i.e., fixing an origin atom and restricting reference atoms in the three directions respectively.

[0205] The calculation model for calculating the elastic modulus of a volume is obtained as follows: Figure 8 As shown, the structure was then optimized using the first-principles software VASP. The calculations employed a PAW pseudopotential and a GGA-PBE exchange-correlation functional, with the plane-wave cutoff energy set to 400 eV and the electron self-consistent energy convergence criterion set to 1 × 10⁻⁶ eV. -5The ion relaxation convergence criterion was set to −0.05 eV / Å. Calculations were performed using spin polarization with ISPIN=2; for the metallic system, the first-order Methfessel-Paxton broadening method was used with ISMEAR=1 and SIGMA=0.2 eV. Structure optimization was performed in two stages. The first stage used ISIF=7 to relax the cell volume while maintaining the cell shape and atomic fraction coordinates, obtaining a preliminary equilibrium volume. The second stage used ISIF=3 to fully relax the cell volume, cell shape, and atomic positions. Ion optimization employed the conjugate gradient algorithm with IBRION=2 and POTIM=0.3.

[0206] After structural optimization, the bulk elastic modulus was further calculated using the energy-volume method. Using the optimized equilibrium structure as a baseline, five lattice scaling factors of 0.96, 0.98, 1.00, 1.02, and 1.04 were applied to the unit cell to calculate the total energy of the system under different volume states. Keeping the unit cell shape and volume unchanged for each volume state, only the internal coordinates of the atoms were relaxed, thus obtaining a set of discrete volume-energy data.

[0207] at last, Figure 9 The results shown are obtained by fitting the volume-energy relationship curve using the third-order Birch-Murnaghan equation of state and extracting the zero-pressure volume elastic modulus from the fitting parameters. The obtained volume elastic modulus serves as a label for the alloy system and, together with the corresponding alloy element composition, site characteristics, and thermodynamic characteristics, constitutes the subsequent machine learning training data.

[0208] Example 3: Data Cleaning, Data Augmentation, and Feature Label Calculation

[0209] In this embodiment, the alloy dataset obtained from Example 2 is cleaned. Since the initial dataset may contain anomalous labels or extreme samples, directly using it for machine learning training may affect the model's generalization ability. Therefore, a cross-validation residual identification strategy is adopted, using k-nearest neighbor regression and Gaussian process regression models as noise detection models to obtain the cross-validation prediction value and residual for each sample.

[0210] A robust z-value is calculated for the residuals of each sample. A sample is identified as a noise sample if it exceeds a preset anomaly threshold in both k-nearest neighbor regression and Gaussian process regression models. Instead of directly deleting noise samples, their bulk elastic modulus label is replaced with the average of the predicted values ​​from both models. This method reduces interference from anomaly labels while preserving the sample's component space coverage information. Figure 10 The predicted label distribution after data cleaning for a dataset with 200 samples.

[0211] After data cleaning, small sample data is augmented. All data augmentation calculations are first performed by centering or equidistant logarithmic ratio transformations of the element mole fractions, mapping them to a space suitable for Euclidean distance calculation. Data augmentation can be achieved using local interpolation methods (denoted as model2), generating new component and bulk modulus samples between adjacent real samples; alternatively, distribution estimation methods (denoted as model3, used only as a control method and not the final preferred augmentation scheme) can be used to fit a Bayesian Gaussian mixture model in the transformed component-label joint space and sample to generate candidate augmented samples. After the generated augmented samples are inversely transformed back to the original component space, component normalization, element number constraints, duplicate sample removal, nearest neighbor density filtering, local consistency filtering, and diversity screening are performed. Figure 11 With candidate augmentation values ​​of 100, 200, 300, 400, 500, 600, and 700, respectively, support vector regression models were used as the evaluation learners to compare the performance evaluation results under different methods, including no augmentation, local interpolation, and distribution estimation.

[0212] In a preferred embodiment, nested cross-validation is further used to compare different enhancement strategies and enhancement amounts. This is achieved by comprehensively considering R... 2 A comprehensive scoring function is constructed using three indicators: RMSE and Coverage, such as... Figure 12 As shown, the interpolation enhancement method based on 500 enhancement values ​​was selected as the best enhancement strategy.

[0213] In the characteristic label calculation stage, the mole fractions of seven elements—Co, Cr, Cu, Fe, Mn, Ni, and V—are first obtained through analysis of the alloy's chemical formula. Subsequently, empirical characteristics of the multi-principal element alloy, including Hmix, delta, Omega, Smix, Tm, and VEC, are calculated based on parameters such as the melting point, atomic radius, valence electron concentration, electronegativity, and enthalpy of mixing of the pure elements. Further calculations are then performed on the weighted average and weighted standard deviation of different physical property parameters, such as sigma_X, sigma_H, sigma_Tm, and sigma_VEC.

[0214] Based on the thermodynamic results of optimal site selection, site-related thermodynamic characteristics such as Sigma(deltaG), Sconfig, and G were also calculated. Finally, a combined characteristic table was formed, including Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G. Figure 13 Methods for calculating all feature labels except element features are given.

[0215] Example 4: SVR Model Training

[0216] In this embodiment, a support vector regression model is used to predict the bulk elastic modulus of a face-centered cubic single-phase multi-principal element alloy. The combined features obtained in Example 3 are used as the input matrix, and the bulk elastic modulus obtained in Example 2 is used as the target column. The dataset used in this embodiment contains a total of 689 valid samples. A random seed of 42 is used to divide the dataset into a training set and an independent test set in a 9:1 ratio, where the training set contains 620 samples and the independent test set contains 69 samples.

[0217] The hyperparameter search for the support vector regression model employs a two-stage search strategy. The first stage involves a random search within a large parameter space to quickly locate the optimal parameter region. The second stage performs a local grid search around the parameter range obtained in the first stage to further determine the optimal parameter combination. When selecting a model, cross-validation R-squared is prioritized. 2 When R 2 When the differences are small, the RMSE and MAE indices are compared. In a preferred embodiment, the optimal hyperparameters of the support vector regression model are the RBF radial basis function kernel, with a penalty coefficient C of 60.0, epsilon of 0.2, and gamma of scale. After determining the model hyperparameters, this embodiment uses permutation importance to evaluate the importance of each input feature, with 5 permutations; during the recursive elimination process, one feature is deleted each time, and cross-validation R² is used as the feature subset evaluation index. Feature selection results show that when 17 input features are retained, the cross-validation R² reaches its maximum value of 0.892117. Figure 14 The figure shows the overall predictive performance of the support vector regression model during the parameter tuning stage and the corresponding feature selection curve.

[0218] The support vector regression model is retrained using the aforementioned 17 features and optimal parameters. The model file is saved as best_model.joblib, the optimal parameters are saved as best_params.json, and the training file name, target column, input feature columns, cross-validation settings, best model name, model performance metrics, and feature selection information are saved as model_metadata.json. The saved model file already includes the preprocessing, feature selection, and final regression model; therefore, standardization and feature selection do not need to be repeated during independent prediction.

[0219] Example 5: High-throughput prediction application

[0220] For each component in the candidate alloy pool, the complete combined features are first calculated according to Example 3. During independent prediction, a new feature table of the alloy to be predicted is read, and the feature_columns recorded in model_metadata.json are loaded. The program first checks whether the prediction table contains all required input feature columns. If any columns are missing, prediction is terminated and the names of the missing features are output; if the feature columns are complete, the input features are extracted according to the order specified by the metadata, and the model pipeline in best_model.joblib is called for prediction.

[0221] After the prediction is completed, the predicted bulk modulus is written to a new column in the table and exported as an Excel file. For large candidate composition pools, this independent prediction process can automatically perform batch predictions to obtain the predicted bulk modulus value for each candidate face-centered cubic single-phase multi-principal element alloy. Figure 15 The bulk elastic modulus prediction results of 4129 candidate FCC multi-principal alloy systems generated by the iterative backtracking method are input into the support vector regression model.

Claims

1. A method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy based on optimal occupancy, characterized in that: First, candidate alloy compositions are generated and FCC single-phase candidate systems are screened. Then, the occupancy fractions of each element in the 1a and 3c sublattices at the target temperature are obtained through thermodynamic phase equilibrium calculations. Based on this, a preferred occupancy supercell structure is constructed, and the bulk elastic modulus label is obtained by calculating the energy-volume relationship obtained through first principles. The samples are further cleaned, enhanced, and feature-constructed. The regression model is trained using elemental composition, physical property parameters, and preferred occupancy thermodynamic features to achieve batch prediction of the bulk elastic modulus of the alloy to be predicted.

2. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 1, characterized in that: Includes the following steps: Step S1: Candidate alloy composition generation and preliminary screening based on empirical criteria; The candidate element set, number of principal components, upper and lower limits of atomic fraction for each principal component, composition step size, and empirical phase formation criterion threshold are defined. The candidate element combinations are traversed, and a composition allocation that satisfies a total atomic fraction of 1 is generated by a loop backtracking method. For each effective composition, the weighted average melting point, mixing entropy, mixing enthalpy, thermodynamic parameter Omega (Ω), atomic size mismatch parameter delta (δ), and valence electron concentration VEC are calculated. The candidate alloy compositions that simultaneously satisfy the empirical criteria are output as the candidate alloy composition pool of the candidate dataset. Step S2, face-centered cubic single-phase classification and screening: The candidate alloy composition obtained in step S1 is converted into a fixed-dimensional element mole fraction feature vector required by the phase classification model, and input into the trained neural network multi-label phase classification model to obtain the probability values ​​of face-centered cubic phase FCC, body-centered cubic phase BCC or intermetallic compound phase IM respectively, and the FCC single-phase candidate alloy is screened according to the probability threshold. Step S3: Construction of a POSCAR structure file based on preferred occupancy. For the FCC single-phase candidate alloys screened in Step S2, thermodynamic calculations and phase equilibrium data are automatically calculated by writing macro command files for the corresponding alloy system using Thermo-Calc thermodynamic and phase equilibrium software. The occupancy fractions of different elements in the 1a and 3c sublattices at the target heat treatment temperature are obtained. Based on the FCC_L12_AuCu3 prototype structure and the set supercell size, the integer number of atoms of each element in each sublattice is calculated. After randomizing the coordinates of each sublattice, the elements are allocated according to the integer number of atoms to generate a POSCAR structure file for first-principles calculations. Step S4, constructing the bulk elastic modulus label; the structure file POSCAR obtained in step S3 is further optimized by first-principles calculation using software, multiple volume scaling states are constructed based on the optimized equilibrium unit cell and the total energy of the system is calculated, the bulk elastic modulus is obtained by fitting the state equation using the energy-volume method, and the bulk elastic modulus is used as the target label for supervised learning in the machine learning task. Step S5, initial data cleaning; perform cross-validation prediction on the initial dataset composed of element composition and bulk elastic modulus labels, calculate the sample residuals and robust z-values ​​using at least two regression detection models respectively, identify noise samples using the intersection judgment method, and perform label replacement or deletion operations on the noise samples to obtain the cleaned sample dataset. Step S6, small sample data augmentation; for the cleaned sample dataset, the mole fraction of alloy composition is mapped to the combined data transformation space to construct local interpolation augmentation candidate samples, and then the augmentation samples are screened through composition constraints, deduplication constraints, nearest neighbor distance constraints, label range constraints and local consistency constraints to obtain the augmented training dataset; Step S7, Machine learning feature construction; parse the alloy chemical formula or read the element mole fraction to generate elemental composition features; calculate the combined features based on elemental physical property parameters, binary mixing enthalpy parameters, electronegativity, and the preferred site thermodynamic database to form a feature table for predicting the bulk elastic modulus. Step S8: Support Vector Regression Model Training; Using the feature table obtained in Step S7 as input and the bulk elastic modulus obtained in Step S4 as the target column, the Support Vector Regression model is trained, and the optimal model is determined through repeated cross-validation, two-stage hyperparameter search, and recursive feature elimination cross-validation; the training results are saved as independently callable model files, parameter files, and metadata files. Step S9, Independent Prediction: Read the feature table of the FCC single-phase multi-principal alloy to be predicted, load the model file and metadata file saved in step S8, verify and extract the features to be predicted according to the feature column order recorded in the metadata file, and call the trained support vector regression model to output the predicted value of the bulk elastic modulus of the alloy to be predicted.

3. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S1, the candidate element set is Co, Cr, Cu, Fe, Mn, Ni, and V; the number of principal elements in the alloy system is 3, 4, or 5; when the number of principal elements is 3, the atomic fraction of each principal element is 30% to 38%; when the number of principal elements is 4, the atomic fraction of each principal element is 20% to 28%; when the number of principal elements is 5, the atomic fraction of each principal element is 16% to 24%; the composition step size is 2%, and elements not involved in the current alloy composition are filled with 0. Step S1 involves generating candidate alloy compositions and conducting preliminary screening based on empirical criteria to generate n-element multi-principal element alloy compositions, including the following steps: Step S11: Initialize the candidate element set, element property dictionary, and binary mixing enthalpy parameter; the candidate element set includes Co, Cr, Cu, Fe, Mn, Ni, and V, the element properties include melting point, atomic radius, and valence electron concentration, and the binary mixing enthalpy parameter is used to calculate the mixing enthalpy characteristics of the candidate alloy; Step S12: Set the number of principal components, the upper and lower limits of the atomic fraction of each principal component, the component step size, and the empirical phase formation criterion threshold; Step S13: Use a combination traversal method to obtain the element combination under the specified number of principal elements, and use a loop algorithm to generate the composition allocation that satisfies the total atomic fraction of 1; fill the elements that do not participate in the current alloy composition with 0, so that each candidate sample forms a composition vector with Co, Cr, Cu, Fe, Mn, Ni and V in a fixed order. Step S14: For each candidate alloy, calculate the weighted average melting point Tm, mixing entropy Smix, mixing enthalpy Hmix, thermodynamic parameter Omega, atomic size mismatch parameter delta, and valence electron concentration VEC based on the element mole fraction. Step S15: Based on empirical phase formation criteria, candidate components are screened, and candidate components that simultaneously satisfy Omega≥1.1, delta≤6.6%, and VEC≥8.0 are retained to form an initial candidate alloy composition pool. Step S16 outputs the filtered candidate alloy composition pool as a data table containing fields such as Co, Cr, Cu, Fe, Mn, Ni, V, formula, Omega, delta, VEC, Smix, and Hmix, providing input for subsequent phase classification and screening.

4. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: Face-centered cubic single-phase multi-principal element alloys refer to candidate alloys that have a tendency to form face-centered cubic single phases and do not have alloy systems that are identified as body-centered cubic phases or intermetallic compound phases. In step S2, the output result is (FCC: 1, BCC: 0, IM: 0), and the phase stability at the target heat treatment temperature is confirmed by thermodynamic phase equilibrium calculation software. The face-centered cubic single-phase alloy is constructed based on the 3×3×3 supercell of the FCC_L12_AuCu3 prototype structure to introduce temperature-dependent ordered preferred site information. When the site fractions of different elements in the 1a and 3c sublattices tend to be consistent, the double sublattice model represents the site state of a near-completely disordered FCC solid solution. Step S2 involves face-centered cubic single-phase classification and screening, including the following steps: Step S21: Read the multi-principal alloy phase structure dataset, take the mole fraction of each element in the alloy as the input feature, and take FCC, BCC and IM as phase labels. If the corresponding phase exists, it is represented by 1, and if it does not exist, it is represented by 0. Multiple phases are allowed to exist in a system at the same time. Step S22: Convert the training data or candidate alloy composition to be predicted into a fixed-dimensional element mole fraction input matrix, and use the Min-Max normalizer to numerically normalize the input features to reduce the impact of different element content scale differences on model training. Step S23: Construct a neural network multi-label phase classification model; the model includes an input layer, a first fully connected hidden layer, a second fully connected hidden layer and an output layer. The hidden layer uses the ReLU activation function and combines Dropout or batch normalization units to suppress overfitting. The output layer uses the Sigmoid activation function to output the probabilities of the three phase labels FCC, BCC and IM respectively. Step S24: The multi-label phase classification model is trained using the binary cross-entropy loss function, Adam optimizer, weight decay and cross-validation to obtain a stable phase classification model; Step S25: Input the initial candidate alloy composition pool obtained in step S1 into the trained phase classification model to obtain the label probabilities of FCC, BCC and IM respectively; when the label probability of the corresponding phase is greater than the classification threshold of 0.5, it is considered that the corresponding phase exists; when the label probability of FCC is greater than 0.5 while the label probabilities of BCC and IM are both less than 0.5, the candidate system is determined as a candidate alloy with a face-centered cubic single phase. Step S26: The selected FCC single-phase candidate alloys are output according to the fields of Co, Cr, Cu, Fe, Mn, Ni, V and formula, providing detailed information of the candidate system for subsequent construction of preferred site structures and high-throughput prediction.

5. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S3, the target heat treatment temperature is set to 973K; the FCC_L12_AuCu3 prototype structure is composed of nested 1a sublattices occupied by Au and 3c sublattices occupied by Cu. The conventional unit cell of the face-centered cubic lattice is composed of lattice sites located at the eight vertices and six face centers of the unit cell. After calculation according to the periodic symmetry of the lattice sites, each unit cell contains one 1a site and three 3c sites; when a 3×3×3 supercell is used, the conventional FCC_L12_AuCu3 unit cell is periodically expanded by 3 times along the three lattice directions to form a supercell with 108 lattice sites, including 27 1a sublattice sites and 81 3c sublattice sites; Step S3 involves constructing the structure file based on optimal placeholders, including the following steps: Step S31: For the FCC single-phase candidate alloys selected in step S2, perform phase equilibrium calculations using thermodynamic and phase equilibrium calculation software based on the existing end-group thermodynamic database file, and generate a YSITE-T.EXP output file containing the occupancy fraction. Step S32: Extract the placeholder fraction rows corresponding to the target heat treatment temperature from the YSITE-T.EXP data; Step S33: Using the FCC_L12_AuCu3 prototype structure as the structural basis, the vertex sites in the conventional unit cell are defined as 1a sublattices, and the face-centered sites are defined as 3c sublattices; the total number of 1a and 3c sublattice sites is determined according to the selected supercell size, and the unit cell contains one 1a sublattice site and three 3c sublattice sites; when constructing the required computational model, a 3×3×3 cell expansion is performed; Step S34: Multiply the occupancy fraction of each element on the corresponding sublattice by the total number of sites in the corresponding sublattice to obtain the theoretical number of atoms occupied by each element in each sublattice; round the theoretical number of atoms occupied, and perform residual iterative correction based on the residual before and after rounding, so that the sum of the number of atoms of all elements in each sublattice is equal to the total number of sites in the corresponding sublattice. Step S35: Fill each element into the corresponding sublattice coordinates according to the integer number of atoms determined in step S34, assign a random number generated by the rand() function of the preset random seed to each sublattice coordinate, and perform random arrangement of the coordinates according to the random number; Step S36: Merge the coordinates and total number of atoms of the same type according to the element symbol order, fill the allocated site information into the standard coordinate file POSCAR, and add a selective dynamic marker at the end of each atomic coordinate to eliminate the rigid body translational degree of freedom.

6. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 5, characterized in that: In step S4, structural optimization includes volume relaxation and full relaxation; based on the optimized equilibrium unit cell, the three lattice vectors are scaled proportionally using multiple lattice linear scaling factors; the energy-volume equation of state is the third-order Birch-Murnaghan equation of state, and the bulk elastic modulus is obtained from the zero-pressure bulk elastic modulus parameters obtained by fitting. The acquisition of the bulk elastic modulus label in step S4 includes the following steps: Step S41: Perform first-principles structure optimization on the POSCAR structure file obtained in step S36 based on the VASP software package, including cell volume, cell shape and atomic position relaxation, to obtain a balanced crystal structure. Step S42: Construct multiple volume scaling states based on the equilibrium volume V0 obtained in step S41; calculate the total energy of the system under different volumes to form an energy-volume dataset. Step S43: The energy-volume data are fitted using the third-order Birch-Murnaghan equation of state to extract the zero-pressure body elastic modulus. Step S44: The bulk elastic modulus label obtained in step S43 is merged with the corresponding alloy composition and preferred occupancy structure information to form basic sample data for model training. Step S5 involves initial sample data cleaning, including the following steps: Step S51: Read the initial data table containing the element composition column and the bulk elastic modulus label column, and set the number of input feature columns, the target column position, the number of cross-validation folds, the random seed, the robust z-value threshold, and the cleaning action; the cleaning action includes deleting noise samples or replacing noise sample labels; Step S52: Construct at least two independent regression detection models; the regression detection models include a k-nearest neighbor regression detection model and a Gaussian process regression detection model, and generate cross-validation prediction values ​​for each sample based on K-fold cross-validation with random seeds; Step S53: Calculate the prediction residuals for each sample under the k-nearest neighbor regression detection model and the Gaussian process regression detection model, and calculate the corresponding robust z-value based on the median and absolute median of the residuals to reduce the impact of extreme residuals on the anomaly detection threshold. Step S54: Noise samples are identified using the intersection determination method. Only when the absolute value of the residual robust z-value of the same sample under the k-nearest neighbor regression detection model and the absolute value of the residual robust z-value under the Gaussian process regression detection model are both greater than the preset threshold, the sample is determined to be a noise sample. Step S55: When the cleaning action is deletion, the noise sample is removed from the initial dataset; when the cleaning action is replacement, the bulk elastic modulus label of the noise sample is replaced with the median or average of the predicted values ​​of the two detection models. Step S56: Output the cleaned sample data table for subsequent small sample data augmentation and model training.

7. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S6, the local interpolation enhancement includes: before performing the logarithmic ratio transformation, performing zero-value replacement and closed-loop normalization on the element mole fraction composition vector, replacing the element mole fractions less than or equal to the preset threshold with the preset threshold, and re-normalizing all element mole fractions after replacement so that the sum of all element mole fractions is 1; during interpolation in the transformation space, interpolating the transformed composition vector and bulk elastic modulus label of the paired samples according to the interpolation coefficients to obtain the candidate enhanced composition vector and candidate enhanced label; after enhancement, inversely transforming the candidate enhanced composition vector to the original composition space, and determining the optional element set of the candidate enhanced sample according to the union of the non-zero element sets in the paired samples, selecting a preset number of elements from the optional element set as the non-zero elements of the candidate enhanced sample, and setting the remaining element mole fractions to 0.

8. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S6, the enhancement scheme and enhancement amount are determined by comparing the final results of the outer layer repeated hold-out validation and the inner layer cross-validation; the candidate enhancement amounts are multiple preset enhancement amounts; the evaluation learner is a support vector regression model; the evaluation index includes the coefficient of determination R. 2 The root mean square error (RMSE) and the actual proportion of true values ​​in the test set falling within the 95% prediction range are used to determine the optimal net augmentation based on the comprehensive evaluation of the three indicators. The true samples and augmented samples together constitute the dataset used for machine learning training. Step S6 involves performing small sample data augmentation, including the following steps: Step S61: Read the cleaned sample dataset obtained in step S5, take Co, Cr, Cu, Fe, Mn, Ni, and V as component feature columns, take Bulk_Modulus as target column, and set the outer layer repeat retention validation seed, outer layer test set ratio, inner layer cross-validation fold number, and candidate net enhancement amount. Step S62, perform zero-value replacement processing on the zero component in the element mole fraction, using 10 -6 Replace the zero components and scale the remaining non-zero components proportionally so that the sum of the mole fractions of each element remains 1 after processing; perform a centered logarithmic ratio transformation or an equidistant logarithmic ratio transformation on the closed normalized component vector to map it to a transformation space that can perform Euclidean distance calculations. Step S63: Select nearest neighbor paired samples for the real sample in the transformation space, generate candidate enhancement components according to the interpolation coefficients, and synchronously interpolate the bulk elastic modulus label to obtain candidate enhancement labels; Step S64: Inversely transform the candidate enhancement components from the transformation space back to the original component space, and perform component normalization and bit retention processing to keep the sum of the mole fractions of each element in the enhancement sample at 1; determine the elements in the real sample and the paired sample whose mole fractions are greater than the non-zero threshold, and determine the union of the non-zero element sets of the two samples as the optional element set of the candidate enhancement sample; according to the preset non-zero principal component range and the size of the mole fraction of each element after inverse transformation, select the corresponding number of elements from the optional element set as the non-zero elements of the candidate enhancement sample, and set the mole fraction of the remaining elements to 0; Step S65: Perform multiple screening on the candidate enhanced samples. The multiple screening includes at least non-zero principal component number constraint, component range constraint, nearest neighbor distance filtering, local label consistency filtering, label quantile range constraint, duplicate sample removal, and component retention number constraint. Step S66: Compare model performance under different augmentation methods and different augmentation amounts through outer layer hold-out validation and inner layer cross-validation; evaluation metrics include the coefficient of determination R. 2 The root mean square error (RMSE) and 95% prediction interval coverage were evaluated, and the final enhancement scheme was determined based on the comprehensive evaluation. in, All three indicators were normalized to the [0,1] interval. R respectively 2 The weights of RMSE and Coverage.

9. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S7, based on the calculation results, the placeholder fraction line corresponding to the target heat treatment temperature in the output file is read for each candidate system to calculate Sigma(deltaG), configuration entropy Sconfig and Gibbs free energy feature G under ordered placeholder conditions. Step S7 involves constructing machine learning input features, including the following steps: Step S71: Read the data table containing alloy chemical formulas or element mole fractions; when the input is a chemical formula, use regular expressions to identify the element symbols and their corresponding values, and extract the mole fractions of seven elements: Co, Cr, Cu, Fe, Mn, Ni and V. Step S72: Normalize the mole fractions of the seven elements in each alloy sample within the row so that the sum of the mole fractions of the elements is 1, and form the element composition characteristics in a fixed element order. Step S73: Based on the element's melting point, atomic radius, valence electron concentration, Pauling electronegativity, and binary mixing enthalpy parameter, calculate the empirical physical properties and thermodynamic combination characteristics, which include at least Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, and X; Step S74: Based on Thermo-Calc thermodynamic and phase equilibrium calculation results, read the occupancy fraction of different elements on the 1a and 3c sublattices at the target heat treatment temperature, and calculate the preferred occupancy thermodynamic characteristics in combination with the end-base thermodynamic database. The preferred occupancy thermodynamic characteristics include Sigma (deltaG), Sconfig and G. Step S75: The elemental composition features obtained in step S72, the empirical property combination features obtained in step S73, and the preferred occupancy thermodynamic features obtained in step S74 are merged to generate a machine learning input feature matrix for predicting the bulk elastic modulus.

10. The method for predicting the elastic modulus of a face-centered cubic single-phase multi-principal element alloy with preferred occupancy as described in claim 2, characterized in that: In step S8, the input feature columns of the support vector regression model include elemental composition features: Co, Cr, Cu, Fe, Mn, Ni, V; physical property features: Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X; and thermodynamic features: Sigma(deltaG), Sconfig, and G. Step S8 involves training and saving the support vector regression model, including the following steps: Step S81: Read the data table containing complete input features and bulk elastic modulus labels, determine Bulk_Modulus as the target column, and determine the input feature columns including Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G; Step S82: Divide the data into a training set and a test set according to the preset test set ratio; use repeated cross-validation to train the model within the training set; Step S83: Construct a two-stage hyperparameter search strategy. The first stage involves a random search within a large parameter space to quickly locate the optimal parameter region. The second stage involves constructing a local mesh around the advantageous parameter region obtained in the first stage and using mesh search to further optimize the hyperparameters. The selection of parameter combinations aims to maximize the cross-validation determination coefficient R. 2 As the main indicator, when R 2 When the differences are small, compare RMSE. 2 When both RMSE and MAE are close, further compare them. Step S84: Construct a support vector regression model using the optimal parameters determined in step S83, determine the importance of each input feature using permutation importance, and perform recursive feature elimination cross-validation. Step S85: Refit the support vector regression model using the filtered input features, and calculate the coefficient of determination R on the independent test set. 2 The root mean square error (RMSE), mean absolute error (MAE), and mean square error (MSE) were evaluated. After the model performance was evaluated, the prediction pipeline, including missing value handling, feature standardization, fixed feature selection, and support vector regression model, was refitted using all sample data to obtain the final model for independent prediction. Step S86: Save the missing value handling and normalization preprocessor, the fixed feature selector established based on the recursive feature elimination results, and the final prediction pipeline of the support vector regression model as best_model.joblib, save the optimal parameters as best_params.json, and write the training file, target column, input feature column, numerical feature column, random seed, test set ratio, cross-validation settings, model selection rules, cross-validation metrics, test set metrics, and feature selection information into model_metadata.json; The prediction method also includes step S9, which involves independently predicting the bulk elastic modulus, including the following steps: Step S91: Load the best_model.joblib model file and model_metadata.json metadata file saved in step S8, and read the new FCC single-phase multi-principal alloy data to be predicted. The input feature columns include the aforementioned Co, Cr, Cu, Fe, Mn, Ni, V, Hmix, delta, Omega, Smix, Tm, VEC, sigma_X, sigma_H, a, sigma_Tm, sigma_VEC, X, Sigma(deltaG), Sconfig, and G; and make appropriate deletions based on the feature filtering results of step S85. Step S92: Extract the features of the sample to be predicted according to the order of the feature_columns field in model_metadata.json, and construct the input matrix; Step S93: Input the input matrix into the prediction pipeline in best_model.joblib. The prediction pipeline automatically completes the same missing value processing, feature scaling and feature filtering as in the training phase to obtain the predicted value of the bulk elastic modulus of the alloy to be predicted, and writes the predicted value into the prediction column. The data containing the original input information and the prediction column is exported to realize the batch prediction output of the bulk elastic modulus of unknown FCC single-phase multi-principal alloy.

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