Material component and performance mapping uncertainty quantification method

Abstract

The invention relates to a material component and performance mapping uncertainty quantification method, and belongs to the technical field of material science. The method comprises the following steps: firstly, collecting material component data from a Materials Project database; then, atomic numbers and fractions in the data are coded into high-dimensional vectors by using a full-connection neural network; thirdly, combining the high-dimensional vectors into an element derived matrix, and inputting the element derived matrix into a Transform module to learn chemical structure information; and finally, performing spectrum normalization residual MLP processing on the learned chemical structure information features, and applying Gaussian process modeling on an output layer to quantify uncertainty. When the method is applied to material property prediction, the prediction value of the material property can be given, the uncertainty of prediction can be quantified, and the distribution of prediction results is provided.

Description

Technical Field

[0001] The present invention relates to an uncertainty quantification method for material composition and performance mapping, and belongs to the cross-technical field of material science and artificial intelligence. Background Art

[0002] The properties of materials are affected by many factors, such as their composition, microstructure, and processing technology. These factors interact with each other, making the prediction of material properties extremely complex. Traditional material development relies on laboratory experiments and trial and error, which is not only time-consuming and labor-intensive, but may not cover all possible material combinations. Machine learning provides a data-driven approach to predict material properties and is expected to accelerate the process of material design. However, existing machine learning models often lack quantification of prediction uncertainty, which limits their reliability and credibility in high-risk applications. The Gaussian process is a powerful nonparametric Bayesian method that can provide uncertainty estimates about predictions and is well suited for dealing with complex systems with uncertainty. The Laplace approximation is a mathematical method used to simplify the calculation of the posterior distribution. The Laplace approximation can make the Gaussian process more feasible in practical applications.

[0003] At the same time, deep learning models are prone to gradient explosion problems during training. Spectral normalization helps control the size of the gradient by limiting the spectral norm of the network layer weights. The Lipschitz constraint ensures that the output of the model will not fluctuate dramatically due to slight changes in the input, improving the stability and generalization ability of the model. The residual network effectively extracts the deep features of the material composition through its residual learning framework. The Transformer model captures the global dependencies of sequence data through the self-attention mechanism, providing a new perspective for material property prediction.

[0004] In summary, by analyzing and evaluating the uncertainty of predictions, we can more accurately understand the credibility of model predictions, thereby guiding experimental design and material discovery projects and avoiding waste of resources and time. This paper introduces a method for quantifying uncertainty in material composition and property mapping, which combines Gaussian processes with a deep learning framework to explore the complex relationship between material composition and its properties through sophisticated data analysis. Gaussian processes are used to quantify uncertainty, thereby providing higher reliability and accuracy in material property predictions. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a method for quantifying uncertainty in material composition and performance mapping, which can quantify the uncertainty in the prediction process and thus solve the above-mentioned problem.

[0006] The technical solution of the present invention is: a method for quantifying uncertainty in material composition and performance mapping, first, collecting material composition data from the Materials Project database; then, using a fully connected neural network to encode the atomic number and score in the data into a high-dimensional vector; then, combining these high-dimensional vectors into an element-derived matrix, and inputting the Transformer module to learn the chemical structure information to obtain chemical structure information features. Finally, the chemical structure information features are processed by spectral normalized residual MLP, and Gaussian process modeling is applied in the output layer to quantify uncertainty.

[0007] The specific steps are:

[0008] Step 1: Collect material composition data from the Materials Project database, including the atomic number of each element and its fraction in the material;

[0009] Step 2: Use a fully connected neural network to encode the atomic number and score into a high-dimensional vector to obtain an element-derived matrix, which contains the relationship information between each element;

[0010] Step 3: Construct a Transformer module, input the element-derived matrix into the Transformer module, and use the self-attention mechanism in the Transformer module to learn the interactions between the elements in the material and the chemical structure information of each element to obtain the chemical structure information features;

[0011] Step 4: Inputting the chemical structure information features into the residual multi-layer perceptron module, designing spectral normalization in the residual multi-layer perceptron module, constraining the mapping of the weight matrix in the residual multi-layer perceptron module, preventing overfitting, and improving the generalization ability of the model;

[0012] Step 5: Construct a Gaussian model, optimize the parameters of the Gaussian model through the logarithmic posterior likelihood and the Hessian matrix, and then input the chemical structure information features in the residual multi-layer perceptron module into the optimized Gaussian model for Gaussian process processing to quantitatively predict the uncertainty of the mapping between material composition and performance.

[0013] The Step 2 is specifically as follows:

[0014] Step 2.1: Perform stratified random sampling on the unpartitioned data set, fix the category ratios in different sets, and finally extract the training set, validation set, and test set required for the experiment;

[0015] Step 2.2: Pass the atomic number of each element through a fully connected neural network layer to generate the corresponding high-dimensional vector representation and obtain the element embedding matrix;

[0016] Step 2.3: Generate the corresponding high-dimensional vector representation of each element through fractional encoding to obtain the fractional embedding matrix;

[0017] Step 2.4: Combine the element embedding matrix and the score embedding matrix to form an element derived matrix to complete the preprocessing of the data set.

[0018] The Transformer module consists of multiple layers, each of which includes a self-attention mechanism and a feedforward neural network;

[0019] Two inter-layer enhancement mechanisms are introduced in the Transformer module: residual connection and layer normalization; the residual connection directly adds the input to the processed output;

[0020] The layer normalization is to standardize the output of each layer, and by standardizing the mean and variance of the features, different batches of data can be trained on a similar scale.

[0021] The Transformer module includes 3 layers of encoders, each of which contains 4 attention heads, which are used to capture contextual information in the input data;

[0022] The encoder's layer-by-layer enhancement mechanism is that each sublayer includes residual connections and layer normalization:

[0023]

[0024] Where X is the feature vector input to this layer, with a dimension of R n×d , Z(X) is the output of the input x after being processed by the previous layer, Z(X)+X represents the residual connection, LayerNorm represents layer normalization, FFN is a feedforward neural network, F l It is the final output encoding feature, which has the same dimension as X.

[0025] The Step 4 is specifically as follows:

[0026] Step 4.1: The residual multilayer perceptron module is composed of different residual blocks, which can be expressed as:

[0027] h=h L-1 ·...·h2·h1 (1)

[0028] h l (F l )=F l +g l (F l ) (2)

[0029] g l (x) = a(W l x+b l ) (3)

[0030] Formula (1) indicates that the forward propagation process of the entire network is composed of multiple layers, namely from h1 to h L-1 combination of;

[0031] In formula (2), F l is the input to the Lth layer, g l (F l ) is the input F in the Lth layer l The transformation performed represents the new features after passing through this layer;

[0032] In formula (3), W l represents the weight matrix of the Lth layer, which is used to linearly transform the input feature x, a is the activation function, b l is the bias term, g l (x) represents the nonlinear transformation of the input x by the Lth layer;

[0033] Step 4.2: Design spectral normalization to transform the weight matrix of each layer in MLP Perform spectral normalization to achieve g l Lipschitz constraint;

[0034] The spectral normalization is to estimate the maximum eigenvalue of the weight matrix through power iteration The weights are normalized using the maximum eigenvalue;

[0035]

[0036] Where c is a hyperparameter used to adjust the upper bound of the spectral norm of the weight matrix so that ||W l ||2≤c,W l is the weight matrix of the current Lth layer, and λ is a threshold, which represents the scalar value used for comparison under specific conditions.

[0037] The Gaussian process processing is specifically as follows:

[0038] The prior distribution of the Gaussian model with the RBF kernel is the multivariate normal distribution:

[0039]

[0040] In the formula, σ 2 and l are hyperparameters, g N×1 is a random vector of length N, representing the output of the Gaussian process, that is, the predicted value, MVN is a multivariate normal distribution, 0 N×1is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero, K N×N is an N×N covariance matrix generated by the RBF kernel function, which represents the correlation between any two observation points. i,j Represents the element in the i-th row and j-th column of the covariance matrix, which describes the similarity between the i-th and j-th input points. represents the square of the Euclidean distance between the i-th and j-th input points, L is the length scale of the RBF kernel, and random feature expansion is used to approximate the prior, defining the kernel matrix K = ΦΦ T , the random characteristic Gaussian process is:

[0041]

[0042] In the formula, g N×1 is a random vector of length N, representing the output of the Gaussian process, i.e. the predicted value; MVN is the multivariate normal distribution; 0 N×1 is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero; is calculated from the random feature matrix Φ, where Φ is an N×D L The matrix represents the feature map generated by the random feature method; Φ i Represents the last layer of the residual network, with dimension D L , W L is a fixed weight matrix, starting from N (0，1) In the independent and identically distributed sampling, b L is a fixed bias term;

[0043] Random feature expansion using Gaussian priors is used to write L And a neural network layer consisting of learnable output weights β:

[0044]

[0045] In the formula, F o is the input feature vector, g(F o ) is the output of the neural network layer, based on the input feature F o Calculated; is a scaling factor, cos(-W L F o +b L ) is the input feature F o After the nonlinear mapping of linear transformation, b L is the bias vector, β conforms to the multivariate normal distribution with mean zero and variance τ, The size is D L ×D L The identity matrix of .

[0046] The logarithmic posterior likelihood is specifically:

[0047]

[0048] In the formula, logp(β|x) represents the logarithmic posterior probability of parameter β given the input data x. It is a term about the likelihood function, which is used to measure the model prediction value g i and the true value y i The difference between is a regularization term, called L2 regularization.

[0049] The Hessian matrix is ​​specifically:

[0050]

[0051] In the formula, It is a D L ×D L The estimated value of the Hessian matrix of i is a feature vector, which is generated in random feature expansion or neural network layer; This term represents the eigenvector Φ i The outer product of L ×D L The matrix of It is a D L ×D L The identity matrix of .

[0052] The beneficial effects of the present invention are:

[0053] 1. Compared with existing uncertainty quantification methods, the present invention provides more accurate material property predictions while being able to generate more reliable uncertainty estimates.

[0054] 2. The present invention does not need to maintain a large number of model parameters, which simplifies the complexity of the model. Even in small data sets, the model of the present invention shows competitive performance, and when resources permit, combined with deep integration methods, it can produce more accurate material property predictions and high-quality uncertainty estimates. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 This is the overall structure diagram of the network that combines Transformer and Gaussian process in the present invention;

[0056] Figure 2 It is a structural diagram of the Transformer encoding module of the present invention;

[0057] Figure 3It is a visualization diagram of uncertainty ranking and error change of different models on a single-atom formation energy attribute data set in an embodiment of the present invention. DETAILED DESCRIPTION

[0058] The present invention will be further described below in conjunction with the accompanying drawings and specific implementation methods.

[0059] Embodiment 1: A method for quantifying uncertainty in material composition and performance mapping, comprising the following steps:

[0060] Step 1: Collect material composition data from the Materials Project (MP) database, which includes the atomic number of each element and its fraction in the material.

[0061] Step 2: Use a fully connected neural network to encode the atomic number and score into a high-dimensional vector to obtain an element-derived matrix, which contains the relationship information between each element, specifically:

[0062] Step 2.1: Perform stratified random sampling on the unpartitioned data set, fix the category ratios in different sets, and finally extract the training set, validation set, and test set required for the experiment;

[0063] Step 2.2: Pass the atomic number of each element through a fully connected neural network layer to generate the corresponding high-dimensional vector representation and obtain the element embedding matrix;

[0064] Step 2.3: Generate the corresponding high-dimensional vector representation of each element through fractional encoding to obtain the fractional embedding matrix;

[0065] Step 2.4: Combine the element embedding matrix and the score embedding matrix to form an element derived matrix to complete the preprocessing of the data set.

[0066] The element-derived matrix combines the properties of each element and its proportion in the material to effectively represent the material composition information.

[0067] Step 3: Construct a Transformer module, input the element-derived matrix into the Transformer module, and through the self-attention mechanism in the Transformer module, learn the interactions between the elements in the material and the chemical structure information of each element to obtain the chemical structure information features.

[0068] The Transformer module consists of multiple layers, each of which includes a self-attention mechanism and a feedforward neural network;

[0069] In order to enhance the stability and effect of model training, two inter-layer enhancement mechanisms are introduced in the Transformer module: residual connection and layer normalization. These mechanisms are designed to ensure that the network can still be effectively trained under deep hierarchical structures.

[0070] The residual connection directly adds the input to the processed output to avoid information being lost or weakened when passing between layers.

[0071] The layer normalization is to standardize the output of each layer, and by standardizing the mean and variance of the features, different batches of data can be trained on a similar scale.

[0072] The Transformer module includes 3 layers of encoders, each of which contains 4 attention heads;

[0073] The attention head is used to capture contextual information in the input data;

[0074] The enhancement mechanism between the layers of the encoder is that each sublayer (self-attention mechanism and feedforward neural network) contains residual connections and layer normalization:

[0075]

[0076] Where X is the feature vector input to this layer, with a dimension of R n×d , Z(X) is the output of the input x after being processed by the previous layer (or part of the current layer), Z(X)+X represents the residual connection, LayerNorm represents layer normalization, FFN is a feedforward neural network, F l It is the encoded feature after multiple layers and has the same dimension as X.

[0077] Step 4: Input the chemical structure information features into the residual multi-layer perceptron (MLP) module, design spectral normalization in the residual multi-layer perceptron module, constrain the mapping of the weight matrix in the residual multi-layer perceptron module, improve its stability, and prevent the collapse of the hidden space features, specifically:

[0078] Step 4.1: The residual multilayer perceptron module is composed of different residual blocks, which can be expressed as:

[0079] h=h L-1 ·...·h2·h1 (1)

[0080] h l (F l )=F l +g l (F l ) (2)

[0081] g l (x) = a(W l x+b l ) (3)

[0082] Formula (1) indicates that the forward propagation process of the entire network is composed of multiple layers, namely from h1 to h L-1 combination of;

[0083] In formula (2), F l is the input to the Lth layer, and can also be regarded as the feature representation passed from the previous layer (or previous layers). l (F l ) is the input F in the Lth layer l The transformation performed represents the new features after passing through this layer;

[0084] In formula (3), W l represents the weight matrix of the Lth layer, which is used to linearly transform the input feature x, a is the activation function, b l is the bias term, g l (x) represents the nonlinear transformation performed by the Lth layer on the input x.

[0085] Step 4.2: Design spectral normalization to transform the weight matrix of each layer in MLP Perform spectral normalization to achieve g l Lipschitz constraint.

[0086] Specifically, the maximum eigenvalue of the weight matrix is ​​estimated by power iteration. The weights are normalized using the largest eigenvalue.

[0087]

[0088] Where c is a hyperparameter used to adjust the upper bound of the spectral norm of the weight matrix so that ||W l ||2≤c,W l is the weight matrix of the current Lth layer, and λ is a threshold, which represents the scalar value used for comparison under specific conditions.

[0089] Step 5: Construct a Gaussian model, optimize the parameters of the Gaussian model through the logarithmic posterior likelihood and the Hessian matrix, and then input the chemical structure information features in the residual multi-layer perceptron module into the optimized Gaussian model for Gaussian process processing to quantitatively predict the uncertainty of the mapping between material composition and performance.

[0090] The Gaussian process processing is specifically as follows:

[0091] The prior distribution of the Gaussian model with the RBF kernel is the multivariate normal distribution:

[0092]

[0093] In the formula, σ 2 and l are hyperparameters, g N×1 is a random vector of length N, representing the output of the Gaussian process, that is, the predicted value, MVN is a multivariate normal distribution, 0 N×1 is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero, K N×N is an N×N covariance matrix generated by the RBF kernel function, which represents the correlation between any two observation points. i,j Represents the element in the i-th row and j-th column of the covariance matrix, which describes the similarity between the i-th and j-th input points. represents the square of the Euclidean distance between the i-th and j-th input points, L is the length scale of the RBF kernel, and random feature expansion is used to approximate the prior, defining the kernel matrix K = ΦΦ T , the random characteristic Gaussian process is:

[0094]

[0095] In the formula, g N×1 is a random vector of length N, representing the output of the Gaussian process, i.e. the predicted value; MVN is the multivariate normal distribution; 0 N×1 is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero; is calculated from the random feature matrix Φ, where Φ is an N×D L The matrix represents the feature map generated by the random feature method; Φ i Represents the last layer of the residual network, with dimension D L , W L is a fixed weight matrix, starting from N (0，1) In the independent and identically distributed sampling, b L is a fixed bias term;

[0096] Random feature expansion using Gaussian priors is used to write L And a neural network layer consisting of learnable output weights β:

[0097]

[0098] In the formula, F o is the input feature vector, g(F o ) is the output of the neural network layer, based on the input feature F o Calculated; is a scaling factor, usually used to normalize random features to ensure that the calculated results are within a reasonable numerical range. L F o +b L ) is the input feature F o After the nonlinear mapping of linear transformation, the cosine function is used for nonlinear expansion. L is a bias vector used to adjust the position in the feature space before nonlinear mapping. β conforms to a multivariate normal distribution with a mean of zero and a variance of τ (where τ is a hyperparameter), The size is D L ×D L The identity matrix of .

[0099] The logarithmic posterior likelihood is specifically:

[0100]

[0101] Where logp(β|x) represents the logarithmic posterior probability of parameter β given the input data x. This part is about the likelihood function, which is used to measure the model prediction value g i and the true value y i The difference between. is a regularization term, called L2 regularization;

[0102] The Hessian matrix is ​​specifically:

[0103]

[0104] In the formula, It is a D L ×D L The estimated value of the Hessian matrix of i is a feature vector, which is generated in random feature expansion or neural network layer; This term represents the eigenvector Φ i The outer product of L ×D L The matrix of It is a D L ×D L The identity matrix of .

[0105] In order to verify the effect of the model of the present invention in quantifying the uncertainty between the material composition and the refractive index, the method proposed by the present invention is compared with several existing uncertainty quantification methods, such as Representation Of Organics, Inorganics, and Synthesized Targets (Roost), Compositionally Restricted Attention-Based Neural Network (CrabNet), etc. The specific steps are:

[0106] First, material composition data is collected from the Materials Project database. Then, a fully connected neural network is used to encode the atomic number and fraction in the data into high-dimensional vectors. Next, these high-dimensional vectors are combined into element-derived matrices and input into the Transformer module to learn chemical structure information and obtain chemical structure information features. Finally, the chemical structure information features are processed by spectral normalized residual MLP, and Gaussian process modeling is applied at the output layer to quantify uncertainty.

[0107] At the same time, in order to verify the effectiveness of the algorithm, the proposed method is compared with existing quantization uncertainty methods such as Roost, CrabNet, and TUPNet (single). The results show that the proposed method TUPNet (ensemble) exhibits lower error and better uncertainty quantization performance. The comparison results are shown in Tables 1 and 2 and the attached Figure 3 As shown. single means that only a single model is used in the training and prediction process. ensemble means an integrated model, that is, a combination of multiple models is used in the training and prediction process, and the prediction results of these models are combined, such as taking the average of the predicted values ​​as the final prediction result.

[0108] Table 1: Prediction accuracy and uncertainty quantification quality assessment on the refractive index property dataset

[0109]

[0110] Table 2: Prediction accuracy and uncertainty quantification quality assessment on the single-atom formation energy property dataset

[0111]

[0112] The specific implementation modes of the present invention are described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above implementation modes, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.

Claims

1. A method for quantifying uncertainty in material composition and performance mapping, characterized in that: The method specifically comprises: Step 1: Collect material composition data from the Materials Project database, including the atomic number of each element and its fraction in the material; Step 2: Use a fully connected neural network to encode the atomic number and score into a high-dimensional vector to obtain an element-derived matrix, which contains the relationship information between each element; Step 3: Construct a Transformer module, input the element-derived matrix into the Transformer module, and use the self-attention mechanism in the Transformer module to learn the interactions between the elements in the material and the chemical structure information of each element to obtain the chemical structure information features; Step 4: Inputting the chemical structure information features into the residual multi-layer perceptron module, designing spectral normalization in the residual multi-layer perceptron module, constraining the mapping of the weight matrix in the residual multi-layer perceptron module, preventing overfitting, and improving the generalization ability of the model; Step 5: Construct a Gaussian model, optimize the parameters of the Gaussian model through the logarithmic posterior likelihood and the Hessian matrix, and then input the chemical structure information features in the residual multi-layer perceptron module into the optimized Gaussian model for Gaussian process processing to quantitatively predict the uncertainty of the mapping between material composition and performance.

2. The uncertainty quantification method for material composition and performance mapping according to claim 1, characterized in that: The Step 2 is specifically as follows: Step 2.1: Perform stratified random sampling on the unpartitioned data set, fix the category ratios in different sets, and finally extract the training set, validation set, and test set required for the experiment; Step 2.2: Pass the atomic number of each element through a fully connected neural network layer to generate the corresponding high-dimensional vector representation and obtain the element embedding matrix; Step 2.3: Generate the corresponding high-dimensional vector representation of each element through fractional encoding to obtain the fractional embedding matrix; Step 2.4: Combine the element embedding matrix and the score embedding matrix to form an element derived matrix to complete the preprocessing of the data set.

3. The uncertainty quantification method for material composition and performance mapping according to claim 1, characterized in that: The Transformer module consists of multiple layers, each of which includes a self-attention mechanism and a feedforward neural network; Two inter-layer enhancement mechanisms are introduced in the Transformer module: residual connection and layer normalization; the residual connection directly adds the input to the processed output; The layer normalization is to standardize the output of each layer, and by standardizing the mean and variance of the features, different batches of data can be trained on a similar scale.

4. The uncertainty quantification method for material composition and property mapping according to claim 1, characterized in that: The Step 4 is specifically as follows: Step 4.1: The residual multilayer perceptron module is composed of different residual blocks, which can be expressed as: h=h L-1 ·...·h2·h1 (1) h l (F l )=F l +g l (F l ) (2) g l (x)=a(W l x+b l ) (3) Formula (1) indicates that the forward propagation process of the entire network is composed of multiple layers, namely from h1 to h L-1 combination of; In formula (2), F l is the input to the Lth layer, g l (F l ) is the input F in the Lth layer l The transformation performed represents the new features after passing through this layer; In formula (3), W l represents the weight matrix of the Lth layer, which is used to linearly transform the input feature x, a is the activation function, b l is the bias term, g l (x) represents the nonlinear transformation of the input x by the Lth layer; Step 4.2: Design spectral normalization to transform the weight matrix of each layer in MLP Perform spectral normalization to achieve g l Lipschitz constraint; The spectral normalization is to estimate the maximum eigenvalue of the weight matrix through power iteration The weights are normalized using the maximum eigenvalue; Where c is a hyperparameter used to adjust the upper bound of the spectral norm of the weight matrix so that ||W l ||2≤c,W l is the weight matrix of the current Lth layer, and λ is a threshold, which represents the scalar value used for comparison under specific conditions.

5. The uncertainty quantification method for material composition and performance mapping according to claim 3, characterized in that: The Transformer module includes 3 layers of encoders, each of which contains 4 attention heads, which are used to capture contextual information in the input data; The encoder's layer-by-layer enhancement mechanism is that each sublayer includes residual connections and layer normalization: Where X is the feature vector input to this layer, with a dimension of R n×d , Z(X) is the output of the input x after being processed by the previous layer, Z(X)+X represents the residual connection, LayerNorm represents layer normalization, FFN is a feedforward neural network, F l It is the final output encoding feature, which has the same dimension as X.

6. The uncertainty quantification method for material composition and property mapping according to claim 1, characterized in that: The Gaussian process processing is specifically as follows: The prior distribution of the Gaussian model with the RBF kernel is the multivariate normal distribution: In the formula, σ 2 and l are hyperparameters, g N×1 is a random vector of length N, representing the output of the Gaussian process, that is, the predicted value, MVN is a multivariate normal distribution, 0 N×1 is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero, K N×N is an N×N covariance matrix generated by the RBF kernel function, which represents the correlation between any two observation points. i,j Represents the element in the i-th row and j-th column of the covariance matrix, which describes the similarity between the i-th and j-th input points. represents the square of the Euclidean distance between the i-th and j-th input points, L is the length scale of the RBF kernel, and random feature expansion is used to approximate the prior, defining the kernel matrix K = ΦΦ T , the random characteristic Gaussian process is: In the formula, g N×1 is a random vector of length N, representing the output of the Gaussian process, i.e. the predicted value; MVN is the multivariate normal distribution; 0 N×1 is a zero vector of length N, indicating that the mean of the multivariate normal distribution is zero; is calculated from the random feature matrix Φ, where Φ is an N×D L The matrix represents the feature map generated by the random feature method; Φ i Represents the last layer of the residual network, with dimension D L , W L is a fixed weight matrix, starting from N (0，1) In the independent and identically distributed sampling, b L is a fixed bias term; Random feature expansion using Gaussian priors is used to write L And a neural network layer consisting of learnable output weights β: In the formula, F o is the input feature vector, g(F o ) is the output of the neural network layer, based on the input feature F o Calculated; is a scaling factor, cos(-W L F o +b L ) is the input feature F o After the nonlinear mapping of linear transformation, b L is the bias vector, β conforms to the multivariate normal distribution with mean zero and variance τ, The size is D L ×D L The identity matrix of .

7. The uncertainty quantification method for material composition and property mapping according to claim 1, characterized in that: The logarithmic posterior likelihood is specifically: In the formula, logp(β|x) represents the logarithmic posterior probability of parameter β given the input data x. It is a term about the likelihood function, which is used to measure the model prediction value g i and the true value y i The difference between is a regularization term, called L2 regularization.

8. The uncertainty quantification method for material composition and property mapping according to claim 1, characterized in that: The Hessian matrix is ​​specifically: In the formula, It is a D L ×D L The estimated value of the Hessian matrix of i is a feature vector, which is generated in random feature expansion or neural network layer; This term represents the eigenvector Φ i The outer product of L ×D L The matrix of It is a D L ×D L The identity matrix of .

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