A Physical-Data Collaborative Driving Based Method for Predicting Hypersurface Spectral Response

Abstract

The present invention discloses a method for predicting the spectral response of a metasurface based on physical-data collaborative driving. According to the physical mechanism of electric field propagation and the electric field boundary conditions of the metasurface structure, a partial differential equation for the interaction between the electric field and the metasurface structure is established. Using the residual of the partial differential equation as the loss function of the U-Net network, by minimizing the residual of the partial differential equation, the electric field output by the encoder of the U-Net network gradually satisfies the partial differential equation. At the same time, the weight parameters of the neurons in the encoder and decoder of the U-Net network are updated by backpropagation to achieve unsupervised learning driven by physics. The U-Net network trained by physical driving can learn the metasurface structure features. On the basis of the physically driven U-Net network, the mutual information method can be used to screen out some metasurface structure features as the input of the extreme learning machine, and the weight parameters of the hidden layer of the extreme learning machine are trained to achieve fast prediction of the spectral response with small samples. The invention of this patent has the characteristics of fast prediction speed, high accuracy, and effectively reducing the data training cost.

Description

Technical Field:

[0001] The present invention belongs to the technical field of hypersurface spectral response prediction, and particularly relates to a method for predicting hypersurface spectral response based on physical-data collaborative driving. Background Art:

[0002] There is a high-dimensional and non-linear relationship between the hypersurface structure and the spectral response. Generally, data-driven methods can only fit the mapping relationship between the hypersurface structure and the spectral response from a large number of data sets. However, overfitting usually occurs, making it difficult to accurately predict the spectral response corresponding to the hypersurface structure.

[0003] Patent No. CN202211157781.6 discloses "a hypersurface transmission spectral response prediction system based on autoencoding". This method uses autoencoding to map the hypersurface structure to the encoding space, and predicts the complete transmission spectrum line through decoding. The application of autoencoding largely balances the order of magnitude of the network input and output, improving the stability of the neural network. However, the neural network training process described in this patent is based on a large amount of data-driven, and data is obtained through simulation, increasing the training cost.

[0004] Patent No. CN114543993A discloses "a hypersurface device, a spectral response measurement device, a spectral response instrument, and a spectral response measurement method". This method collects the light intensity through a photoreceptor, inversely calculates the spectral response intensity of the incident light from the light intensity, and uses an optimization algorithm to reconstruct the spectral response intensity. The reconstructed spectral response intensity is unique and close to the true spectral response intensity, but it takes a certain amount of time in the process of reconstructing the spectral response intensity.

[0005] Therefore, there is an urgent need for a method for predicting hypersurface spectral response that can accelerate the prediction speed while reducing the cost of collecting data.

[0006] The information disclosed in this background art section is only intended to enhance the overall understanding of the present invention,

[0007] and should not be regarded as an admission or any form of implication that this information constitutes prior art already known to those of ordinary skill in the art. Summary of the Invention:

[0008] The purpose of the present invention is to provide a method for predicting hypersurface spectral response based on physical-data collaborative driving. By using the collaborative driving method of the U-Net network and the ELM model, the U-Net network extracts the hypersurface structure features and updates the weight parameters of the encoder and decoder neurons, and the ELM model predicts the spectral response, thereby overcoming the defects in the above-mentioned prior art.

[0009] To achieve the above object, the present invention provides a method for predicting the spectral response of a metasurface based on physical-data collaborative driving, and the steps are as follows:

[0010] S01. Use the historical data of the metasurface structure features and the corresponding historical data of the spectral response in the historical database, and adopt the mutual information method to calculate the estimated values of the mutual information between the historical data of the metasurface structure features and the historical data of the spectral response respectively. Set a threshold, and screen the historical data of the metasurface structure features with the estimated value of the mutual information greater than the threshold and add them to the set L;

[0011] S02. Construct a U-Net network, and use the set L as the input of the U-Net network. The U-Net network includes two parts: an encoder and a decoder. The encoder is used for dimensionality reduction processing and feature learning of the metasurface structure, and extracting metasurface structure features. The decoder is used for outputting the electric field;

[0012] S03. According to the physical mechanism of the electric field propagation of the metasurface structure and the electric field boundary conditions, establish a partial differential equation for the interaction between the electric field and the metasurface structure, and perform unsupervised training on the U-Net network that only requires providing input. Input the two-dimensional pixel images intercepted transversely from different metasurfaces into the U-Net network, and use the residual of the partial differential equation as the loss function to constrain the U-Net network. Through backpropagation, that is, continuously reducing this loss function to make the electric field gradually satisfy the partial differential equation, and at the same time update the parameters of the encoder and decoder neurons of the U-Net network until the electric field converges. At this time, fix the parameters of the encoder and decoder neurons of the U-Net network, and determine the mapping relationship between the metasurface and the electric field through the U-Net network;

[0013] S04. When the unsupervised training of the U-Net network is completed, the encoder learns the metasurface structure feature information from multiple dimensions, and designs an ELM characterization model based on small-sample training; the ELM characterization model is composed of extreme learning machines. After small-sample training of the extreme learning machine, input the metasurface two-dimensional image, and the ELM characterization model outputs the predicted spectral response.

[0014] Preferably, in the technical solution, in step S01, the historical database contains the historical data of the metasurface structure features simulated by several numerical methods and the corresponding historical data of the spectral response;

[0015] Assume that the metasurface has N structural features, where the type of the metasurface structure feature is a i (i = 1, 2, 3... N), the historical data of the metasurface structure feature is A i (i = 1, 2, 3... N), the historical data of the metasurface structure feature corresponding to the same subscript of the metasurface structure feature type, the historical data of the spectral response corresponding to the historical data of the metasurface structure feature is S, and S is composed of several data points; for A iPerform normalization processing, and the historical data of the metasurface structure features after normalization processing is:

[0016]

[0017] wherein, max(A i ) represents the maximum value of the historical data of the i-th metasurface structure feature, min(A i ) represents the minimum value of the historical data of the i-th metasurface structure feature, and mean(A i ) represents the average value of the historical data of the i-th metasurface structure feature;

[0018] Calculate the mutual information value between each piece of historical data of the normalized metasurface structure feature and S respectively The mutual information value is:

[0019]

[0020] wherein, p(x) and p(y) respectively represent the distribution functions of the historical data of the spectral response and the historical data of the metasurface structure feature, and p(x, y) is the joint probability distribution function of the historical data of the spectral response and the historical data of the metasurface structure feature;

[0021] Put the mutual information value obtained in formula (2) into set Q, and the length of Q is N, then Q is expressed as: Set a threshold θ, and put the historical data of the metasurface structure feature corresponding to the mutual information value greater than the threshold θ in set Q into set L, and the length of L is d. Set L is:

[0022]

[0023] Preferably, in the technical solution, in step S02, the d metasurface structure features of set L are used as the input of the U-Net network. Here, the metasurface structure feature is a two-dimensional cross-sectional picture of the metasurface. The encoder encodes the two-dimensional cross-sectional picture of the metasurface and compresses the high-dimensional information of the metasurface into feature information in a low-dimensional space; the decoder outputs a two-channel electric field pixel image, and the two channels are the real part and the imaginary part of the electric field.

[0024] Preferably, in the technical solution, in step S03, the partial differential equation is:

[0025]

[0026] wherein, represents the Hamiltonian operator, r represents the shape of the metasurface in space, E(r) represents the electric field distribution of the metasurface, ε r represents the dielectric constant distribution of the metasurface, and k0 represents the wave number; λ represents the wavelength;

[0027] The function that satisfies the boundary constraints of the partial differential equation is:

[0028]

[0029] where u represents the solution of the partial differential equation that satisfies the physical process of constructing the partial differential equation, x represents the independent variable, U is the spatial region, represents the boundary that satisfies the spatial region;

[0030] The partial differential equation residual L PDE is:

[0031]

[0032] where T p is the residual point of the partial differential equation loss term;

[0033] The boundary loss term L BC is obtained by taking the residual of the boundary constraint function:

[0034]

[0035] where T BC is the residual point of the boundary loss term training;

[0036] By reducing the total loss L PDE and L BC of the total loss L total constantly update the parameters of the encoder and decoder neuron weights of the U-Net network until the electric field converges, and fix the parameters of the U-Net network;

[0037] The total loss L total is:

[0038] L total = r PDE L PDE + r BC L BC (8);

[0039] where r PDE is the balance factor of L PDE and r BC is the balance factor of L BC ;

[0040] Preferably, in the technical solution, in step S04, the extreme learning machine is a single-hidden layer neural network composed of an input layer with d neurons, a hidden layer with m neurons, and an output layer with v neurons;

[0041] The weight matrix W of the input layer = [w 11 w12 …… w 1m …… w 2m …… w dm T , the threshold matrix B of the input layer = [b1 b2 …… b m T , the weight matrix C of the hidden layer = [c 11 c 12 …… c 1v …… c 2v …… c mv T , the vector matrix Y output by the output layer = [y1 y2 …… y v T , where T represents the transpose operation of the matrix;

[0042] During the small-sample training process of the extreme learning machine, d pairs of data are selected from the historical database. This data pair contains the historical data of the metasurface structure features and the corresponding historical data of the metasurface spectral response. Among them, the historical data of the metasurface structure features is used as the input X of the ELM characterization model; The corresponding spectral response historical data is SS ii , ii (ii = 1, 2, 3... d) represents the ii-th historical data;

[0043] Input X and SS ii into the extreme learning machine. When the number of neurons in X is equal to the number of neurons in the input layer, randomly initialize the weights and thresholds of the input layer. The output matrix G of the hidden layer is:

[0044]

[0045] Among them, represents the tanh activation function, represents the transpose of the connection weight matrix between the n-th neuron in the input layer and the m-th neuron in the hidden layer in the extreme learning machine, represents the n-th metasurface structure feature of the ii-th input data, b n represents the threshold of the n-th neuron in the hidden layer of the extreme learning machine. Among them, the range of n is [1, d]; the output vector matrix Y of the output layer is composed of the spectral response historical data SS ii and satisfies:

[0046] Y = G T ·C(10),

[0047] Among them, G T is the transpose of the output matrix of the hidden layer;

[0048] ​​​​By performing an inverse operation on the output vector matrix Y of the output layer, the weight matrix C of the hidden layer is:

[0049] C = Pinv(G T )·Y(11);

[0050] where Pinv(*) represents the pseudo-inverse of the output matrix G of the hidden layer;

[0051] According to the obtained weight matrix C of the hidden layer, given the input layer weights and thresholds, finally, the two-dimensional hypersurface cross-section image is input into the U-Net model to obtain the predicted spectral response. The ELM characterization model outputs v data points representing the spectral response, and the final spectral response can be obtained by concatenating these data points.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] By using the U-Net model based on physics-driven training, the complexity of the assumed space will be greatly reduced, and an ELM characterization model for training the hypersurface structure features with small samples can be realized. Collaborating physical-driven and data-driven deep learning can solve problems such as data limitation, curse of dimensionality, and overfitting.

[0054] The established partial differential equation reveals the relationship between the hypersurface and the electric field. Compared with the traditional method that needs to search for physical information in a large amount of data, the residual of the partial differential equation introduces prior knowledge into the U-Net network, and combined with the U-Net network, it can quickly converge the electric field and fix the neuron weight parameters of the encoder and decoder, realizing unsupervised learning that only requires providing input, effectively reducing the data acquisition cost and improving the network training speed.

[0055] As a single-hidden-layer neural network, the weights and thresholds of the input layer neurons of the extreme learning machine are all randomly initialized. Only the weights of the hidden layer need to be solved, and the matrix inversion method can be used to solve the weight parameters of the hidden layer of the extreme learning machine. Compared with the backpropagation gradient calculation, the extreme learning machine has advantages in speed and accuracy. When the number of training data is equal to the number of hidden layer neurons, the linear equations can be accurately solved to obtain the weights of the hidden layer neurons. Description of the drawings:

[0056] Figure 1 It is the schematic diagram of the prediction method for the hypersurface spectral response based on the physical-data collaborative drive of the present invention;

[0057] Figure 2 It is the flowchart of the prediction method for the hypersurface spectral response based on the physical-data collaborative drive of the present invention;

[0058] Figure 3This is the architecture diagram of the prediction method for the hypersurface spectral response based on physical-data collaborative driving of the present invention. Specific implementation manner:

[0059] The following describes the specific implementation manner of the present invention in detail, but it should be understood that the protection scope of the present invention is not limited by the specific implementation manner.

[0060] Unless otherwise clearly stated, throughout the specification and claims, the term "comprising" or its variations such as "comprises" or "including" etc. will be understood to include the stated elements or components, without excluding other elements or other components.

[0061] As Figures 1-3 shown, the present invention provides a prediction method for the hypersurface spectral response based on physical-data collaborative driving, and its steps are as follows:

[0062] S01. Assume that the hypersurface has N structural features, where the hypersurface structural feature type is a i (i = 1, 2, 3...N), the hypersurface structural feature historical data is A i (i = 1, 2, 3...N), the hypersurface structural feature historical data corresponding to the same subscript of the hypersurface structural feature type, and the spectral response historical data corresponding to the hypersurface structural feature historical data is S, and S consists of several data points;

[0063] Select 50 pieces of hypersurface structural feature historical data and the corresponding spectral response historical data from the historical database as samples, and use the mutual information method to calculate the mutual information estimation values of the hypersurface structural feature historical data and the spectral response historical data respectively;

[0064] Perform normalization processing on A i , and the normalized hypersurface structural feature historical data is:

[0065]

[0066] where, max(A i ) represents the maximum value of the i-th hypersurface structural feature historical data in the sample, min(A i ) represents the minimum value of the i-th hypersurface structural feature historical data in the sample, and mean(A i ) represents the average value of the i-th hypersurface structural feature historical data in the sample;

[0067] Calculate the mutual information values of each normalized hypersurface structural feature historical data and S respectively The mutual information value is:

[0068]

[0069] Among them, p(x) and p(y) respectively represent the distribution functions of the historical data of spectral response and the historical data of the characteristics of the metasurface structure, and p(x, y) is the joint probability distribution function of the historical data of spectral response and the historical data of the characteristics of the metasurface structure;

[0070] Put the mutual information value obtained in formula (2) into the set Q. The length of Q is N, then Q is expressed as: Set a threshold θ, and put the historical data of the metasurface structure characteristics corresponding to the mutual information value greater than the threshold θ in the set Q into the set L. The length of L is d, and the set L is:

[0071]

[0072] S02. Construct a U-Net network. Take the d metasurface structure characteristics in the set L as the input of the U-Net network. Here, the metasurface structure characteristics are two-dimensional cross-sectional pictures of the metasurface. The U-Net network includes two parts: an encoder and a decoder. The encoder encodes the two-dimensional pixel picture of the metasurface cross-section and compresses the high-dimensional information of the metasurface into the feature information in the low-dimensional space. The decoder outputs a two-channel electric field pixel image, and the two channels are the real part and the imaginary part of the electric field;

[0073] S03. According to the physical mechanism of the electric field propagation of the metasurface structure and the electric field boundary conditions, establish a partial differential equation PDE for the interaction between the electric field and the metasurface structure. The partial differential equation is:

[0074]

[0075] Among them, represents the Hamiltonian operator, r represents the shape of the metasurface in space, E(r) represents the electric field distribution of the metasurface, and ε r represents the dielectric constant distribution of the metasurface, and k0 represents the wave number; λ represents the wavelength;

[0076] The function that satisfies the boundary constraint of the partial differential equation is:

[0077]

[0078] Among them, u represents the solution of the partial differential equation that satisfies the physical process of constructing the partial differential equation, x represents the independent variable, U is the spatial region, represents the boundary that satisfies the spatial region;

[0079] The partial differential equation residual L PDE is:

[0080]

[0081] Among them, T p is the residual point of the loss term of the partial differential equation;

[0082] The residual of the boundary constraint function is obtained to get the boundary loss term L BC :

[0083]

[0084] Among them, T BC is the residual point for training the boundary loss term;

[0085] By reducing the total loss L PDE and L BC of the total loss L total continuously update the parameters of the encoder and decoder neurons' weights of the U-Net network until the electric field converges, and fix the parameters of the U-Net network;

[0086] The total loss L total is:

[0087] L toal = r PDE L PDE + r BC L BC (8);

[0088] Among them, r PDE is the balance factor of L PDE , r BC is the balance factor of L BC ;

[0089] S04. When the unsupervised training of the U-Net network is completed, the encoder learns the hypersurface structure feature information from multiple dimensions, and designs an ELM characterization model based on small-sample training; the ELM characterization model consists of an extreme learning machine, and the extreme learning machine is a single-hidden layer neural network composed of an input layer with d neurons, a hidden layer with m neurons, and an output layer with v neurons; the weight matrix of the input layer W = [w 11 w 12 …… w 1m …… w 2m …… w dm T , the threshold matrix of the input layer B = [b1 b2 …… b m T , the weight matrix of the hidden layer C = [c 11 c 12 …… c 1v …… c 2v …… c mv T ​​​, the vector matrix Y output by the output layer is Y = [y1 y2 …… y v T , where T represents the transpose operation of the matrix, w dm represents the connection weight between the d-th neuron in the input layer and the m-th neuron in the hidden layer in the extreme learning machine, b m represents the threshold of the m-th neuron in the hidden layer, c m represents the connection weight between the m-th neuron in the hidden layer and the v-th neuron in the output layer;

[0090] During the small-sample training process of the extreme learning machine, d pairs of data are selected from the historical database. This data pair contains the historical data of the metasurface structure features and the corresponding historical data of the metasurface spectral response. Among them, the historical data of the metasurface structure features is used as the input X of the ELM characterization model; The corresponding spectral response historical data is SS ii , ii (ii = 1, 2, 3 …… d) represents the ii-th historical data;

[0091] Input X and SS ii into the extreme learning machine. When the number of neurons in X is equal to that in the input layer, randomly initialize the weights and thresholds of the input layer. The output matrix G of the hidden layer is:

[0092]

[0093] Among them, represents the tanh activation function, represents the transpose of the connection weight matrix between the n-th neuron in the input layer and the m-th neuron in the hidden layer in the ELM, represents the n-th metasurface structure feature of the ii-th input data, b n represents the threshold of the n-th neuron in the ELM hidden layer. Among them, the range of n is [1, d], and the range of o is [1, m]; the output vector matrix Y of the output layer is composed of the spectral response historical data SS ii and satisfies:

[0094] Y = G T ·C(10),

[0095] Among them, G T is the transpose of the output matrix of the hidden layer;

[0096] By performing an inverse operation on the output vector matrix Y of the output layer, the weight matrix C of the hidden layer is:

[0097] C = Pinv(G T )·Y(11);

[0098] ​wherein, inv(*) represents the pseudo-inverse of the output matrix G of the hidden layer;

[0099] According to the obtained weight matrix C of the hidden layer, with the known input layer weights and thresholds, finally, the two-dimensional hypersurface cross-sectional image is input into the U-Net model to obtain the predicted spectral response. The ELM characterization model outputs v data points representing the spectral response, and the final spectral response can be obtained by concatenating these data points.

[0100] The foregoing description of the specific exemplary embodiments of the present invention is for purposes of illustration and exemplification. These descriptions are not intended to limit the invention to the precise forms disclosed, and it is apparent that many changes and variations are possible in light of the above teachings. The purpose of selecting and describing the exemplary embodiments is to explain the specific principles of the present invention and its practical applications, so that those skilled in the art can implement and utilize various different exemplary embodiments of the present invention, as well as various different selections and changes. The scope of the present invention is intended to be defined by the claims and their equivalents.

Claims

1. A prediction method for the spectral response of a metasurface based on physical-data collaborative driving, the steps of which are as follows: S01. Use the historical data of the metasurface structure features and the corresponding spectral response historical data in the historical database, and adopt the mutual information method to calculate the estimated values of the mutual information between the metasurface structure feature historical data and the spectral response historical data respectively. Set a threshold, and screen the metasurface structure feature historical data with the estimated value of mutual information greater than the threshold and add it to the set L; S02. Construct a U-Net network, use the set L as the input of the U-Net network. The U-Net network includes two parts: an encoder and a decoder. The encoder is used for dimensionality reduction processing and feature learning of the metasurface structure, and extracting metasurface structure features. The decoder is used to output the electric field; S03. According to the physical mechanism of the electric field propagation of the metasurface structure and the electric field boundary conditions, establish a partial differential equation for the interaction between the electric field and the metasurface structure, and perform unsupervised training on the U-Net network. Unsupervised training only needs to provide the input of the U-Net network. The content of unsupervised training is: input the two-dimensional pixel pictures intercepted transversely from different metasurfaces into the U-Net network, use the residual of the partial differential equation as the loss function to constrain the U-Net network. Through backpropagation, that is, continuously reducing this loss function to make the electric field gradually satisfy the partial differential equation, and at the same time update the parameters of the encoder and decoder neurons of the U-Net network until the electric field converges. At this time, fix the parameters of the encoder and decoder neurons of the U-Net network, and determine the mapping relationship between the metasurface and the electric field through the U-Net network; S04. When the unsupervised training of the U-Net network is completed, the encoder learns the metasurface structure feature information from multiple dimensions, and design an ELM characterization model based on small-sample training. The ELM characterization model is composed of extreme learning machines. After small-sample training of the extreme learning machine, input the two-dimensional image of the metasurface, and the ELM characterization model outputs the predicted spectral response.

2. The method for predicting the hypersurface spectral response based on physical-data collaborative driving according to claim 1, wherein: In step S01, the historical database contains the historical data of the metasurface structure features simulated by several numerical methods and the corresponding spectral response historical data; Assume that the metasurface has N structural features, where the type of metasurface structural feature is a i (i = 1, 2, 3…N), and the historical data of the metasurface structural feature is A i (i = 1, 2, 3…N), the historical data of the metasurface structural feature corresponding to the type of metasurface structural feature with the same subscript, and the historical spectral response data corresponding to the historical data of the metasurface structural feature is S, and S consists of several data points; for A i Perform normalization processing, and the historical data of the metasurface structural feature after normalization processing is: Among them, max(A i ) represents the maximum value of the historical data of the i-th metasurface structure feature, min(A i ) represents the minimum value of the historical data of the i-th metasurface structure feature, and mean(A i ) represents the average value of the historical data of the i-th metasurface structure feature; Calculate the mutual information values between the normalized historical data of each metasurface structure feature and S respectively Mutual information value It is: Among them, p(x) and p(y) respectively represent the distribution functions of the spectral response historical data and the metasurface structure feature historical data, and p(x,y) is the joint probability distribution function of the spectral response historical data and the metasurface structure feature historical data; Put the mutual information value obtained in formula (2) into set Q. The length of Q is N, then Q is expressed as: Set a threshold θ, and put the historical data of the metasurface structure features corresponding to the mutual information values greater than the threshold θ in set Q into set L. The length of L is d, and set L is:

3. A physical-data collaborative-driven hypersurface spectral response prediction method according to claim 2, characterized in that: In step S02, use the d metasurface structure features of the set L as the input of the U-Net network. Here, the metasurface structure feature is the two-dimensional metasurface transverse cross-section picture. The encoder encodes the two-dimensional metasurface transverse cross-section picture and compresses the high-dimensional information of the metasurface into the feature information in the low-dimensional space; The decoder outputs a two-channel electric field pixel image, and the two channels are the real part and the imaginary part of the electric field.

4. A method for predicting the hypersurface spectral response based on physical-data collaborative driving according to claim 3, characterized in that: In step S03, the partial differential equation is: where, ▽ represents the Hamiltonian operator, r represents the shape of the metasurface in space, E(r) represents the metasurface electric field distribution, and ε r represents the metasurface permittivity distribution, and k0 represents the wave number; λ represents the wavelength; The boundary constraint function that satisfies the partial differential equation is: Among them, \(u\) represents the solution of the partial differential equation that satisfies the physical process of constructing the partial differential equation, \(x\) represents the independent variable, \(U\) is the spatial region, represents the boundary that satisfies the spatial region; Partial differential equation residual L PDF is as follows: Among them, T p is the residual point of the partial differential equation loss term; The residual of the boundary constraint function is obtained to get the boundary loss term L BC : Among them, T BC is the residual point trained for the boundary loss term; By reducing L PDE and L BC The total loss L total Continuously update the parameters of the encoder and decoder neuron weights of the U-Net network until the electric field converges, and then fix the parameters of the U-Net network; Total loss L total is as follows: L total = r PDE L PDE + r BC L BC (8); Among them, r PDE is the balance factor of L PDE , and r BC is the balance factor of L BC .

5. A physical-data collaborative driven hypersurface spectral response prediction method according to claim 4, characterized in that: In step S04, the extreme learning machine is a single-hidden layer neural network composed of an input layer with d neurons, a hidden layer with m neurons, and an output layer with v neurons; The weight matrix W of the input layer = [w 11 w 12 …… w 1m …… w 2m …… w dm T , and the threshold matrix B of the input layer = [b1 b2 …… b m T , the weight matrix C of the hidden layer = [c 11 c 12 …… c 1v …… c 2v …… c mv T , the vector matrix Y output by the output layer = [y1 y2 …… y v T , where T represents the transpose operation of the matrix;​​​​ During the small-sample training process of the extreme learning machine, d data pairs are selected from the historical database. The data pair contains the historical data of the metasurface structure features and the corresponding historical data of the metasurface spectral response. Among them, the historical data of the metasurface structure features is used as the input X of the ELM characterization model; The corresponding historical spectral response data is SS ii , where ii (ii = 1, 2, 3... d) represents the ii-th historical data; Input X and SS ii Input them into the extreme learning machine. When the number of neurons in the input layer is equal to X, randomly initialize the weights and thresholds of the input layer. The output matrix G of the hidden layer is as follows: Among them, represents the tanh activation function, is represented as the transpose of the connection weight matrix between the n-th neuron in the input layer and the m-th neuron in the hidden layer in the extreme learning machine, is represented as the n-th metasurface structure feature of the ii-th input data, b n represents the threshold of the n-th neuron in the hidden layer of the extreme learning machine; where the range of n is [1, d]; the output vector matrix Y of the output layer consists of the spectral response historical data as SS ii is composed of, satisfying: Y = G T ·C(10), Among them, G T is the transpose of the output matrix of the hidden layer; By performing an inverse operation on the output vector matrix Y of the output layer, the weight matrix C of the hidden layer is: C = Pinv(G T )·Y (11); where Pinv(*) represents the pseudo-inverse of the output matrix G of the hidden layer; Based on the obtained weight matrix C of the hidden layer, given the input layer weights and thresholds, finally, the two-dimensional hypersurface cross-section image is input into the U-Net model to obtain the predicted spectral response. The ELM characterization model outputs v data points representing the spectral response, and the final spectral response can be obtained by concatenating these data points.

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