Parametric Modeling Method for Electromagnetic Coding Metasurface Unit and Its Electromagnetic Behavior

Abstract

The present invention discloses a parametric modeling method for the electromagnetic response of coded metamaterials that combines a deep neural network and a pole-residue transfer function. By constructing a deep neural network to learn the mapping relationship between the pole-residues of the transfer function and the geometric structure of the coded metamaterials, rapid and accurate prediction of the electromagnetic response is achieved. According to the parametric model constructed by this method, the pole-residue parameter values predicted for the surface geometric structure of the metamaterials in the test set, and the electromagnetic response S-parameter values generated by the transfer function are highly fitted to the full-wave simulation results of the electromagnetic response of the metamaterials, and the accuracy and real-time performance are significantly better than other methods.

Description

Technical Field

[0001] The present invention relates to the field of electromagnetic metamaterial design, and more particularly to a parametric modeling method for electromagnetic behavior of coded metasurfaces based on multi-task learning and transfer functions. Background Art

[0002] Metamaterials are artificial materials composed of periodic sub-wavelength metal or dielectric unit structures that can resonate and couple with electrical and magnetic components in an electromagnetic field. Metamaterials are described by effective medium parameters such as permittivity and permeability. By flexibly designing and arbitrarily arranging meta-atoms, metamaterials with custom effective medium parameters can be fabricated to manipulate electromagnetic waves. Metasurfaces are the 2D equivalent form of metamaterials and are the most promising development and application in the field of metamaterials. Coded metasurfaces are an important branch of metasurfaces and can be controlled using existing digital information technologies. The design of coded metasurfaces breaks the original situation where the research and design of metamaterials were only limited to physical properties and related performances, and brings the research of metasurfaces into the torrent of digital information development. The traditional method for calculating the electromagnetic behavior response of metasurfaces is through numerical analysis and iterative calculation of Maxwell's equations. The traditional method for calculating electromagnetic responses requires a large amount of time and computing resources.

[0003] Deep neural networks are capable of learning complex function mapping relationships. Since the 1990s, researchers have applied deep neural networks in the field of metamaterial design. The process of taking the parameters that can characterize the geometric information of the metasurface unit structure as the input of the deep neural network and making the deep neural network output the electromagnetic response is called forward prediction. The deep neural network learning model learns the non-linear relationship between the network input and output through methods such as non-linear activation functions and backpropagation. Implementing forward prediction through a deep neural network can efficiently solve Maxwell's equations to obtain the electromagnetic response of the metasurface. Compared with other metasurface structures, the coding matrix used to characterize the properties of coded metasurfaces has a higher degree of freedom. Therefore, the function mapping relationship between the coding matrix and the electromagnetic response is more complex, and simple deep neural networks can no longer achieve forward prediction of coded metasurfaces.

[0004] With the in - depth understanding of electromagnetic response S - parameters by researchers, it has become possible to combine the description of the pole - residue transfer function and deep neural networks. Using the pole - residue transfer function for parameter modeling has natural advantages in data dimensionality reduction. Compared with the method of equally - spaced sampling of electromagnetic response S - parameters adopted by traditional deep neural networks, the pole - residue can more accurately characterize the electromagnetic response through the transfer function. However, due to the different geometric structures of the coded metasurface units, the effective orders of the transfer functions corresponding to their electromagnetic responses are different, and it is necessary to introduce the pole - residue tracking technology to homogenize the different orders existing in the dataset to make the orders of each sample consistent. However, pole splitting during the homogenization process inevitably causes multiple sets of valid solutions for the pole - residue, that is, there is a one - to - many inconsistency problem between the pole - residue and the electromagnetic response, which has an adverse impact on the subsequent training of the deep neural network. Summary of the Invention

[0005] Object of the Invention: The object of the present invention is to provide a parametric modeling method for electromagnetic metasurface units and their electromagnetic behaviors in view of the deficiencies of the prior art, design a parametric method for multi - task learning and pole - residue transfer functions, solve the one - to - many inconsistency problem between the pole - residue and the electromagnetic response, and improve the accuracy and real - time performance of forward prediction of the coded metasurface.

[0006] Technical Solution: The electromagnetic coded metasurface unit described in the present invention includes an upper layer, a middle layer, and a lower layer. The upper layer is the coding pattern area; the width of the coding pattern area of the upper layer is 8 mm, and the coding pattern area can be divided into several randomly generated and centrosymmetric 16×16 coding matrices with a side length of 0.5 mm. Metal copper patches are pasted in the matrix, and the conductivity σ of the metal copper patch is 5.8×10⁷ S / m, and the thickness t2 = 0.017 m.

[0007] Preferably, the middle layer of the electromagnetic coded metasurface unit is a dielectric substrate, and its material is F4B; the lower layer is covered with all - metal copper.

[0008] Preferably, the amplitude A and phase of the electromagnetic response of the electromagnetic coded metasurface unit have the following corresponding relationships with the real part Re and imaginary part Im of the electromagnetic response:

[0009]

[0010]

[0011] The parametric modeling method for the electromagnetic behavior of the above - mentioned electromagnetic coded metasurface unit of the present invention includes the following steps:

[0012] S1. Obtain electromagnetic response S - parameters and pole - residues:

[0013] Send the coded metasurface unit structure into the electromagnetic simulation software to batch obtain the S-parameters corresponding to the coded metasurface, and export all the coded metasurfaces and the corresponding spectral response data; the S-parameters obtained by the m-th coded metasurface sample through the simulation software are H m (s), and obtain a set of pole vectors p′ m and residue vectors r′ m ;

[0014] S2. Construct a data set;

[0015] Perform data processing on the obtained pole vectors p′ m and residue vectors r′ m to obtain a data set, which is represented as {Meta m , Re m , Im m , p m , r m}; where Meta m is a 32×32 new coding matrix, which is obtained by horizontally and vertically replicating and expanding the original 16×16 coding matrix, and is used to represent the coding matrix of the m-th coded metasurface; let the electromagnetic response H m corresponding to the m-th coded metasurface Meta s (m) have a real part vector Re m , and an imaginary part vector Im m ; after data processing, the complex pole vector p m and residue vector r m are represented by their real part vectors and imaginary part vectors, then the pole vector p m and residue vector r m are finally represented as:

[0016]

[0017]

[0018] Among them, m ∈ [1, S], S is the number of data set samples, and 80% of the data set is split as the training set {Meta m} train , and 20% is used as the test set {Meta m} test , and the training samples Meta m in the data set are 32×32 coding matrices, corresponding to a 1001-dimensional real part vector Re m , a 1001-dimensional imaginary part vector Im m , a 12-dimensional pole vector p m and a 12-dimensional residue vector r m ;

[0019] S3. Construct the pole network PNet and the residue network RNet:

[0020] Both the pole network PNet and the residue network RNet take as input the coding matrix Meta that characterizes the coded metasurface m , and both network structures adopt the ResNet18 structure, with the optimizer being Adam. PNet outputs the pole vector, and RNet outputs the residue vector;

[0021] Parametrically model the electromagnetic behavior response of the coded metasurface, and use the pole vector and residue vector output from the PNet and RNet network structures as parameters to construct the transfer function, that is, the Transform layer, to calculate Meta m The corresponding electromagnetic response and obtain the real part vector of the response and the imaginary part vector of the response

[0022] S4. Input the training set {Meta m} train into the pole network PNet and the residue network RNet respectively for multi-task learning. The specific methods include:

[0023] S41. The pole network PNet and the residue network RNet train tasks T P and T R :

[0024] Input the {Meta m , p m} train dataset into the pole network PNet, and let the pole network PNet output the predicted pole vector with {p m} train as the label, denoted as task T P ; Input the {Meta m , r m} train dataset into the residue network RNet, and let the residue network RNet output the predicted residue vector with {r m} train as the label, denoted as task T R ;

[0025] Input the coded metasurface structure design Meta m into the pole network PNet and the residue network RNet, where m ∈ [1, S]. After passing through the pole network PNet, the predicted pole vector is output, and the residue network RNet outputs the predicted residue vector The two networks use the mean square error as the network error, task T P and task TR The loss functions are respectively Then

[0026]

[0027]

[0028] The calculation formula of the mean square error is:

[0029]

[0030] where y is the true value, is the predicted value, M is the dimension, and the network weights and biases in the pole network PNet and the residue network RNet are represented as θ;

[0031] Task T P , T R The training objectives are denoted as:

[0032]

[0033]

[0034] To represent the final learning effects of tasks T P , T R The normalized mean square errors E P , E R are used to represent:

[0035]

[0036]

[0037] S42, the pole network PNet and the residue network RNet simultaneously train task T Hs :

[0038] Input the {Meta m , Re m , Im m} train dataset into the pole network PNet, and let the pole network PNet output the predicted pole vector with {Re m , Im m} train as the label Input the {Meta m , Re m , Im m} train dataset into the residue network RNet, and let the residue network RNet output the predicted pole vector with {Re m , Im m} trainResidue vector for label output prediction

[0039] After each round of training, two sub-networks are loaded simultaneously, and a set of predicted pole vectors is output and residue vector Then is input to the Transform layer to calculate the spectral response Obtain the real part vector of the spectral response and the imaginary part vector Backpropagate to the network to calculate the reconstruction error; the task of outputting a set of pole vectors that can calculate the correct spectral response H s (m) from the two sub-networks PNet and RNet Residue vector is denoted as T Hs , with the mean square error as the network error, and the loss function of task T Hs is Then

[0040]

[0041] S43. Update the parameters of the pole network PNet through the backpropagation algorithm:

[0042] According to the loss function of task T Hs Update the parameters of each variable in the pole network PNet; after learning task T , the pole network PNet continues to learn task T P , and the update of network weights and biases can be denoted as: Hs

[0043]

[0044] According to Obtain the gradients of each variable in the pole network, send them to the pole network optimizer, set the Adam optimizer, and update the pole network parameters through the backpropagation algorithm;

[0045] S44. Update the parameters of the residue network RNet through the backpropagation algorithm:

[0046] According to the loss function of task T Hs Update the parameters of each variable in the residue network RNet; after learning task T , the residue network RNet continues to learn task T R , and the update of network weights and biases can be denoted as: Hs

[0047]

[0048]

[0048] According to Obtain the gradients of each variable in the residue network, and feed them into the residue network optimizer. Set the Adam optimizer, and update the parameters of the residue network through the backpropagation algorithm;

[0049] S45. Iteratively loop through multiple loss functions until the network converges:

[0050] In the pole network PNet, T P and T Hs have been trained once respectively, and in the residue network RNet, T R and T Hs have been trained once respectively. This is defined as the network having completed one training cycle P. After one training cycle P, the pole network PNet and the residue network RNet will replace the training set again and learn tasks T P and T R respectively. At this time, for the pole network PNet and the residue network RNet, the updates of the network weights and biases can be respectively recorded as:

[0051]

[0052]

[0053] The pole network PNet obtains the gradients of each variable in the pole network according to , feeds them into the pole network optimizer, sets the Adam optimizer, and updates the parameters of the pole network through the backpropagation algorithm; the residue network RNet obtains the gradients of each variable in the residue network according to , feeds them into the residue network optimizer, sets the Adam optimizer, and updates the parameters of the residue network through the backpropagation algorithm;

[0054] The overall network training is counted in terms of training cycles. After training the pole network PNet and the residue network RNet for multiple training cycles, the pole network PNet and the residue network RNet will converge. When converging, the pole network PNet outputs the predicted pole vector and the residue network RNet outputs the predicted residue vector

[0055] A further preferred technical solution of the present invention is that in step S1, the encoded metasurface unit structure is fed into the electromagnetic simulation software, the operating frequency range is set to 8 GHz - 12 GHz, and relevant parameters are set to batch obtain the S parameters corresponding to the encoded metasurface.

[0056] Preferably, in step S1, the method for obtaining a set of pole vectors p′ m and residue vectors r′ m from the S parameters is to use the vector fitting method to obtain the pole-residue transfer function of order N m , that is, Nm Describe H m (s) corresponding set of pole vectors, residue vectors [p′ m , r′ m with length [N m , N m , and in this transfer function H m (s) and the corresponding pole vector p′ m and residue vector r′ m have the following corresponding relationship:

[0057]

[0058] where s = j × 2πf, j is the imaginary unit, and f is the operating frequency range of the coded metasurface.

[0059] Preferably, in step S2, the data processing of the pole vector p′ m and the residue vector r′ m includes two parts, namely:

[0060] A. Preprocess the pole vector p′ m and the residue vector r′ m ;

[0061] B. Apply pole-residue tracking to the preprocessed pole vector p′ m and the residue vector r′ m to make them homogeneous;

[0062] Among them, the method for preprocessing the pole vector p′ m and the residue vector r′ m is as follows:

[0063] I. Delete the complex poles with negative imaginary parts in the pole vector p′ m and their corresponding residues;

[0064] II. Sort the pole vector p′ m and its corresponding residue vector r′ m in ascending order according to the imaginary part of the pole vector p′ m to obtain the ordered pole vector p′ m and the residue vector r′ m ;

[0065] The method for applying pole-residue tracking to the preprocessed pole vector p′ m and the residue vector r′ m to make them homogeneous is as follows:

[0066] I. Select the nearest neighbor metasurface After preprocessing, for any coded metasurface Meta in the datasetm Corresponding transfer function order N m ∈ [N min , N max , where N min represents the lowest order of the transfer function after data processing of all electromagnetic responses in the dataset, and N max represents the highest order of the transfer function after data processing of all electromagnetic responses in the dataset;

[0067] At the transfer function order of N m +1, for multiple coded metasurface sets T r , use the L1 norm to select the nearest neighbor for the coded metasurface Meta m The process of picking the nearest neighbor is defined as:

[0068]

[0069] The L1 norm is defined as the sum of the absolute values of each element in the vector; i and j represent the row and column pixel coordinates in the coding matrix, and x ij represents the pixel value in the coding matrix, and p is the maximum value of the rows and columns in the coding matrix;

[0070] II. Select the split pole p′ M , and split the N m poles in the coded metasurface Meta m in sequence. Compare the N m +1 poles after splitting in Meta m with the N +1 poles in the nearest neighbor m for the degree of difference; the degree of difference D m is obtained by calculating the Euclidean norm using the imaginary parts of the two sets of poles. This process is denoted as:

[0071]

[0072]

[0073] After N m times of degree of difference comparison, determine that the split pole in Meta m is p′ M .

[0074] Preferably, apply pole residue tracking to the preprocessed pole vector p′ m and residue vector r′ m When they are homogeneous, the splitting in step II is to split the original pole and residue vectors [p′ m , N m with lengths of [N m , r′m , increasing to [N m +1, N m +1]; the N m +1th pole is equal to the split pole p′ M , while the N m +1th residue is the split pole p′ M corresponding residue r′ M halved. At the same time, the split residue r′ M is also reduced to half of its original value; denoted as:

[0075]

[0076]

[0077]

[0078] Among them, for the transfer function order N m ≠N max in the dataset, this step is performed on all samples, so that the transfer function order corresponding to all coded metasurfaces is N max .

[0079] Preferably, in step S3, the construction of the Transform layer includes supplementing the negative imaginary part poles and their corresponding residues to construct the transfer function to calculate the spectral response denoted as:

[0080]

[0081] s = j×2πf

[0082] where j is the imaginary unit and f is the working frequency range of the coded metasurface; and the Transform layer does not update the parameters.

[0083] Beneficial effects: (1) Based on the pole-residue transfer function, the present invention constructs two deep neural networks with the same structure, namely the pole network PNet and the residue network RNet, to respectively predict the pole and residue vectors obtained after the spectral response passes through vector fitting and data processing, realizing the parametric modeling of the coded metasurface. Modeling with poles and residues as parameters can ensure high restoration of the spectral response at a low dimension.

[0084] (2) In the two sub-networks, the present invention respectively uses two loss functions to alternately constrain the model to predict a correct set of pole and residue vectors, introducing a multi-task learning method to solve the many-to-one inconsistency problem between the pole residue and the spectral response brought by the pole residue tracking technology, realizing the fast forward prediction of the coded metasurface.

[0085] (3) The two sub-networks designed in the present invention can flexibly select to output the poles, residue vectors of the coding metasurface, or the real and imaginary parts of the spectral response. It can not only characterize the parameter characteristics but also the frequency domain characteristics, and can characterize the electromagnetic behavior response from multiple aspects, having a wider application value.

[0086] (4) Generally speaking, the parametric modeling method of the electromagnetic response of the coding metamaterial combining the deep neural network and the pole-residue transfer function realizes the rapid and accurate prediction of the electromagnetic response by constructing a deep neural network to learn the mapping relationship between the pole residues of the transfer function and the geometric structure of the coding metamaterial.

[0087] (5) In addition, for different geometric structures of the coding metamaterial units, the effective orders of their electromagnetic response corresponding transfer functions are different. It is necessary to introduce the pole-residue tracking technology to homogenize the different orders existing in the dataset to make the orders of each sample consistent. However, the pole splitting during the homogenization process inevitably causes multiple groups of valid solutions for the pole residues, that is, the one-to-many inconsistency problem between the pole residues and the electromagnetic response. For this reason, the present invention introduces multi-task learning. By designing a multi-objective loss function and alternately iteratively training multiple loss objective functions during the neural network optimization process, the model learning is effectively constrained to achieve the full learning and optimization of the neural network. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 is the structural diagram of the network model of the present invention;

[0089] Figure 2 is the schematic diagram of the electromagnetic metasurface unit structure and the coding matrix of the present invention;

[0090] Figure 3 is the diagram of the alternating iteration process of two loss functions of the pole network multi-task learning of the present invention;

[0091] Figure 4 is the diagram of the alternating iteration process of two loss functions of the residue network multi-task learning of the present invention;

[0092] Figure 5 is the diagram of the decreasing trend of multiple loss functions during the network training process in the example of the present invention;

[0093] Figure 6 is the comparison diagram of the difference between the predicted value and the true value of a random case in the example of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0094] The technical solution of the present invention will be described in detail below with reference to the drawings, but the protection scope of the present invention is not limited to the described embodiments.

[0095] Example: An electromagnetic metasurface unit includes an upper layer, a middle layer, and a lower layer. The upper layer is an encoding pattern area with a width l2 = 8 mm. The encoding pattern area can be divided into a 16×16 pixel matrix with a side length of l3 = 0.5 mm. Metal copper patches are pasted in the matrix. The conductivity σ of the metal copper patches is 5.8 + 007 S / m, and the thickness t2 = 0.017 mm. The middle layer is a dielectric substrate made of F4B, and the lower layer is covered with all-metal copper. A centrosymmetric 16×16 encoding matrix is randomly generated. The electromagnetic response of the above electromagnetic metasurface unit depends on the style of the encoding pattern. Taking the encoding matrix as the encoding pattern, the corresponding electromagnetic metasurface unit is constructed, and the frequency range is set to 8 GHz–12 GHz, and the corresponding electromagnetic response is obtained through electromagnetic simulation software. The amplitude A and phase of the electromagnetic response have the following corresponding relationships with the real part Re and the imaginary part Im of the electromagnetic response:

[0096]

[0097]

[0098] The above electromagnetic coding metasurface unit-based parametric modeling method for the electromagnetic behavior of coding metasurfaces based on multi-task learning and transfer functions includes the following steps:

[0099] S1. Obtain the electromagnetic response S parameters and pole residues:

[0100] Send the encoding metasurface unit structure into the electromagnetic simulation software, set the working frequency range to 8 GHz–12 GHz and set relevant parameters, and batch obtain the S parameters corresponding to the encoding metasurface. Export all the encoding metasurfaces and the corresponding spectral response data. Using the vector fitting method, obtain a set of pole vectors p′ m , residue vectors r′ m . The S parameter obtained by the m-th encoding metasurface sample through the simulation software is H m (s). Using the vector fitting method, obtain a pole residue transfer function of order N m , that is, N m can describe a set of poles and residue vectors [p′ m (s) corresponding to [p′ m , r′ m with a length of [N m , N m . The transfer function describes the following corresponding relationship between H m (s) and the corresponding pole and residue vectors [p′ m , r′ m :

[0101]

[0102] where \(s = j\times2\pi f\), \(j\) is the imaginary unit, and \(f\) is the operating frequency range of the coded metasurface.

[0103] S2. Construct a data set;

[0104] Perform data processing on the obtained pole vector \(p'\) m and residue vector \(r'\) m , which includes two parts: preprocessing and introducing the pole-residue tracking technique for homogenization; after data processing, a data set is obtained, and the data set is represented as \(\{Meta\) m , \(Re\) m , \(Im\) m , \(p\) m , \(r\) m \}; where \(Meta\) m is a new 32×32 coding matrix, which is obtained by horizontally and vertically replicating and expanding the original 16×16 coding matrix, and is used to represent the coding matrix of the \(m\)th coded metasurface; let the electromagnetic response \(H\) m corresponding to the \(m\)th coded metasurface \(Meta\) s \((m)\) have a real part vector \(Re\) m , and an imaginary part vector \(Im\) m ; after data processing, the complex pole vector \(p\) m and residue vector \(r\) m are represented by their real part vectors and imaginary part vectors, then the pole vector \(p\) m and residue vector \(r\) m are finally represented as:

[0105]

[0106]

[0107] where \(m\in[1,S]\), \(S\) is the number of data set samples, and 80% of the data set is divided as the training set \(\{Meta\) m \}\) train , and 20% is used as the test set \(\{Meta\) m \}\) test , and the training samples \(Meta\) m in the data set are 32×32 coding matrices, corresponding to a 1001-dimensional real part vector \(Re\) m , a 1001-dimensional imaginary part vector \(Im\) m , a 12-dimensional pole vector \(p\) m and a 12-dimensional residue vector \(r\) m .

[0108] The data processing of the pole vector \(p'\) m and residue vector \(r'\) m in step S2 includes two parts, which are respectively:

[0109] A. Preprocess the pair of poles vector p′ m and the residue vector r′ m ;

[0110] B. Apply pole-residue tracking to the preprocessed poles vector p′ m and the residue vector r′ m to make them homogeneous;

[0111] Among them, the method for preprocessing the pair of poles vector p′ m and the residue vector r′ m is as follows:

[0112] I. Delete the complex poles with negative imaginary parts in the poles vector p′ m and their corresponding residues;

[0113] II. Sort the poles vector p′ m and its corresponding residue vector r′ m in ascending order according to the magnitude of the imaginary part of the poles vector p′ m to obtain the ordered poles vector p′ m and the residue vector r′ m ;

[0114] The method for applying pole-residue tracking to the preprocessed poles vector p′ m and the residue vector r′ m to make them homogeneous is as follows:

[0115] I. Select the nearest neighbor hypersurface After preprocessing, for any encoded hypersurface Meta m in the dataset, the corresponding transfer function order N m ∈[N min , N max , where N min represents the lowest order of the transfer function after data processing for all electromagnetic responses in the dataset, and N max represents the highest order of the transfer function after data processing for all electromagnetic responses in the dataset;

[0116] Among the multiple sets T m of encoded hypersurfaces with transfer function order N r +1, use the L1 norm to select the nearest neighbor for the encoded hypersurface Meta m The process of selecting the nearest neighbor is defined as:

[0117]

[0118] ​The L1 norm is defined as the sum of the absolute values ​​of each element in the vector; i and j represent the row and column pixel coordinates in the encoding matrix, respectively, and x ij Represents the pixel value in the encoding matrix, and p is the maximum value of the row and column in the encoding matrix;

[0119] II. Selecting the splitting point p′ M , split the encoding hypersurface Meta in turn m N m Meta m The split N m +1 point with nearest neighbor N m +1 extreme point for difference comparison; difference D m It is obtained by using two sets of extreme imaginary parts to calculate the Euclidean norm. The process is recorded as:

[0120]

[0121]

[0122] After N m After comparing the differences, determine Meta m The splitting point is p′ M .

[0123] The above split is to split the original length [N m ,N m ], the poles and residue vectors [p′ m ,r′ m ], increasing to [N m +1,N m +1]; Nth m +1 pole Same splitting point p′ M are equal, and the Nth m +1 Remaining Point is the splitting point p′ M Corresponding residue r′ M half of the splitting residue r′ M It is also reduced to half of its original size; recorded as:

[0124]

[0125]

[0126]

[0127]

[0128] Among them, the transfer function order N in the data setm ≠N max For all samples of ≠N, this step is executed to make the order of the transfer function corresponding to all encoded metasurfaces be N max 。

[0129] S3. Construct the pole network PNet and the residue network RNet:

[0130] The inputs of both the pole network PNet and the residue network RNet are the encoding matrix Meta representing the encoded metasurface m , and the structures of both networks adopt the ResNet18 structure. The optimizer is Adam. The PNet outputs the pole vector, and the RNet outputs the residue vector;

[0131] Parametrically model the electromagnetic behavior response of the encoded metasurface, and use the pole vector and residue vector output from the PNet and RNet network structures as parameters to construct the transfer function, that is, the Transform layer, to calculate the electromagnetic response corresponding to Meta m and obtain the real part vector of the response and the imaginary part vector of the response

[0132] The construction of the Transform layer includes supplementing the negative imaginary part poles and their corresponding residues Construct the transfer function to calculate the spectral response Denoted as:

[0133]

[0134] s = j×2πf

[0135] And the parameters of the Transform layer are not updated.

[0136] S4. Input the training set {Meta m} train into the pole network PNet and the residue network RNet respectively for multi-task learning. The specific method includes:

[0137] S41. The pole network PNet and the residue network RNet train tasks T P and T R :

[0138] Input {Meta m , p m} train data set into the pole network PNet, and let the pole network PNet output the predicted pole vector with {p m} train as the label, denoted as task T P; Input {Meta m , r m} train dataset into the residue network RNet, and let the residue network RNet output the predicted residue vector with {r m} train as the label, denoted as task T R ;

[0139] Input the encoded metasurface structure design Meta m , m ∈ [1, S], into the pole network PNet and the residue network RNet. After passing through the pole network PNet, output the predicted pole vector The residue network RNet outputs the predicted residue vector The two networks use the mean square error as the network error. The loss functions of task T P and task T R are respectively Then

[0140]

[0141]

[0142] The calculation formula of the mean square error is:

[0143]

[0144] where y is the true value, is the predicted value, M is the dimension, and the network weights and biases in the pole network PNet and the residue network RNet are represented as θ;

[0145] The training objectives of task T P , T R are denoted as:

[0146]

[0147]

[0148] To represent the final learning effects of tasks T P , T R , the normalized mean square errors E P , E R are used to represent:

[0149]

[0150]

[0151] S42. The pole network PNet and the residue network RNet simultaneously train task T Hs :

[0152] Input {Meta m , Re m , Im m} train data set into the pole network PNet, and let the pole network PNet output the predicted pole vector with {Re m , Im m} train as the label Input {Meta m , Re m , Im m} train data set into the residue network RNet, and let the residue network RNet output the predicted residue vector with {Re m , Im m} train as the label

[0153] After each round of training, load the two sub-networks simultaneously and output a set of predicted pole vectors and residue vectors Then input them into the Transform layer to calculate the spectral response Obtain the real part vector of the spectral response and the imaginary part vector Backpropagate to the network to calculate the reconstruction error; the task of outputting a set of pole vectors s (m) and residue vectors from which the correct spectral response H can be calculated in the two sub-networks PNet and RNet is denoted as T Hs . Taking the mean square error as the network error, the loss function of task T Hs is Then

[0154]

[0155] Task T P , task T R predicted pole vectors residue vectors and task T Hs predicted pole vectors residue vectors are not the same

[0156] S43. Update the parameters of the pole network PNet through the backpropagation algorithm:

[0157] According to the loss function of task T Hs ​Update the variable parameters of the pole network PNet; after the learning of task T P the pole network PNet continues to learn task T Hs , and the update of network weights and biases can be recorded as:

[0158]

[0159] According to obtain the gradients of each variable of the pole network, send them into the pole network optimizer, set the Adam optimizer, and update the pole network parameters through the backpropagation algorithm;

[0160] S44. Update the parameters of the residue network RNet through the backpropagation algorithm:

[0161] According to the loss function of task T Hs update and calculate the variable parameters of the residue network RNet; after the learning of task T the residue network RNet continues to learn task T R , and the update of network weights and biases can be recorded as: Hs

[0162]

[0163] According to obtain the gradients of each variable of the residue network, send them into the residue network optimizer, set the Adam optimizer, and update the residue network parameters through the backpropagation algorithm;

[0164] S45. Iteratively loop through multiple loss functions until the network converges:

[0165] After training T P once and T Hs in the pole network PNet respectively, and training T R once and T Hs in the residue network RNet respectively, it is defined as the network has gone through a training cycle P; after a training cycle P, the pole network PNet and the residue network RNet will change the training set again and learn tasks T P and T R respectively; at this time, for the pole network PNet and the residue network RNet, the updates of network weights and biases can be recorded as:

[0166]

[0167]

[0168] The pole network PNet is based on The gradients of each variable of the pole network are obtained and sent to the pole network optimizer. The Adam optimizer is set, and the parameters of the pole network are updated through the backpropagation algorithm; the residue network RNet is based on The gradients of each variable of the residue network are obtained and sent to the residue network optimizer. The Adam optimizer is set, and the parameters of the residue network are updated through the backpropagation algorithm;

[0169] The training of the overall network is counted by training epochs. After training the pole network PNet and the residue network RNet for multiple training epochs, the pole network PNet and the residue network RNet will converge. When converging, the pole network PNet outputs the predicted pole vector The residue network RNet outputs the predicted residue vector

[0170] When parametrically modeling the electromagnetic behavior of the electromagnetic coding metasurface unit in this embodiment according to the above method, 3361 16×16 centrosymmetric coding matrices are randomly generated in the embodiment, and the corresponding electromagnetic metasurface unit structure {Meta m} is sent into the electromagnetic simulation software for numerical simulation. According to step S1, the spectral responses corresponding to all coded metasurfaces are obtained through simulation, and the pole vector p′ m corresponding to the coded metasurface {Meta m , the residue vector r′ m are obtained. According to step S2, the obtained pole vector p′ m , the residue vector r′ m are preprocessed and the pole residue tracking technology is introduced into two homogeneous parts. And the 16×16 coding matrix is replicated and expanded horizontally and vertically into a 32×32 coding matrix. The dataset {Meta m , Re m , Im m , p m , r m} is obtained and divided into a training set of size 2690 {Meta m , Re m , Im m , p m , r m} train , and a test set of size 671 {Meta m , Re m , Im m , p m , r m} test . Among them, Re m , Im m represent the coding matrix Meta mThe real and imaginary parts corresponding to the spectral response are both 1001 in dimension; p m , r m represent the poles and residue vectors obtained after the spectral response undergoes vector fitting, data preprocessing, and pole residue tracking technology, both of which are 12 in dimension; Re m , Im m , p m , r m are all labeled with the coding matrix Meta m label.

[0171] It should be noted that in Figure 2 , "1" indicates that there is a copper patch at the corresponding position, and "0" indicates that the corresponding position is air.

[0172] Construct the pole network PNet and the residue network RNet according to step S3. Both sub-networks adopt the ResNet18 structure, the optimizer is Adam, and the learning rate is 0.001. Both networks input the 32×32 coding matrix Meta m , PNet outputs a 12-dimensional pole vector, and RNet outputs a 12-dimensional residue vector; introduce the pole residue transfer function and build the Transform layer.

[0173] Determine that the two sub-networks are trained for 10 training cycles according to step S4. During each training cycle, the pole network PNet trains and outputs the predicted pole vector with {p m} train as the label, that is, task T , iterate 10 rounds, and the pole network PNet outputs the predicted pole vector with {Re P , Im m} m as the label, that is, task T train , iterate 100 rounds; during each training cycle, the residue network RNet trains and outputs the predicted pole vector with {r } Hs as the label, that is, task T m} train as the label, that is, task T , iterate 10 rounds, and the residue network RNet outputs the predicted residue vector with {Re R , Im m} m as the label, that is, task T train , iterate 100 rounds. Multi-task learning is carried out separately in the pole network PNet and the residue network RNet, and multiple loss functions are iteratively looped. After 10 training cycles, the simultaneous convergence of the pole network PNet and the residue network RNet is achieved. That is, task T Hs , iterate 100 rounds. Multi-task learning is carried out separately in the pole network PNet and the residue network RNet, and multiple loss functions are iteratively looped. After 10 training cycles, the simultaneous convergence of the pole network PNet and the residue network RNet is achieved.

[0174] As shown in Table 1, taking the second-order pole residue transfer function as an example, a unique set of pole vectors p obtained after homogeneous transformation by introducing the pole residue tracking technology is described. m And residue vectors r m As well as the pole network PNet and residue network RNet trained with {Re m , Im m} train As labels to predict the inconsistency problems of pole vectors Residue vectors . This illustrates the many-to-one problem among poles, residue vectors, and spectral responses after homogeneous transformation. In the table, taking the selected split pole p1 as an example, for transfer functions of other orders, the pole residue tracking technology is used strictly according to the steps described in S2. In this embodiment, only simple orders are listed to illustrate the many-to-one problems among poles, residue vectors, and spectral responses existing in the prior art.

[0175]

[0176] Table 1. Schematic table of the many-to-one relationship among poles, residue vectors, and spectral responses

[0177] Referring to Figure 5 , the loss diagrams of each task show that the losses of tasks T P , T R , and T Hs Overall show a downward trend. At each alternating iteration, there are spikes in the loss, which are caused by the above-mentioned many-to-one inconsistency problems among poles, residues, and spectral responses. However, finally, both the pole network PNet and the residue network RNet converge. The specific loss values are shown in Table 2. It can be seen from the loss that the test set tends to converge and has high accuracy.

[0178]

[0179] Table 2. Loss table of poles, residues, real part of responses, and imaginary part of responses

[0180] The simulation results show that for the parametric model constructed according to this method, the pole residue parameter values predicted for the surface geometric structure of the metamaterial in the test set, and the electromagnetic response S parameter values generated by the transfer function are highly fitted with the full-wave simulation results of the metamaterial electromagnetic response. The accuracy and real-time performance are significantly better than other methods.

[0181] As described above, although the present invention has been shown and described with reference to specific preferred embodiments, it should not be construed as a limitation to the present invention itself. Various changes can be made in its form and details without departing from the spirit and scope of the present invention defined by the appended claims.

Claims

1. A parametric modeling method for the electromagnetic behavior of an electromagnetic coding metasurface unit. The electromagnetic coding metasurface unit includes an upper layer, a middle layer, and a lower layer, where the upper layer is the coding pattern area; the width of the coding pattern area in the upper layer is 8 mm, and the coding pattern area can be divided into a number of randomly generated and centrosymmetric 16×16 coding matrices with a side length of 0.5 mm. Metal copper patches are pasted on the matrices, and the conductivity σ of the metal copper patches is 5.8+007 S / m, and the thickness t2 = 0.017 mm; The middle layer of the electromagnetic coding metasurface unit is a dielectric substrate made of F4B; the lower layer is covered with all-metal copper; The amplitude A and phase of the electromagnetic response of the electromagnetic coding metasurface unit have the following corresponding relationships with the real part Re and imaginary part Im of the electromagnetic response: It is characterized in that The method includes the following steps: S1. Obtain electromagnetic response parameters and pole residues: Send the coded metasurface unit structure into the electromagnetic simulation software, batch obtain the electromagnetic response parameters corresponding to the coded metasurface, and export all the coded metasurfaces and the corresponding electromagnetic response data; the electromagnetic response obtained by the m-th coded metasurface sample through the simulation software is H m (s), and obtain a set of pole vectors p′ m and residue vectors r′ m ; S2. Construct a data set; For the obtained pole vector p′ m and residue vector r′ m , data processing is performed to obtain a data set, which is represented as {Meta m , Re m , Im m , p m , r m}; where Meta m is a new 32×32 coding matrix, which is obtained by horizontally and vertically replicating and expanding the original 16×16 coding matrix, and is used to represent the coding matrix of the m-th coding metasurface; let the real part vector of the electromagnetic response H m (s) corresponding to the m-th coding metasurface Meta m be Re m , and the imaginary part vector be Im m ; after data processing, the complex pole vector p m and residue vector r m are represented by their real part vectors and imaginary part vectors, then the pole vector p m and residue vector r m are finally represented as: Among them, m ∈ [1, N], where N is the number of samples in the dataset. 80% of the dataset is segmented as the training set {Meta m} train , and 20% is used as the test set {Meta m} test . The training samples Meta m in the dataset are 32×32 encoding matrices, and the corresponding labels are the real part vector Re m of 1001 dimensions, the imaginary part vector Im m of 1001 dimensions, the pole vector p m of 12 dimensions, and the residue vector r m of 12 dimensions; S3. Construct a pole network PNet and a residue network RNet: The input of both the pole network PNet and the residue network RNet is the coding matrix Meta that characterizes the coding metasurface m , and the structures of both networks adopt the ResNet18 structure. The optimizer is Adam. PNet outputs the pole vector, and RNet outputs the residue vector; Parametrically model the electromagnetic behavior response of the coded metasurface, construct a transfer function, i.e., the Transform layer, using the pole vector and residue vector output from the PNet and RNet network structures as parameters, and calculate the corresponding electromagnetic response H m (s), and obtain the real part vector of the response m and the imaginary part vector of the response ​ S4. Input the training set {Meta m} train into the pole network PNet and the residue network RNet respectively for multi-task learning. The specific method includes: S41. The pole network PNet and the residue network RNet are respectively trained for tasks T P and T R : Input {Meta m , p m} train dataset into the pole network PNet, and let the pole network PNet output the predicted pole vector with {p m} train as the label, denoted as task T P ; Input {Meta m , r m} train dataset into the residue network RNet, and let the residue network RNet output the predicted residue vector with {r m} train as the label, denoted as task T R ; Input the encoded metasurface structure design Meta into the pole network PNet and the residue network RNet m , m ∈ [1, N], and output the predicted pole vector through the pole network PNet The residue network RNet outputs the predicted residue vector The two networks use the mean square error as the network error, task T P and task T R The loss functions of are respectively Then T P : T R : The calculation formula for the mean square error is: where y is the true value, is the predicted value, M is the dimension, and the network weights and biases in the pole network PNet and the residue network RNet are added to the parameter set θ; Task T P , T R The training objective is denoted as: T P ∶ T R ∶ To characterize the final learning effect of task T P and T R the normalized mean square error E P and E R are used to represent: T P ∶ T R ∶ S42. The pole network PNet and the residue network RNet simultaneously train the task T Hs : Input the {Meta m , Re m , Im m} train dataset into the pole network PNet, and let the pole network PNet output the predicted pole vector with {Re m , Im m} train as the label Input the {Meta m , Re m , Im m} train dataset into the residue network RNet, and let the residue network RNet output the predicted residue vector with {Re m , Im m} train as the label After each round of training, two sub-networks are loaded simultaneously, and a set of predicted pole vectors are output and residue vectors Then is input into the Transform layer to calculate the electromagnetic response H m (s), and the real part vector of the electromagnetic response is obtained and the imaginary part vector Backpropagate to the network to calculate the reconstruction error; the task of outputting a set of pole vectors from the two sub-networks PNet and RNet that can calculate the correct electromagnetic response H m (s) is denoted as T residue vectors The mean square error is used as the network error, and the loss function of task T Hs is Hs Then Then T Hs : S43. Update the parameters of the pole network PNet through the backpropagation algorithm: According to task T Hs 's loss function Update and calculate the variable parameters of each pole network PNet; After learning task T P , the pole network PNet continues to learn task T Hs , and the update of network weights and biases can be recorded as: T Hs ∶ According to The gradients of each variable of the pole network are obtained and sent to the pole network optimizer. The Adam optimizer is set, and the parameters of the pole network are updated through the backpropagation algorithm; S44. Update the parameters of the residue network RNet through the backpropagation algorithm: According to task T Hs 's loss function Update and calculate the variable parameters of each residue network RNet; After learning task T R 's learning, the residue network RNet continues to learn task T Hs , and the update of network weights and biases can be recorded as: T Hs : According to The gradients of each variable in the residue network are obtained and sent to the residue network optimizer. The Adam optimizer is set, and the parameters of the residue network are updated through the backpropagation algorithm; S45. Iteratively loop multiple loss functions until the network converges: Once trained separately in the pole network PNet for T P and T Hs , and once trained separately in the residue network RNet for T R and T Hs , it is defined that the network has undergone one training cycle P; after one training cycle P, the pole network PNet and the residue network RNet will replace the training set again and learn tasks T P and T R respectively; at this time, for the pole network PNet and the residue network RNet, the updates of the network weights and biases can be respectively denoted as: T P ∶ T R ∶ The pole network PNet is based on to obtain the gradients of each variable in the pole network, and send them to the pole network optimizer. The Adam optimizer is set, and the parameters of the pole network are updated through the backpropagation algorithm; the residue network RNet is based on to obtain the gradients of each variable in the residue network, and send them to the residue network optimizer. The Adam optimizer is set, and the parameters of the residue network are updated through the backpropagation algorithm; The overall network training is counted in training cycles. After training the pole network PNet and the residue network RNet for multiple training cycles, the pole network PNet and the residue network RNet will converge. When converging, the pole network PNet outputs a predicted pole vector The residue network RNet outputs a predicted residue vector 2. The parametric modeling method for the electromagnetic behavior of the electromagnetic coding metasurface unit according to claim 1, characterized in that In step S1, the structure of the coding metasurface unit is sent into an electromagnetic simulation software, the working frequency range is set to 8 GHz - 12 GHz, and relevant parameters are set to batch obtain the electromagnetic response parameters corresponding to the coding metasurface.

3. The parametric modeling method for the electromagnetic behavior of the electromagnetic coding metasurface unit according to claim 1, wherein In step S1, a set of pole vectors p′ is obtained from the electromagnetic response parameters m and residue vectors r′ m by using the vector fitting method to obtain an N m th-order pole-residue transfer function, that is, N m describing a set of pole vectors and residue vectors [p′ m , r′ m with lengths of [N m , N m , N m in the transfer function. There is the following correspondence between H m (s) and the corresponding pole vectors p′ m and residue vectors r′ m : Among them, s = j×2πf, where j is the imaginary unit and f is the working frequency range of the coding metasurface.

4. The parametric modeling method for the electromagnetic behavior of the electromagnetic coding metasurface unit according to claim 1, characterized in that, In step S2, for the pole vector p′ m and the residue vector r′ m Data processing includes two parts, namely: A. Preprocess the pole pair vector p′ m and the residue vector r′ m ; B. Apply the pole residue tracking to the preprocessed pole vector p′ m and the residue vector r′ m Homogeneous; Among them, the method for preprocessing the antipodal vector p′ m and the residue vector r′ m is as follows: I. Delete the pole vector p′ m which contains complex poles with negative imaginary parts and their corresponding residues; II. According to the imaginary part magnitude of the pole vector p′ m sort the pole vector p′ m and its corresponding residue vector r′ m from small to large, and obtain the ordered pole vector p′ m and residue vector r′ m ; Apply the pole residue tracking to the preprocessed pole vector p′ m and the residue vector r′ m The homogeneous method is as follows: I. Select the nearest neighbor metasurface After preprocessing, any encoded metasurface Meta in the dataset m corresponds to the order N of the transfer function m ∈[N min , N max , where N min represents the lowest order of the transfer function after data processing of all electromagnetic responses in the dataset, and N max represents the highest order of the transfer function after data processing of all electromagnetic responses in the dataset; Among multiple sets T of coded metasurfaces with transfer function order N m +1, the L1 norm is used to select the nearest neighbor for the coded metasurface Meta r The process of selecting the nearest neighbor is defined as: m select the nearest neighbor The process of picking the nearest neighbor is defined as: The L1 norm is defined as the sum of the absolute values of the elements in the vector; i and j represent the row and column pixel coordinates in the encoding matrix, respectively, and x ij represents the pixel value in the encoding matrix, and p is the maximum value of the rows and columns in the encoding matrix; II. Select the splitting pole p′ M , and split the N m poles in the coding metasurface Meta m one by one. Compare the N m + 1 split poles in Meta m with the N + 1 poles in the nearest neighbor m for the degree of difference comparison. The degree of difference D m is obtained by calculating the Euclidean norm using the imaginary parts of the two sets of poles. This process is denoted as: After N m times of difference degree comparison, determine the split pole in Meta m to be p′ M .

5. The parametric modeling method for the electromagnetic behavior of the electromagnetic coding metasurface unit according to claim 4, characterized in that Apply the pole residue tracking to the preprocessed pole vector p′ m and the residue vector r′ m In the homogeneous case, the splitting in step II is to increase the original pole and residue vectors [p′ m ,N m of length [N m ,r′ m to [N m +1,N m +1]; the (N m +1)-th pole is equal to the split pole p′ M , while the (N m +1)-th residue is half of the residue r′ M corresponding to the split pole p′ M . At the same time, the split residue r′ M is also reduced to half of its original value; denoted as: Among them, for the samples where the order N of the transfer function in the dataset m ≠N max this step is executed for all of them, so that the order of the transfer function corresponding to all coding metasurfaces is N max ; s = j×2πf, where j is the imaginary unit and f is the operating frequency range of the coding metasurface.

6. The parametric modeling method for the electromagnetic behavior of the electromagnetic coding metasurface unit according to claim 1, characterized in that In step S3, the construction of the Transform layer includes supplementing negative imaginary part poles and their corresponding residues Construct the transfer function to calculate the electromagnetic response H m (s), denoted as: Among them, s = j×2πf, where j is the imaginary unit and f is the working frequency range of the coding metasurface; and the Transform layer does not update parameters.

Images