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Abstract
Since inverse design is an ill-conditioned problem of mapping from low dimensions to high dimensions, inverse design is challenging, especially for design problems with many degrees of freedom (DOFs). Traditional deep learning methods and optimization methods cannot readily calculate the inverse design of meta-atoms with high DOFs. In this paper, a new method combining deep learning and genetic algorithm (GA) methods is proposed to realize the inverse design of meta-atoms with high DOFs. In this method, a predicting neural network (PNN) and a variational autoencoder (VAE) generation model are constructed and trained. The generative model is used to constrain and compress the large design space, so that the GA can jump out of the local optimal solution and find the global optimal solution. The predicting model is used to quickly evaluate the fitness value of each offspring in the GA. With the assistance of these two machine learning models, the GA can find the optimal design of meta-atoms. This approach can realize, on demand, inverse design of meta-atoms, and opens the way for the optimization of procedures in other fields.
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1. Introduction
Metamaterials are composite materials designed and synthesized by human beings. They have some characteristics that natural materials do not have. Their properties are determined by the internal microstructure rather than by the chemical composition of natural materials. The key to the manufacture of metamaterials is to use artificially designed and manufactured nanostructure units to achieve the expected performance and function. Metasurfaces, the two-dimensional (2D) versions of metamaterials, are planar devices composed of subwavelength structures, called meta-atoms [1,2]. By interacting with the local electromagnetic field, according to the generalized Snell’s law [3], metasurfaces can tailor phase, amplitude, polarization, and angular momentum of incident waves [4,5]. Metasurfaces have been widely used in perfect absorption [6], super resolution imaging [7,8], beam steering [9,10], and nonlinear optical generation [11,12].
To prepare metasurfaces with the desired optical properties, the spectra and the geometry of individual meta-atoms must be accurately predicted and designed. However, the complex relation between meta-atom structures and their spectral responses (SR) cannot be resolved by generalized theory, so traditional design methods rely on empirical reasoning or experimental trial-and-error [10,13]. Since this method involves numerous full-wave numerical simulations (e.g., finite-difference time-domain (FDTD) method, finite integration technique (FIT) and finite-component method (FEM)), the efficiency is low and the results are often unsatisfactory; using these methods to find a suitable set of meta-atom structures for a specific design is time-consuming and laborious. Thanks to rapid development of optimization methods and artificial intelligence (AI), some traditionally intractable problems have recently been solved by these methods [14–18]. The field of optics is no exception – it presents many intensely difficult problems – and in order to speed up the design process without the need for computationally intensive numerical and analytical methods, Maxwell’s equations and data-driven approaches, especially deep neural networks (DNNs), have gradually been incorporated into the design process for microwave and nano-photonics devices [19,20].
When DNNs were first applied to meta-atom inverse design, they were used to fit the mapping relation between the structural parameters of some simple structures (such as ring [21], H shape [22], V shape [23], etc.) and their SRs. Then, they were used to fit the mapping relation between the arrangement and combination of several fixed structures and their SRs [24,25]. The common idea in these contributions was to model the relation between design parameters and SR as a bidirectional map, which can only handle a few design parameters in a small range of applications. Treating meta-atom inverse design as a regression problem of one-to-one mapping contradicts a physical phenomenon: very different meta-atom structures may yield very similar SRs; this is the “one-to-many” mapping problem, which exists in all inverse engineering scenarios. Later, some theoretically feasible methods were proposed. The earliest adopted a tandem network (TN) structure [20–33] to combine a pre-trained simulator with another model generator, overcoming the problem of non-unique solutions and allowing the neural network to converge stably. Subsequently, references [34,26] proposed an inverse design method of meta-atoms based on conditional generative adversarial network (CGAN), which achieved inverse design by constructing a generative model and a discriminative model to play against each other and train each other alternately. Because the generative model can convert random noise and input conditional spectra into meta-atom images, the problem of one-to-many mapping in inverse design can be solved by inputting the combination of the same conditional spectrum and different interfering noises. Reference [35] proposed a meta-atom inverse design method based on a deep generative model of Variational Auto-Encoders (VAE) from a probabilistic perspective. Since the inverse design is performed by random sampling in the latent space and combined with the conditional spectrum, the problem of one-to-many mapping can also be solved to a certain extent like the generative adversarial network. In reference [36], the inverse design of broadband metasurface absorber have been realized based on VAE variants. The methods of combining forward prediction with optimization method [37–30], which are different from the methods of directly realize inverse design using unique neural network structures (such as TN, VAE, and GAN, reference [41] benchmarked these three networks and gave its advantages, respectively), can also solve the “one-to-many” mapping problem. In reference [41], the inverse design of circular and H-shaped meta-atoms were realized by combining forward prediction and genetic algorithm. However, these works have a common limitation: the inversely designed meta-atoms do not have high DOFs. The lack of existing high-DOF solutions limits the functionality and performance of the designed meta-atoms. Now, the most advanced works on inverse design of high DOFs meta-atoms should be the references [42,43] (it is also the only one at present). They realized the inverse design of high DOFs meta-atoms by combining the CGAN with the Wasserstein Generative Adversarial Networks (WGAN) to stabilize the training process, which is really amazing. However, this method of directly using deep learning for inverse design has defects in theory, especially for the inverse design of high DOFs meta-atoms. Neural networks are unlikely to fit the mapping relation of “one to many”, and the problem of “one to many” mapping is particularly prominent in the inverse design of high DOFs meta-atoms. Therefore, if the neural network is directly used for the inverse design of high DOFs meta-atoms, it will make it difficult for the neural network to converge. If the network is forced to converge, it will inevitably limit (or destroy) the objective “one to many” mapping. So, for the inverse design of high DOFs meta-atoms, the combination of optimization algorithm and forward prediction network (there is no “one to many” mapping, but “many to one” mapping) is more reasonable. Unfortunately, the performance of the method is limited due to the backwardness of the graphics coding method. In its supporting information, reference [41] gave a forward prediction network with complex structures, but failed to inverse design it, presumably due to the lack of an appropriate coding scheme. Traditional shape coding falls into two categories: (a) parametric coding [44], and (b) pixilation [45]. The former can only generate a very limited set of shapes, which severely limits the performance of the optimization method, while the latter will generate a larger space set, but a large part of it is infeasible, increasing the difficulty of method optimization. Here, we present a new scheme to alleviate this problem.
This paper combines deep learning and optimization methods to study an intelligent deep optimization model to realize the inverse design of meta-atoms with high DOFs. The method uses the generative model in deep learning to help coding, to compress the solution space, and to eliminate most of the infeasible solutions in the solution space, so that the optimization method can quickly find the solution. A predicting neural network (PNN) is used to replace the traditional numerical simulation to calculate the SR and shorten the calculation time of the whole method. Through the combination of optimization method and PNN, the problems of one-to-many mapping and the difficulty in generating meta-atom structures on demand by inverse design of meta-atoms with high DOFs are solved. We have successively verified on the test set and on custom data, and the results are good. The method we proposed can be applied not only to the inverse design of meta-atoms but also to other electromagnetic devices, such as antennas, integrated optical circuit devices, etc. In addition, the new coding idea will be helpful to better use optimization algorithms in any other field, and exposes a new opportunity for digital coding metasurfaces [46–48].
2. Dataset
The dataset [49] in this paper is composed of the structural information of all-dielectric meta-atoms and their corresponding SRs, as shown in Fig. 1. In this dataset, the meta-atoms consist of a stack of two layers of a dielectric component with different refractive indices, where the bottom layer is a square low-refractive-index dielectric component (this substrate layer is assigned a refractive index of 1.4), and the upper layer is a high-index dielectric component of arbitrary symmetrical shape; the all-dielectric components are only generated in the top-left quadrant of each unit cell and then symmetrically replicated along x and y axes to form the entire image. The all-dielectric meta-atom structural information can be expressed as 2D structural images and one-dimensional (1D) structural parameters. The 2D images are a top view of a high-refractive-index dielectric component (an image composed of 1-bit pixels and of size 64 × 64 pixels), and there are three 1D structural parameters (forming a 1 × 3 matrix): refractive index∈ [3.5, 5], thickness ∈ [0.5, 1] and lattice size ∈ [2.5, 3] (all lengths are in microns). The SR is the real and imaginary parts of the transmission coefficient, the frequency range is 30-60 THz (wavelength is 5-10 µm); the SR is discretized into 301 points (framing 300 0.1 THz intervals). The dataset comprises 174,883 sets of 2D images, structural parameters and their corresponding SRs.
Fig. 1. Dataset presentation. (a) 3D stereogram of a randomly generated meta-atom. White represents high-index dielectric components, while black represents the low-index substrate. (b) The meta-atom is expressed by 2D structural images and structural parameters. (c) The SR of the corresponding meta-atom.
3. System model structure
For the method of constructing neural network for direct inverse design, when the DOFs of meta-atoms is high, the one-to-many problem becomes more prominent, making it difficult to train the neural network; the forward prediction network can be better trained without the influence of this problem. Therefore, this paper proposes to combine the forward prediction network with the optimization method to realize the inverse design of meta-atoms with high DOFs. However, the performance of the traditional optimization method is limited due to the backwardness of the graphics coding method. We propose a new deep-learning-aided coding method to improve the performance of the method. As shown in Fig. 2, the system model is composed of two parts: the PNN and an intelligent optimization module based on GA. The PNN [49] can achieve accurate and fast spectral prediction of meta-atoms with high DOFs. The intelligent optimization module combines VAE decoding with GA to realize the inverse design of meta-atoms with high DOFs, a procedure that does not have the limitations of traditional machine learning or optimization methods. In the intelligent optimization module, the decoder is composed of a binary decoder and a pre-trained VAE model, which is used to decode the binary code (representing the individual) generated in the initialization and genetic process into the corresponding 2D structural image and 1D structural parameters. After inputting the 2D structural images and structural parameters, the PNN can quickly and accurately predict its optical response, so it applies to evaluate the fitness value of each offspring in the genetic method.
3.1 PNN structure
Calculation with the traditional FDTD method is time-consuming. If the SR were calculated by the FDTD method, it would slow down the entire optimization process. The method presented in this paper uses a PNN to replace the traditional numerical calculation method, speeding the optimization process of the method. The PNN is a data-driven neural network that can quickly and accurately predict the SR of the meta-atoms. Neural networks are composed of many neurons, and have shown great potential for representing very complex mapping relations [50,51]. Since the relation between the meta-atoms and their SRs is complicated, a neural network is a good choice for calculating the latent relation between the meta-atoms and their SRs.
In the PNN, we use the structural image and structural parameters to predict the SR [49]. The network structure is shown in Fig. 3. The 2D structural image outputs 64 feature maps of size 8 × 8 by use of a three-layered convolutional neural network (CNN). The three structural parameters are expanded to 1 × 64 through the Neural Tensor Network (NTN) [41] layer and each of these 64 numbers is copied to the size of 8 × 8 (the output size is 8 × 8 × 64). Finally, the outputs of the CNN and NTN are spliced and output to the SR through an additional three-layer CNN and a two-layer fully connected neural network. The SR is discretized into 301 points, and the real and imaginary prediction networks have exactly the same structure but with different parameters. The operations of the NTN layer are:
where e (size is 3 × 1) is the structural parameter, W (size is 3 × 3) is a trainable weight matrix with k = 64, b (size is 1 × 3) is the bias term, and f is the ReLU activation function. In the entire network structure, except the last layer, the batch normalization layer (BN) and ReLU activation function are applied to the output tensors of each layer, and the activation function of the last layer is tanh.3.2 Codec model
Since complex meta-atoms are expressed as structural parameters and structural images, their design space is high dimensional and manifests heterogeneity. Therefore, the inverse design of complex meta-atoms is a highly ill-conditioned problem. However, the traditional GA usually adopts a simple coding method and, due to the one-to-many mapping problem, the method does not perform well in on-demand optimization in the entire solution space. In order to solve this problem, this paper adopts an encoding and decoding scheme based on a deep generative model. The specific encoding and decoding scheme is shown in Fig. 4. For the structural images, we first use the neural network to learn the regular shape information of the structural images, compress it into a high-dimensional latent space, and then perform binary encoding on the points in the latent space, and directly for the structural parameters, finally splicing the two binary codes to obtain the final encoding. Decoding is the opposite.
To learn the regular shape information of the meta-atom structural images and encode it onto the latent space, we construct a VAE model. A VAE is derived from the autoencoder (AE). An AE is an unsupervised learning method where the basic idea is to learn the optimal encoding and decoding scheme by simultaneously training the encoder and the decoder; the encoder reduces the dimension of the input object to a certain point in the latent space to achieve encoding, and the decoder restores the encoded point. The general structure of AE is shown in Fig. 5(a). The essence of VAE is to learn its probability distribution through existing data samples, and then randomly sample on the learned probability distribution and restore, so that new samples with the characteristics (regular and symmetrical) of the dataset can be obtained. In VAE, the points in its latent space are related and obey the same distribution. We assume that it is a Gaussian mixture model (GMM), and its mean and variance are exactly the output of the encoder network. Compared to AE, VAE can generate content. The parametric form of the input and output of the VAE is shown in Fig. 5(b), and the details of the DNNs for the VAE’s encoder and decoder are shown in Fig. 5(c). Here, we set the dimension of the latent space to 20. The output of the VAE’s encoder is the mean and covariance tensor describing the standard normal distribution. Based on the mean and covariance tensor, the latent encoding vector can be obtained by a reparameterization method [52], and then the latent encoding vector is input to the VAE’s decoder, which restores it as a 2D structural image of meta-atom.
Since each dimension of the latent space of the VAE is approximately a standard normal distribution (when training VAE model, the latent space will meet this characteristic due to the constraint of Kulback-Leibler (KL) divergence), we take the interval of each dimension [-10, 10] for encoding (in terms of probability this contains almost all the points in the latent space). When coding, considering that the span sp is 20, and there is a showing that the individual accuracy ac in the genetic method should be about 0.01, the binary digit n must satisfy the following rule:
After calculation, each of the 20 dimensions is encoded with an 11-bit (at least) binary number. Similarly, according to the value ranges of the three structural parameters and the corresponding accuracy requirements, we encode the parameters into 6-, 6-, and 8-bit binary numbers, respectively. Finally, we encode structural images and structural parameters of the meta-atoms together into a 240-bits binary code (11 × 20 + 6 + 6 + 8).3.3 Intelligent optimization module
In order to realize the inverse design of meta-atoms with high DOFs, we combine the above-mentioned the PNN, the codec model and the GA. The inverse design process is shown in Fig. 6. Randomly initializing the binary population is the key to realizing one-to-many inverse design. By running this method multiple times on the same target spectrum, with random initialization, our method can find multiple approximate solutions scattered throughout the latent space. Through the decoder, each individual (a binary number) can be decoded into the 2D structural images and the three structural parameters of the meta-atom (that is, the input of the PNN). To speed up the evaluation of individuals in each generation of population, in the evaluation step, the SR of individuals are quickly predicted and evaluated using a PNN. After evaluating each individual, the method decides whether an approximate optimal solution has been found. A sign of finding a near-optimal solution is that the evaluated spectral error is less than a threshold. If the optimal solution is found, the structural image and structural parameters will be output, otherwise, a new population will be generated by GA and sent to PNN after decoding. Such operations are repeated until an approximate optimal solution is found or the specified number of iterations is reached.
4. Results and analysis
The following is a detailed simulation analysis from three aspects: forward prediction network, generative model, and inverse design. We remark that all simulation experiments are performed on a workstation with an Intel Xeon Silver 4210R @2.40 GHz & 2.39 GHz (two processors) central processing unit, NVIDIA GeForce GTX 3080 (double), and 128 GB access memory; and the deep learning frameworks Tensor Flow and Keras are used to design and train the networks (PNN and VAE).
4.1 Forward prediction results and analysis
For the PNN training process, the mean square error (MSE) is used as the loss function. The loss function is expressed as:
The PNN training process is as follows: first, the dataset is divided into training set, validation set, and test set according to the ratio of 80%, 10%, and 10%; second, the training set and validation set are used to train the network. At this time the validation set is only used to judge whether the training can be terminated early; the batch size is 128, iterations are 100; finally, the trained network model is applied to the test set. The mean square error between the prediction result and the simulation result is 0.0031 for the real part (Fig. 7(a)), 0.0021 for the imaginary part (Fig. 7(b)), and the average error rate of the prediction result is 1.30% for the real part. The imaginary part is 1.38%, and its calculation formula is:Fig. 7. (a) Training error for the PNN with real part component as targets. (b) Training error for the PNN with imaginary part component as targets. (c) The prediction results of the PNN on the test set. The first row is the structural image and structural parameters to be predicted. The second and third rows are the real and imaginary parts of the corresponding SR, respectively. In the last row, the red curves are the phase profiles, while the blue curves refer to the amplitude responses. All four meta-atoms presented are randomly selected from the test data.
We demonstrate the validity of the well-trained PNN with four meta-atom samples (Fig. 7(c)) that were randomly selected from the test dataset. Additionally, the amplitude and phase are calculated by the following formulas:
In Fig. 7(c), their transmission coefficients (red curves) were evaluated and compared with the results derived from FEM-based simulations (blue curves). The simulation results show that the PNN prediction results agreed well with the full-wave electromagnetic simulations. It is significant that it takes only 0.6 seconds (including the time to load network model parameters (real part training takes 2.3 hours and imaginary part takes 2.2 hours)) for the PNN to get the SR of a meta-atom, but 8 minutes for the numerical calculation method.4.2 Generate model simulation results
For the VAE model, we use all the 2D structural images in the dataset for training. The loss function used to train the VAE is defined as:
where a and b are the weight parameters (the values in the reported research are 100 and 1, respectively), and $los{s_{re}}$ is a reconstruction term, that is, the mean square error between the input structural image and the reconstructed structural image, to ensure the performance of the encoding-decoding scheme, which is expressed as follows: $los{s_{KL}}$ is the KL divergence between the returned distribution and the standard Gaussian distribution, which aims to restrict the Gaussian mixture distribution in the latent space, so that the mean and variance of each dimension are close to 0 and 1 as far as possible. It is defined as:After training the VAE model, the $los{s_{re}}$ is 0.0826 and $los{s_{KL}}$ is 5.4002. The test effect is shown in Fig. 8(a). The reconstructed image is basically consistent with the original image, but it is fuzzy at the boundary; this is because the last layer of the VAE model uses the sigmoid activation function, which obtains a number between 0 and 1, while the pixel value of the original image is not 0 or 1. Therefore, we add a processing layer after the VAE model to judge the pixel value of the reconstructed image. If it is larger than 0.5, it will be 1, otherwise it will be 0. The effect after processing is shown in Fig. 8(b). Although there are some differences in some details, the effect of overall reconstruction is still ideal.
Fig. 8. Verify the performance of VAE model (a) VAE model generation effect diagram after trained, with input diagram on the left and reconstruction diagram on the right. (b) VAE model generation effect diagram after adding processing layer, with input diagram on the left and reconstruction diagram on the right. (c) Structural image generated by expanding sampling values of different multiples through decoder in VAE (the expansion ratio is 10 times and 5 times respectively from left to right).
Because the genetic method will continue to produce new individuals, our model is required to decode the new individuals (i.e., any point in the latent space) into a regular structural image. Now we randomly generate 121 (11 × 11) 20-dimensional latent space data from the standard normal distribution, expand them by different multiples (i.e., equivalent to random sampling from different ranges), and generate the structural image through the coder model of VAE, respectively. The result is shown in Fig. 8(c). It can be seen from the figure that the points randomly sampled from the latent space can be well mapped into a structural image with regular shape. Therefore, this coding method can be well integrated into the genetic method.
4.3 Inverse design results and analysis
Traditional GA usually adopts pixilated coding for meta-atoms with high DOFs. As shown in Fig. 9, the traditional GA combined with the PNN is used for inverse design, and the final structural image is disorganized (but its SR is very similar to the target spectrum). This kind of structural image, difficult to fabricate, is obviously not the structural image we need. In order to get a structural image that is easy to fabricate (regular shape structural image), we introduce constraints to the solution space, so that the genetic method is optimized in the constrained solution space, which can make the genetic method jump out of the local optimal expectation and find the global optimal.
Fig. 9. The result of the inverse design of using pixelated coding. (a),(b) The blue curve is the target SR, and the red dots is the SR of the predicted meta-atom. (c) The red curves represent the phase profiles, while the blue curves refer to the amplitude responses. Dots represent data generated by the PNN, while solid curves are data obtained from numerical simulations. (d) The structural image and structural parameters obtained by the inverse design.
The first method attempted is to find the structure with the target spectrum, while also restricting the structure itself. Considering that the structural images in the dataset are regular symmetrical graphs, the number of connected domains in the structural image is relatively small, so the initial attempt was to use the number of connected domains in the structural image as a constraint that affects the fitness value in the GA. The simulation results are shown in Fig. 10. The effect of decreasing the number of connected domains is not obvious (the number of connected domains in Fig. 9 is 577, and in Fig. 10 is 470), but the average error between the predicted spectrum and the target spectrum has increased several times (the MSE is 0.014 for Fig. 9 and 0.038 for Fig. 10).
Fig. 10. The results of the inverse design of the GA after adding the connected domain constraints. (a),(b) The blue curve is the target SR, and the red dots is the SR of the predicted meta-atom. (c) The red curves represent the phase profiles, while the blue curves refer to the amplitude responses. Dots represent data generated by the PNN, while solid curves are data obtained from numerical simulations. (d) The structural image and structural parameters obtained by the inverse design.
Finally, we use a deep learning model to constrain the solution space, through a set of neural networks to learn the regular shape features of the structural image, and compress them into a high-dimensional latent space (i.e., constrained solution space), and then restore the points in the high-dimensional latent space to a structural image through another set of neural networks. In this way, the GA only needs to optimize in the latent space with low capacity. On the one hand, compressing the solution space can speed up the method to find the optimal solution, on the other hand, the final optimized structural image can meet our expectations.
In order to demonstrate the effectiveness of the proposed method, we use the same target spectrum as Fig. 9 and Fig. 10, and use this method for inverse design. The population size was set to 200 and the maximum number of iterations was 100. As shown in Fig. 11 (the MSE is 0.001), our method successfully inversed the design of the SRs in the test set, and obtained a regular structural image and its structural parameters, and the predicted structural image and structural parameter have a spectrograph very similar to the target SR. The whole inverse design process (including loading PNN, VAE model and optimization process) only takes 134 seconds. However, it takes about 10 minutes for the traditional digital simulation software to calculate the spectral response of a complex structure. So compared with the traditional inverse design, our method greatly improves the efficiency.
Fig. 11. Evaluation of the proposed method. (a),(b) The blue curve is the target SR, and the red dots is the SR of the predicted meta-atom. (c) The red curves represent the phase profiles, while the blue curves refer to the amplitude responses. Dots represent data generated by the PNN, while solid curves are data obtained from numerical simulations. (d) The structural image and structural parameters obtained by the inverse design.
Next, we will use this method to inverse design the same SR multiple times to test whether it can solve the “one-to-many” mapping problem in the inverse design. The result is shown in Fig. 12. It can be seen from the figure that our method can retrieve any symmetrical shape, and after multiple optimizations, we can obtain multiple inverse design results with completely different structural images but similar SR. We can choose one of the easiest structures to make from multiple groups of designs (as shown in the fourth line (left) and second line (right) in Fig. 12).
Fig. 12. Performance for handling “one-to-many” problem. The first line is the target SR. The next few lines result from inverse prediction. The lines are the target SR, and the dots are the SR of the predicted meta-atom.
Although our method inverse-engineered well on test datasets, practical applications often require on-demand inverse engineering of meta-atoms with artificially defined SR for specific tasks. Therefore, we arbitrarily generate some target amplitude curves by the following formula:
Fig. 13. Inverse design renderings according to a specific spectrum. The first line is the target SR, and the next few lines result from the inverse prediction. The blue curves are the target SR, and the red curves are the SR of the predicted meta-atom.
5. Conclusion
In this paper, we propose a method for realizing the inverse design of meta-atoms with high DOFs. The method combines PNN and the GA based on VAE coding, and uses the GA to search for the optimal meta-atom in the latent space (i.e., the constrained solution space) encoded by the VAE. A PNN was constructed and trained to reveal the relationship between the meta-atoms and its SR, and it can quickly and accurately predict the SR of the meta-atoms. Compared with the traditional meta-atoms inverse design method, our inverse design method avoids extensive and time-consuming full-wave numerical simulation and shortens the time spent in inverse design. Compared with the method of inverse design using neural network only in recent years, our method breaks through the bottleneck of low DOFs of design objects. The method proposed in this paper can be applied not only to the inverse design of meta-atoms but also to other electromagnetic devices, such as antennas, integrated optical circuit devices, etc. In addition, the new coding ideas are crucial to better use optimization algorithms in any other field and our idea of “adding constraints” can also be applied to the field of deep learning. By considering some characteristics of practical problems, artificially adding some constraints to the neural network improves its results as compared with the general data-driven neural network.
Funding
National Natural Science Foundation of China (61963016, 61967007); Key Research and Development Program of Jiangxi Province (20201BBF61012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Ref. [49].
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