Analysis of Dielectric Post-Wall Waveguide-based Passive Circuits using Recurrent Neural Network Saba Kobakhidze 1, Elguja Archemashvili 1, Vakhtang Jandieri 2*, Kiyotoshi Yasumoto 3, Hiroshi Maeda 4, Wonbin Hong 5, Douglas H. Werner 6 and Daniel Erni 2 1 2 Department of Electrical and Computer Engineering, Free University of Tbilisi, 0159 Tbilisi, Georgia General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, Germany 3 Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka 819-0395, Japan 4 Department of Information and Communication Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan 5 Pohang University of Science and Technology (POSTECH), Pohang, 37673, Republic of Korea 6 Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA Corresponding Author: Vakhtang Jandieri* (vakhtang.jandieri@uni-due.de) Abstract—The dielectric post-wall waveguide-based passive circuits are analyzed using a surrogate model represented by an attention-based recurrent neural network. The predicted results from this much simpler, trained model show a good agreement with those obtained by a corresponding full-wave computational electromagnetics analysis with a fully independent EM commercial software package. Index Terms—Deep learning, recurrent neural network, post-wall waveguide, circuits. I. INTRODUCTION Recently, we have developed a self-consistent and efficient analytical method [1, 2] applicable to the evaluation of Post-Wall Waveguide (PWW) and PWW-based passive circuits in a very short computation time (Fig. 1). The PWW was composed of periodic arrays of perfect electric conductor (PEC) posts placed on both sides of the guiding channel and the circuits have been realized by introducing additional posts into the guiding region. We have shown that in case of PEC rods as the wall elements, the PWW demonstrates very similar guiding properties to those of the conventional rectangular waveguide and therefore, the simulation process has been considerably simplified, which has led to a dramatic increase in computation efficiency [1]. Our proposed method was based on the lattice-sums technique combined with the transition matrix method and the method of images [1], [3]-[5]. Dielectric materials, which can be easily produced by 3-D printers, have attracted an increasing interest for all-dielectric waveguide structures and integration schemes. Dielectric structures are well suited to reduce losses that make them ideal for applications at very high frequencies, such as in the range from 30 GHz to 300 GHz. Such materials can replace the metallic posts to extend the operating frequency of the device beyond mm-waves towards the THz range. It should be noted that unlike the metal structures (PEC rods as wall elements), in the case of dielectric rods, the confinement is achieved through the electro-magnetic band-gap behavior Fig. 1. Schematic view of the dielectric PWW-based circuit. The wall elements of the dielectric rods are marked by gray and the loaded elements in the guiding region are labelled red. similar to Bragg reflection in a periodic layered structure (see Fig. 1). Our original method [3]-[5] can accurately and efficiently analyze dielectric PWWs, including the phase and attenuation constants, and define the frequency range suitable for relatively low-loss guidance. However, the method cannot be directly applied to the dielectric PWWbased circuits. We need a modification of our method [1], which has been originally developed for the analysis of PEC rods as the wall elements. Commercial software is mainly used to study the Sparameters of the circuits, but it is time-consuming, especially when engineers are concerned with the structural optimization where a large amount of data is required. In this contribution, in order to circumvent cumbersome computational EM analysis, the dielectric post-wall waveguidebased passive circuits are analyzed using a surrogate model represented by an attention-based recurrent neural network. Firstly, we define the structural parameters of the PWW using our original method [3]-[5], namely, we determine the number of layers N of the wall elements, a distance between adjacent layers p, a period h, a width of the waveguide a, a radius r and a relative permittivity ε of the rods for a lowloss propagation in the desired frequency range [6], [7]. This paper's copyright is held by the author(s). It is published in these proceedings and included in any archive such as IEEE Xplore under the license granted by the "Agreement Granting EurAAP Rights Related to Publication of Scholarly Work." Then, we train the attention-based recurrent neural network based on the collected data using CST corresponding to the S-parameter frequency responses for different configurations. Our preliminary numerical experiments were found a good agreement between the predicted results by the trained attention-based neural network and the results obtained based on the CST [8]. The goal of this work is not to design compact and functional dielectric PWW-based circuits, but to build a well-trained neural network with the aim of optimizing in a very short computation time. II. A SCHEME OF THE ATTENTION-BASED RECURRENT NEURAL NETWORK Without loss of generality, the operating frequency range is chosen as 54 GHz < f < 68 GHz. Firstly, our analysis has shown that the 2-layered (N=2) dielectric PWW (see Fig. 1) having a width a = 3.92 mm, a radius of the dielectric cylindrical posts (as wall elements) of r = 0.353 mm, a period of h = 1.96 mm, p = 1.96 mm and a relative dielectric permittivity of the rods of ε = 11.56 [6], [7] can be considered as one of the best optimized design for low-loss guidance in the PWW. The dispersion diagram of the corresponding PWW - the phase constant and the attenuation constant - calculated by our original method are shown in [6], [7] and the readers may refer to these references for the details of the calculation procedure for the complex modes. Next, the Fig. 2. Attention model, which consists of an Attention Block, BiDirectional GRU, and RNN. Fig. 3. GRU cell of a scheme shown in Fig. 2. data collection for the PWW-based circuits has been carried out using the CST commercial software package [8]. We consider up to five circular rods as the loaded elements. Next, the locations of the rods, their radii and relative dielectric permittivities are randomly changed and their corresponding S-parameters are calculated. We use an attention-based recurrent neural network based on the model shown in Fig. 2. It is composed of an Attention Block, GRU (Gated Recurrent Unit) and RNN (Recurrent Neural Network) blocks. It should be mentioned that in the beginning of the numerical experiments, we only considered GRU based BRNN (Bi-directional Recurrent Neural Network). However, in order to improve the accuracy in the whole frequency spectrum, we added an attention block [9]. Similar models are often used in the processing of natural languages such as synchronous translations and voice recognition systems. The Attention Model contains the block shown in Fig. 3 which is called a GRU. The current block has two entries and one exit. The entry denoted by ht-1 is called a hidden layer, and the current input data - structural parameters of the cylinders - is denoted by Xt, so the given block output is also a hidden layer ht, which is passed to the next block. This block also contains activation functions: sigm (Sigmoid function) and tanh (tangent hyperbolic function), which participate in the internal computing processes of the block. The hidden layer (ht) transfers the information to the next block regarding how important the settings in the previous blocks are. In other words, the values of the hidden layer carry information about the dependence of the predictive S-parameters of a given block on the S-parameters of the previous blocks. The GRU block requires the function Zt. It acts as a gate, i.e. it carries or does not carry the input values of the hidden layers. Our model uses the Bi-Directional GRU model, which predicts the values of the S-parameters at a given Fig. 4. The first configuration used to test a trained attention-based neural network. This paper's copyright is held by the author(s). It is published in these proceedings and included in any archive such as IEEE Xplore under the license granted by the "Agreement Granting EurAAP Rights Related to Publication of Scholarly Work." Fig. 5. Frequency response of the S11 of the configuration shown in Fig. 4: CST (solid line); predicted result using attention-based recurrent neural network (dashed line). Fig. 6. The second configuration used to test a trained attention-based neural network. Fig. 8. The third configuration used to test a trained attention-based neural network. Fig. 9. Frequency response of the S11 of the configuration shown in Fig. 8: CST (solid line); predicted result using attention-based recurrent neural network (dashed line). III. Fig. 7. Frequency response of the S11 of the configuration shown in Fig. 6: CST (solid line); predicted result using attention-based recurrent neural network (dashed line). frequency using its values at the previous and subsequent frequencies. The given blocks receive constant values as inputs from the X matrix (see Fig. 3). The size of the input X matrix is batch-size multiplied by 16, where the batch-size is the numberof different input values. In our case these input values are: the five elements of each loaded cylinder’s radius, its x and y coordinates, and the sixteenth parameter is the dielectric permittivity of the cylinders. Our proposed model also contains a simple RNN block (the uppermost block in Fig. 3). It behaves in a similar way as the GRU, but unlike the GRU, it does not contain gates, i.e. it cannot increase or decrease the previous hidden layer (ht-1) values. We could use GRU instead, but after several numerical experiments we have found that the use of a RNN substantially reduces the computation time. NUMERICAL RESULTS We collected about 8500 data sets corresponding to the S-parameter frequency responses for different configurations using CST and trained the attention-based recurrent neural network. Three configurations (see Figs. 4, 6 and 8) of the dielectric PWW-based circuits are presented in the manuscript to demonstrate a comparison between the results obtained based on the trained attention-based recurrent neural network and the CST full-wave simulations. The S11 parameters are illustrated in Figs. 5, 7 and 9, respectively. A good agreement is demonstrated for the first two configurations. Namely, the root-mean-square error is 0.0035 for the first configuration (Fig. 5) and 0.0027 for the second configuration (Fig. 7). During our studies, we have conducted several numerical tests and for most of them the agreement between the predicted results and the results based on CST was good. However, for some particular arrangements of the rods inside the guiding layer (Fig. 8), we have observed some disagreements (Fig. 9). In order to improve the results, more data is needed and this problem is currently under investigation. The results will be presented in due course. ACKNOWLEDGEMENT Parts of this work were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 196 MARIE under Grant 287022738 (project M03). The work was supported by Shota Rustaveli This paper's copyright is held by the author(s). It is published in these proceedings and included in any archive such as IEEE Xplore under the license granted by the "Agreement Granting EurAAP Rights Related to Publication of Scholarly Work." National Science Foundation of Georgia (SRNSFG) [grant number: FR-19-4058]. 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