Metamaterial Parameter Estimation by Machine Learning Method

Metamaterial Parameter Estimation by Machine Learning Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Metamaterial Parameter Estimation by Machine Learning Method Shipra Tiwari, Pramod Sharma, Shoyab Ali This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4650387/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Artificial neural network modeling is used to synthesize the metamaterial unit cell. Artificial neural networks are powerful tools to establish the relation between inputs and outputs parameters under highly nonlinear conditions. Artificial neural networks captured the synaptic weights according to their training data set. In artificial neural networks, the back propagation technique is the fastest learning method, which reduces the computer’s processing time and provides the best results under the nonlinear relationship between input and output. This work is divided into three parts. In the first part, we design a metamaterial unit cell, which is in the shape of square split rings. This shape is widely used to realize a metamaterial unit cell. In the second part, we develop a regression model using artificial neural networks to estimate the output resonance frequency when design parameters are used as input of artificial neural networks. In the last part, we use three different machine learning method to estimate the output parameter and then do the comparison in between them. Therefore, the objective of this research work is to develop a hypothesis using feed forward backpropagation method, Bayesian regularization and Elman backpropagation method, to find the resonance frequency when dimension of the metamaterial unit cell is given. Metamaterial machine learning artificial neural network backpropagation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction The work in this paper is to establish the relationship between the unique field of machine learning and electromagnetics. This relation between machine learning and electromagnetic and microwave or antenna applications reduces the computational cost and suppresses the need of designing software and packages, which are commercially available for Computer-aided design [1, 2]. Artificial Neural networks are the subset of machine learning technique. This technique is effective when information is extracted from nonlinear environment. ANN technique is inspired from human brain. As the human brain is, learn or train by supervisor and adapt the information from environment. Similarly, ANN train by labelled input and update its weight. After training, ANN can deal with any unknown input of similar kind of data. ANN are machine-learning tools that are very effective when we have data, which is experimentally generated. ANN are used where we have no mathematical formula to calculate the output for a certain input and there is no linear relationship between the input and output parameters, so the ANN's are the best tool for the prediction and for the Optimization [3–5] of output parameters. In this kind of scenario, ANN reduces the complex calculations [6–8] and reduces the computational cost. Metamaterials are engineering materials, which are not found in nature. In general, when MM are placed with any microwave device or antennas, they disturbed the EM field, which causes alteration in the property of the material. These alteration form several advantages like gain enhancement, wide banding multi banding etc. There are several applications of metamaterials like the one they are used in Microwave devices, antennas to improve performance, Cloaking, to design super lens, perfect absorber etc. Metamaterials are the combination of split rings, which generates the rotating current and thin wires, which generates the perpendicular magnetic field. The thin wires are arranged periodically in the form of an array. Metamaterials have the property of negative permittivity and permeability, which is not found in any natural material. Metamaterials are engineering materials, which are not found in nature. Metamaterials are the combination of split rings, which generates the rotating current and thin wires, which generates the perpendicular magnetic field. The thin wires are arranged periodically in the form of an array. Metamaterials have the property of negative permittivity and permeability [9], which is not found in any natural material. Rotating current generates the negative permeability and thin wire generates the negative permittivity. The general shape of the split ring is circular or square-shaped but some researchers also worked on different geometries [10–13]. Researchers are always interested in calculating the dimension of split rings and thin wire to design metamaterial unit cells for a certain resonant frequency but there is no straightforward formula to calculate the dimensions of a unit cell. There are a lot of research papers in which researchers used machine learning techniques to explore the MM but no one works on to calculate the resonant frequency for a given dimension of MM. Figure 1 is showing the input-output mapping using artificial neural networks. The above diagram maps the nonlinear input into a linear output. The network is trained with the labelled data or training data set. The performance measure is the mean square error. To minimize the mean square error the optimization algorithm is used which is the steepest gradient descent algorithm. This algorithm updates the weight function in each iteration to minimize the MSE. Finally, the weight functions are frozen for optimum output. After this process, the network is ready to predict the output for an unknown input or test data. 2. Metamaterial unit cell Metamaterials have special properties, which are not found in natural materials. They have negative permittivity and permeability for a certain frequency band. Because of negative permittivity and permeability, the reflection Coefficient is also negative. The negative reflection Coefficient indicates the backward waves. MM is formed by the combination of split rings and thin wires. Metallic portions of the split rings have inductive properties and the gap between the split rings have capacitive properties. Resonance frequency is given by $${f}_{0}= \frac{1}{2\pi \sqrt{{L}_{0}{C}_{0}}}$$ The frequency range negative index is given by $${\varepsilon }^{R}{\mu }^{I}+ {\varepsilon }^{I}{\mu }^{R} >0$$ $${\varepsilon }^{R}= \frac{h}{\frac{2\pi \upsilon {\varepsilon }_{0}pV}{I{e}^{j(x-y-\frac{\pi }{2})}}}-2{z}_{1}$$ This L and C combination generates negative permeability and permittivity. For the demonstration of MM and to collect the data we use the design from the reference [13]. The structure of SRR is shown in Fig. 2 . The design parameters of the SRR are L = length of the SRR = 2.5 mm W = width of the SRR = 2.5 mm D = width of the SRR = 0.2 mm G = gap between the SRR = 0.3 mm S = spacing between SRR = 0.15 mm P = width of thin wire = 0.14 mm. Simulation of the metamaterial unit cell has been done on the HFSS software. We use the wave port analysis to extract the parameters of the metamaterial unit cell. To generate the data we use the optometric tool of the HFSS. In optometric we make the outer length and in a length of both split rings variable and vary the gap size and width of the thin wire. By wearing these six input variables, we found corresponding resonant frequencies. By this process, we collect sixty samples. The first five sample values are shown in the table one. These samples are used to develop artificial neural network models in the Matlab software. Table 1 Five sample values from dataset. Inner length of outer SRR (mm) Inner length of inner SRR (mm) Gap (mm) Inner width of outer SRR (mm) Inner width of inner SRR (mm) Width of thin wire (mm) Resonant frequency (GHz) 1.9 1.2 0.1 1.9 1.2 0.1 11.9 2.2 1.2 0.1 1.9 1.2 0.1 12.9 2.2 1.2 0.1 1.9 1.5 0.2 14 2.2 1.5 0.3 1.9 1.2 0.14 13.1 2.2 1.5 0.1 1.9 1.5 0.1 10.1 3. Implementation of ANN Artificial neural networks are powerful tools to establish the relation between in-puts and outputs parameters under highly nonlinear conditions. Artificial neural networks captured the synaptic weights according to their training data set. Artificial neural network modeling is used to synthesize the metamaterial unit cell. In artificial neural networks, the back propagation technique is the fastest learning method, which reduces the computer’s processing time and provides the best results under the nonlinear relationship between input and output. Here's we have six inputs i.e. Inner length of outer SRR (l1), Inner length of inner SRR (l2), Inner width of outer SRR (w1), Inner width of inner SRR(w2), Gap of SRR (g), Width of thin wire (p) and one output that has been predicted is Resonant frequency (f). Neural network architecture is design with the following steps. Each first six values are fed as input to each neuron of the first layer. The input layer neurons are fed to the hidden layer neurons, in our case there are 16 neurons in the hidden layer. The hidden layer neurons map the non-linearity of the input into the linear output. Most of the processing has done in the hidden layer neurons. All the input neurons connect with each hidden layer neurons and make a dense network. All the connections are associated with the weight function. These weight functions are updated with every epoch. In the output layer, the neuron with the highest value fires and determines the output the values are probable. Error is calculated at each layer of the neural network; both forward and backpropagation take place during the training process of a neural network Most of the data processing is carried out in the hidden layers. All the hidden neurons also contain a bias function. Bias function is similar to the IQ level of human brain. Output of the hidden neurons are fad at the output stage. At this output stage, we make the estimation of the actual output. The difference between actual output and estimated output is the error. Mean square error is a performance criterion of the ANN. To minimize the error we use steepest gradient Descent method that is an Optimization method. To reduce the error, network adjust the weight functions and this process is repeated until we got the minimum error. In every epoch the weights are updated and output error reduced, this process is called learning method. In our network, the learning method is LM backpropagation. The back propagation is a fast algorithm, which reduce the computation time and computational cost. Figure 3 is showing the ANN architecture to predict the resonant frequency. Detail parameters of ANN architecture are as follows: Number of input: 6 Number of output: 1 Number of hidden layer: 1 Number of neurons in the hidden layer: 16 Training algorithms: LM backpropagation Learning function: Gradient descent Performance measure: Mean square error (MSE) 4. Results discussion Metamaterials have a great scope in the future electromagnetics and photonics application. These materials are engineering materials and here we are trying to find out the resonant frequency for the given input dimensions. To perform this experiment first we simulate a metamaterial unit cell in the HFSS software and collect the data by optometric method. We make the length and width of both split rings variable and vary the gap and width of the thin wire. Through this process, we collect 60 samples. To predict the resonant frequency we use machine learning. Machine learning is a very powerful tool when a nonlinear relationship exists between multiple Inputs and multiple outputs. Here we use ANN method and use ANN tool in MATLAB for curve fitting application. 4.1 Levenberg Marquardt backpropagation method Figure 4 is showing the generalize backpropagation model. This model contain input and output layer along with a hidden layer. We have 60 samples and out of 60, 55 samples have been used for the training and the rest 5 samples are used for the testing purpose. The performance criteria were mean square error. In the neural network architecture, we have six inputs and one output. We have used one hidden layer with 60 neurons the training algorithm is Levenberg Marquardt backpropagation and the optimization method is gradient descent performance of the network is shown in Fig. 5 where we can see the mean square error is 0.7 which is the best validation performance of the network. Figure. 6 is showing the neural network training regression curve where we can see that the goodness of fit value (R) for training is 0.95 and the goodness of fit value for validation and testing is 0.79 and 0.84 respectively. The overall performance of the network is R = 0.87. It is quite a good network performance. Training performance is better than testing performance, which shows that our network is not over-fitted. 4.2 Backpropagation method with Bayesian regularization algorithm In the next process to predict the output, we choose Bayesian regularization (BR) training algorithm for backpropagation method. Schematic diagram of the model will remain same as Fig. 4 the only difference is the training algorithm. BR algorithm works better when data is noisy and small. The only drawback of this algorithm is that, it is time consuming. Figure 7 is showing the Performance curve of ANN BR Backpropagation method By Fig. 7 , we can see that training mean square error is 1.52 at 5 epoch and the training performance is better than the testing performance so the model is not over fitted. Regression diagram is shown in Fig. 8 . The goodness of fit value is 0.9 for training, it is 0.77 for validation, and the testing goodness of fit is 0.95, which represents the excellent performance of the model. 4.3 Elman Backpropagation method Figure 9 is showing the Elman back propagation model. In this model, there is a feedback system in the hidden layer and one more layer is exist in between hidden layer and output layer. Figure 10 is showing the mean square error vs epoch curve. By Fig. 10 , we can say that best performance of the model or the mean square error is 3.14 at 6 epoch. Finally, we validate our results by passing unknown inputs to the model and getting the predicted output. This performance is shown in Table 2 where we passed five samples from the back propagation model and their actual output and predicted output were shown in Table 2 along with their corresponding error, so we can conclude from the table that the actual and predicted output are very similar and the model performance is far better. Table 2 Predicted output of the test data S.No. Actual output Predicted output Error 1 14.7 14.56423998 0.135760018 2 10.6 10.47184845 0.12815155 3 14.4 14.23083319 0.169166808 4 11.2 11.65548216 -0.455482163 5 14.8 14.39433851 0.405661488 Table 3 is comparing the performance of all three models. Performance criteria is mean square error. By Table 3 , we can conclude that most accurate method is Bayesian regularization method having lowest MSE i.e. 1.527 and the fastest method is LM backpropagation method, which train the network in only two epoch. Performance wise a very less difference in LM back prop and BR back prop method. On the other hand the Elman model having large MSE as well epoch. Table 3 Performance of different machine learning techniques S.No. Name of model MSE No. of epoch 1 Levenberg Marquardt backpropagation 1.74093 2 2 Bayesian regularization Backpropagation 1.527 5 3 Elman Backpropagation 3.1467 6 5. Conclusion In this research work, artificial neural network modeling is used to synthesize the metamaterial unit cell. In the first part, we design a metamaterial unit cell, which is in the shape of square split rings. This shape is widely used to realize a metamaterial unit cell. In the second part, we develop a regression model using artificial neural networks to estimate the output resonance frequency when design parameters are used as input of artificial neural networks. In the last part, we use three different machine learning method to estimate the output parameter and then do the comparison in between them. Therefore, the objective of this research work is to develop a hypothesis using feed forward backpropagation method, Bayesian regularization and Elman backpropagation method, to find the resonance frequency when dimension of the metamaterial unit cell is given. We validate our results by passing unknown inputs to the model, getting the predicted output. We passed five samples from the back propagation model, and calculate the mean square error for each sample. Therefore, we can conclude that the actual and predicted output are very similar and the model performance is far better. Another conclusion is that most accurate method is Bayesian regularization method having lowest MSE i.e. 1.527 and the fastest method is LM backpropagation method, which train the network in only two epoch. Declarations Author Contribution S.T. Wrote main manuscript.P.S. Reviewed the manuscript. 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Commun. 69(10), 1453–1463 (2015) Belfore, L.A., Arkadan, A.A., Lenhardt, B.M.: ANN inverse mapping technique applied to electromagnetic design. IEEE Trans. Mag. 37(5), 3584–3587 (2001) Mishra, R.K., Patnaik, A.: Designing rectangular patch antenna using the neurospectral method. IEEE Trans. Antennas Propag. 51(8), 1914–1921 (2003) Rawat, A., Yadav, R.N., Shrivastava, S.C.: Neural network applications in smart antenna arrays: a review. AEU Int. J. Electron. Commun. 66(11), 903–912 (2012) Zhou, H, et al.: An improved method of determining permittivity and permeability by S parameters. In: PIERS Proceedings, Beijing, China pp. 768–773 (2009) Pendry, J.B., et al.: Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999) Engheta, N., Ziolkowski, R.W. (eds.): Metamaterials: Physics and Engineering Explorations. Wiley, New York (2006) Mishra, D., Poddar, D.R., Mishra, R.K.: Deformed omega array as LHM. In: Recent Advances in Microwave Theory and Applications, 2008. International Conference on MICROWAVE 2008. Smith, D. R. and Vier, D. C. and Koschny, Th. and Soukoulis, C. M. Electromagnetic parameter retrieval from inhomogeneous metamaterials. Phys. Rev. E, volume 71, issue 3, page 11, March 2005. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4650387","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":323298633,"identity":"9dd9032b-bedb-4383-9d22-c71f95e6ff74","order_by":0,"name":"Shipra 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Introduction","content":"\u003cp\u003eThe work in this paper is to establish the relationship between the unique field of machine learning and electromagnetics. This relation between machine learning and electromagnetic and microwave or antenna applications reduces the computational cost and suppresses the need of designing software and packages, which are commercially available for Computer-aided design [1, 2]. Artificial Neural networks are the subset of machine learning technique. This technique is effective when information is extracted from nonlinear environment. ANN technique is inspired from human brain. As the human brain is, learn or train by supervisor and adapt the information from environment. Similarly, ANN train by labelled input and update its weight. After training, ANN can deal with any unknown input of similar kind of data. ANN are machine-learning tools that are very effective when we have data, which is experimentally generated. ANN are used where we have no mathematical formula to calculate the output for a certain input and there is no linear relationship between the input and output parameters, so the ANN's are the best tool for the prediction and for the Optimization [3\u0026ndash;5] of output parameters. In this kind of scenario, ANN reduces the complex calculations [6\u0026ndash;8] and reduces the computational cost.\u003c/p\u003e \u003cp\u003eMetamaterials are engineering materials, which are not found in nature. In general, when MM are placed with any microwave device or antennas, they disturbed the EM field, which causes alteration in the property of the material. These alteration form several advantages like gain enhancement, wide banding multi banding etc. There are several applications of metamaterials like the one they are used in Microwave devices, antennas to improve performance, Cloaking, to design super lens, perfect absorber etc.\u003c/p\u003e \u003cp\u003eMetamaterials are the combination of split rings, which generates the rotating current and thin wires, which generates the perpendicular magnetic field. The thin wires are arranged periodically in the form of an array. Metamaterials have the property of negative permittivity and permeability, which is not found in any natural material. Metamaterials are engineering materials, which are not found in nature. Metamaterials are the combination of split rings, which generates the rotating current and thin wires, which generates the perpendicular magnetic field. The thin wires are arranged periodically in the form of an array. Metamaterials have the property of negative permittivity and permeability [9], which is not found in any natural material. Rotating current generates the negative permeability and thin wire generates the negative permittivity. The general shape of the split ring is circular or square-shaped but some researchers also worked on different geometries [10\u0026ndash;13]. Researchers are always interested in calculating the dimension of split rings and thin wire to design metamaterial unit cells for a certain resonant frequency but there is no straightforward formula to calculate the dimensions of a unit cell. There are a lot of research papers in which researchers used machine learning techniques to explore the MM but no one works on to calculate the resonant frequency for a given dimension of MM.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is showing the input-output mapping using artificial neural networks. The above diagram maps the nonlinear input into a linear output. The network is trained with the labelled data or training data set. The performance measure is the mean square error. To minimize the mean square error the optimization algorithm is used which is the steepest gradient descent algorithm. This algorithm updates the weight function in each iteration to minimize the MSE. Finally, the weight functions are frozen for optimum output. After this process, the network is ready to predict the output for an unknown input or test data.\u003c/p\u003e"},{"header":"2. Metamaterial unit cell","content":"\u003cp\u003eMetamaterials have special properties, which are not found in natural materials. They have negative permittivity and permeability for a certain frequency band. Because of negative permittivity and permeability, the reflection Coefficient is also negative. The negative reflection Coefficient indicates the backward waves. MM is formed by the combination of split rings and thin wires. Metallic portions of the split rings have inductive properties and the gap between the split rings have capacitive properties. Resonance frequency is given by\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$${f}_{0}= \\frac{1}{2\\pi \\sqrt{{L}_{0}{C}_{0}}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe frequency range negative index is given by\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$${\\varepsilon }^{R}{\\mu }^{I}+ {\\varepsilon }^{I}{\\mu }^{R} \u0026gt;0$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$${\\varepsilon }^{R}= \\frac{h}{\\frac{2\\pi \\upsilon {\\varepsilon }_{0}pV}{I{e}^{j(x-y-\\frac{\\pi }{2})}}}-2{z}_{1}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis L and C combination generates negative permeability and permittivity. For the demonstration of MM and to collect the data we use the design from the reference [13]. The structure of SRR is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The design parameters of the SRR are\u003c/p\u003e \u003cp\u003eL\u0026thinsp;=\u0026thinsp;length of the SRR\u0026thinsp;=\u0026thinsp;2.5 mm\u003c/p\u003e \u003cp\u003eW\u0026thinsp;=\u0026thinsp;width of the SRR\u0026thinsp;=\u0026thinsp;2.5 mm\u003c/p\u003e \u003cp\u003eD\u0026thinsp;=\u0026thinsp;width of the SRR\u0026thinsp;=\u0026thinsp;0.2 mm\u003c/p\u003e \u003cp\u003eG\u0026thinsp;=\u0026thinsp;gap between the SRR\u0026thinsp;=\u0026thinsp;0.3 mm\u003c/p\u003e \u003cp\u003eS\u0026thinsp;=\u0026thinsp;spacing between SRR\u0026thinsp;=\u0026thinsp;0.15 mm\u003c/p\u003e \u003cp\u003eP\u0026thinsp;=\u0026thinsp;width of thin wire\u0026thinsp;=\u0026thinsp;0.14 mm.\u003c/p\u003e \u003cp\u003eSimulation of the metamaterial unit cell has been done on the HFSS software. We use the wave port analysis to extract the parameters of the metamaterial unit cell. To generate the data we use the optometric tool of the HFSS. In optometric we make the outer length and in a length of both split rings variable and vary the gap size and width of the thin wire. By wearing these six input variables, we found corresponding resonant frequencies. By this process, we collect sixty samples. The first five sample values are shown in the table one. These samples are used to develop artificial neural network models in the Matlab software.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFive sample values from dataset.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInner length of outer SRR\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInner length of inner SRR\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGap\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eInner width of outer SRR\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eInner width of inner SRR\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eWidth of thin wire\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eResonant frequency\u003c/p\u003e \u003cp\u003e(GHz)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e11.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e12.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e13.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e10.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"3. Implementation of ANN","content":"\u003cp\u003eArtificial neural networks are powerful tools to establish the relation between in-puts and outputs parameters under highly nonlinear conditions. Artificial neural networks captured the synaptic weights according to their training data set. Artificial neural network modeling is used to synthesize the metamaterial unit cell. In artificial neural networks, the back propagation technique is the fastest learning method, which reduces the computer\u0026rsquo;s processing time and provides the best results under the nonlinear relationship between input and output.\u003c/p\u003e \u003cp\u003eHere's we have six inputs i.e. Inner length of outer SRR (l1), Inner length of inner SRR (l2), Inner width of outer SRR (w1), Inner width of inner SRR(w2), Gap of SRR (g), Width of thin wire (p) and one output that has been predicted is Resonant frequency (f).\u003c/p\u003e \u003cp\u003eNeural network architecture is design with the following steps. Each first six values are fed as input to each neuron of the first layer. The input layer neurons are fed to the hidden layer neurons, in our case there are 16 neurons in the hidden layer. The hidden layer neurons map the non-linearity of the input into the linear output. Most of the processing has done in the hidden layer neurons. All the input neurons connect with each hidden layer neurons and make a dense network. All the connections are associated with the weight function. These weight functions are updated with every epoch.\u003c/p\u003e \u003cp\u003eIn the output layer, the neuron with the highest value fires and determines the output the values are probable. Error is calculated at each layer of the neural network; both forward and backpropagation take place during the training process of a neural network Most of the data processing is carried out in the hidden layers. All the hidden neurons also contain a bias function. Bias function is similar to the IQ level of human brain. Output of the hidden neurons are fad at the output stage. At this output stage, we make the estimation of the actual output. The difference between actual output and estimated output is the error. Mean square error is a performance criterion of the ANN. To minimize the error we use steepest gradient Descent method that is an Optimization method. To reduce the error, network adjust the weight functions and this process is repeated until we got the minimum error. In every epoch the weights are updated and output error reduced, this process is called learning method. In our network, the learning method is LM backpropagation. The back propagation is a fast algorithm, which reduce the computation time and computational cost.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e is showing the ANN architecture to predict the resonant frequency. Detail parameters of ANN architecture are as follows:\u003c/p\u003e \u003cp\u003eNumber of input: 6\u003c/p\u003e \u003cp\u003eNumber of output: 1\u003c/p\u003e \u003cp\u003eNumber of hidden layer: 1\u003c/p\u003e \u003cp\u003eNumber of neurons in the hidden layer: 16\u003c/p\u003e \u003cp\u003eTraining algorithms: LM backpropagation\u003c/p\u003e \u003cp\u003eLearning function: Gradient descent\u003c/p\u003e \u003cp\u003ePerformance measure: Mean square error (MSE)\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"4. Results discussion","content":"\u003cp\u003eMetamaterials have a great scope in the future electromagnetics and photonics application. These materials are engineering materials and here we are trying to find out the resonant frequency for the given input dimensions. To perform this experiment first we simulate a metamaterial unit cell in the HFSS software and collect the data by optometric method. We make the length and width of both split rings variable and vary the gap and width of the thin wire. Through this process, we collect 60 samples.\u003c/p\u003e \u003cp\u003eTo predict the resonant frequency we use machine learning. Machine learning is a very powerful tool when a nonlinear relationship exists between multiple Inputs and multiple outputs. Here we use ANN method and use ANN tool in MATLAB for curve fitting application.\u003c/p\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Levenberg Marquardt backpropagation method\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e is showing the generalize backpropagation model. This model contain input and output layer along with a hidden layer. We have 60 samples and out of 60, 55 samples have been used for the training and the rest 5 samples are used for the testing purpose. The performance criteria were mean square error. In the neural network architecture, we have six inputs and one output. We have used one hidden layer with 60 neurons the training algorithm is Levenberg Marquardt backpropagation and the optimization method is gradient descent performance of the network is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e where we can see the mean square error is 0.7 which is the best validation performance of the network.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure. 6 is showing the neural network training regression curve where we can see that the goodness of fit value (R) for training is 0.95 and the goodness of fit value for validation and testing is 0.79 and 0.84 respectively. The overall performance of the network is R\u0026thinsp;=\u0026thinsp;0.87. It is quite a good network performance. Training performance is better than testing performance, which shows that our network is not over-fitted.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Backpropagation method with Bayesian regularization algorithm\u003c/h2\u003e \u003cp\u003eIn the next process to predict the output, we choose Bayesian regularization (BR) training algorithm for backpropagation method. Schematic diagram of the model will remain same as Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e the only difference is the training algorithm. BR algorithm works better when data is noisy and small. The only drawback of this algorithm is that, it is time consuming.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e is showing the Performance curve of ANN BR Backpropagation method\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBy Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, we can see that training mean square error is 1.52 at 5 epoch and the training performance is better than the testing performance so the model is not over fitted.\u003c/p\u003e \u003cp\u003eRegression diagram is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The goodness of fit value is 0.9 for training, it is 0.77 for validation, and the testing goodness of fit is 0.95, which represents the excellent performance of the model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Elman Backpropagation method\u003c/h2\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e is showing the Elman back propagation model. In this model, there is a feedback system in the hidden layer and one more layer is exist in between hidden layer and output layer.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e is showing the mean square error vs epoch curve. By Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e, we can say that best performance of the model or the mean square error is 3.14 at 6 epoch.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFinally, we validate our results by passing unknown inputs to the model and getting the predicted output. This performance is shown in Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e where we passed five samples from the back propagation model and their actual output and predicted output were shown in Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e along with their corresponding error, so we can conclude from the table that the actual and predicted output are very similar and the model performance is far better.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePredicted output of the test data\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS.No.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eActual output\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePredicted output\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eError\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14.56423998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.135760018\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e10.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10.47184845\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.12815155\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14.23083319\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.169166808\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e11.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e11.65548216\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.455482163\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14.39433851\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.405661488\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e is comparing the performance of all three models. Performance criteria is mean square error. By Table \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, we can conclude that most accurate method is Bayesian regularization method having lowest MSE i.e. 1.527 and the fastest method is LM backpropagation method, which train the network in only two epoch. Performance wise a very less difference in LM back prop and BR back prop method. On the other hand the Elman model having large MSE as well epoch.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePerformance of different machine learning techniques\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS.No.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eName of model\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo. of epoch\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLevenberg Marquardt backpropagation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.74093\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBayesian regularization Backpropagation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.527\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eElman Backpropagation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.1467\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eIn this research work, artificial neural network modeling is used to synthesize the metamaterial unit cell. In the first part, we design a metamaterial unit cell, which is in the shape of square split rings. This shape is widely used to realize a metamaterial unit cell. In the second part, we develop a regression model using artificial neural networks to estimate the output resonance frequency when design parameters are used as input of artificial neural networks. In the last part, we use three different machine learning method to estimate the output parameter and then do the comparison in between them. Therefore, the objective of this research work is to develop a hypothesis using feed forward backpropagation method, Bayesian regularization and Elman backpropagation method, to find the resonance frequency when dimension of the metamaterial unit cell is given.\u003c/p\u003e \u003cp\u003eWe validate our results by passing unknown inputs to the model, getting the predicted output. We passed five samples from the back propagation model, and calculate the mean square error for each sample. Therefore, we can conclude that the actual and predicted output are very similar and the model performance is far better. Another conclusion is that most accurate method is Bayesian regularization method having lowest MSE i.e. 1.527 and the fastest method is LM backpropagation method, which train the network in only two epoch.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eS.T. Wrote main manuscript.P.S. Reviewed the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eYildiz, C., T\u0026uuml;rkmen, M.: A CAD approach based on artificial neural networks for shielded multilayered coplanar waveguides. Int. J. Electron. Commun. 58(4), 284 (2004)\u003c/li\u003e\n\u003cli\u003eYildiz, C., et al.: Neural models for quasi-static analysis of conventional and supported coplanar waveguides. AEU Int. J. Electron. Commun. 61(8), 521\u0026ndash;527 (2007)\u003c/li\u003e\n\u003cli\u003eLee, Y., et al.: Design and optimisation of one-port RF MEMS resonators and related integrated circuits using ANN-based macromodelling approach. IEE Proc. Circuits Devices Syst. 153(5), 480\u0026ndash;488 (2006)\u003c/li\u003e\n\u003cli\u003eSarmah, K., Sarma, K.K., Baruah, S.: ANN based optimization of resonating frequency of split ring resonator. In: 2014 IEEE Symposium on Computational Intelligence for Communication Systems and Networks (CIComms) (2014)\u003c/li\u003e\n\u003cli\u003eSotiroudis, S.P., Siakavara, K.: Mobile radio propagation path loss prediction using artificial neural networks with optimal input information for urban environments. AEU Int. J. Electron. Commun. 69(10), 1453\u0026ndash;1463 (2015)\u003c/li\u003e\n\u003cli\u003eBelfore, L.A., Arkadan, A.A., Lenhardt, B.M.: ANN inverse mapping technique applied to electromagnetic design. IEEE Trans. Mag. 37(5), 3584\u0026ndash;3587 (2001)\u003c/li\u003e\n\u003cli\u003eMishra, R.K., Patnaik, A.: Designing rectangular patch antenna using the neurospectral method. IEEE Trans. Antennas Propag. 51(8), 1914\u0026ndash;1921 (2003)\u003c/li\u003e\n\u003cli\u003eRawat, A., Yadav, R.N., Shrivastava, S.C.: Neural network applications in smart antenna arrays: a review. AEU Int. J. Electron. Commun. 66(11), 903\u0026ndash;912 (2012)\u003c/li\u003e\n\u003cli\u003eZhou, H, et al.: An improved method of determining permittivity and permeability by S parameters. In: PIERS Proceedings, Beijing, China pp. 768\u0026ndash;773 (2009)\u003c/li\u003e\n\u003cli\u003ePendry, J.B., et al.: Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech. 47(11), 2075\u0026ndash;2084 (1999)\u003c/li\u003e\n\u003cli\u003eEngheta, N., Ziolkowski, R.W. (eds.): Metamaterials: Physics and Engineering Explorations. Wiley, New York (2006)\u003c/li\u003e\n\u003cli\u003eMishra, D., Poddar, D.R., Mishra, R.K.: Deformed omega array as LHM. In: Recent Advances in Microwave Theory and Applications, 2008. International Conference on MICROWAVE 2008.\u003c/li\u003e\n\u003cli\u003eSmith, D. R. and Vier, D. C. and Koschny, Th. and Soukoulis, C. M. Electromagnetic parameter retrieval from inhomogeneous metamaterials. Phys. Rev. E, volume 71, issue 3, page 11, March 2005.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Metamaterial, machine learning, artificial neural network, backpropagation","lastPublishedDoi":"10.21203/rs.3.rs-4650387/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4650387/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eArtificial neural network modeling is used to synthesize the metamaterial unit cell. Artificial neural networks are powerful tools to establish the relation between inputs and outputs parameters under highly nonlinear conditions. Artificial neural networks captured the synaptic weights according to their training data set. In artificial neural networks, the back propagation technique is the fastest learning method, which reduces the computer\u0026rsquo;s processing time and provides the best results under the nonlinear relationship between input and output. This work is divided into three parts. In the first part, we design a metamaterial unit cell, which is in the shape of square split rings. This shape is widely used to realize a metamaterial unit cell. In the second part, we develop a regression model using artificial neural networks to estimate the output resonance frequency when design parameters are used as input of artificial neural networks. In the last part, we use three different machine learning method to estimate the output parameter and then do the comparison in between them. Therefore, the objective of this research work is to develop a hypothesis using feed forward backpropagation method, Bayesian regularization and Elman backpropagation method, to find the resonance frequency when dimension of the metamaterial unit cell is given.\u003c/p\u003e","manuscriptTitle":"Metamaterial Parameter Estimation by Machine Learning Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-07-19 11:09:14","doi":"10.21203/rs.3.rs-4650387/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9acf5050-1c08-4619-92a8-e38a7b132b94","owner":[],"postedDate":"July 19th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-08-23T11:00:17+00:00","versionOfRecord":[],"versionCreatedAt":"2024-07-19 11:09:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4650387","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4650387","identity":"rs-4650387","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}