Machine Learning Based Method of Moments (ML-MoM) He Ming Yao and Li Jun Jiang* Yu Wei Qin Department of Electrical and Electronic Engineering The University of Hong Kong Hong Kong yaohmhk@hku.hk, jianglj@hku.hk* Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA, USA yuweiq@andrew.cmu.edu The popular cloud computing platform ? Amazon Web Service (AWS) is utilized to support the computation [7-8]. Abstract?This paper proposes a novel method by rethinking the method of moments (MoM) solving process into a machine learning training process. Based on the artificial neural network (ANN), the conventional MoM matrix is treated as the training data set, based on which machine learning training process becomes conventional linear algebra MoM solving process. The trained result is the solution of MoM. The multiple linear regression (MLR) is utilized to train the model. Amazon Web Service (AWS) is used as the computations platform to utilize the existing hardware and software resources for machine learning. To verify the feasibility of the proposed new machine learning based method of moments (ML-MoM), we choose the static parasitic capacitance extraction and dynamic electromagnetics scattering as examples. The proposed novel idea opens a new gateway between conventional computational electromagnetics and machine learning algorithms with various application potentials. II. ML-MOM TRAINING PROCESS The equation (1) is a conventional MoM matrix equation based on the static integral equation [9-10], where the charge density ? is discretized using the pulse basis function and the corresponding excitation vector is represented by ?. S11 [ ? SM1 or (1) ???= ? where M is the number of field point, N is the unknown number for vector ?, and the vector V has N terms. The elastance matrix S has the dimension M by N. While S, ? and V are real numbers in static field problem, they are complex for dynamic problems. The elastance element Smn could be tested using Galerkin or point correlation methods while the Green?s function is used as the integration kernel for Smn. When M=N, Eqn (1) is conventional MoM matrix system. Training 1 Training i V1 ? ? . . . . . . S12 . . . S11 ?1 ? 2 . . . ?N S1N Si1 Si 2 ?N . . . ?1 ? 2 ?N . . . ?2 . . . ?1 VM ? . . . . . . As today?s most popular research field, machine learning (ML) aims at learning from known information and make predictions upon unknown data [5]. Due to large quantities of applications, machine learning technologies have enormous computing resources ranging from software algorithms to hardware platforms, which potentially could benefit other scientific computation fields. Hence, these sources motivate us to build the direct connection between the machine learning and MoM so that we could use ML resources to serve CEM. Training M Vi S iN SM1 . . . The method of moments (MoM) [1] is a popular computational electromagnetics (CEM) method applicable to parasitic capacitance extractions, electromagnetic scattering and radiation, signal and power integrity [2-4], etc. . . . . . . I. INTRODUCTION . . . . . . Keywords? MoM; Machine Learning; Artificial Neural Network; Capacitance Extraction; Electromagnetic Scattering SM 2 S MN S Fig. 1. Artificial nerual network representation of the machine learning based MoM (ML-MoM). ?? is the unknown, ?? , is the excitation of the MoM equation, and ??? is the matrix coefficient of MoM. In this paper, the conventional MoM is reinterpreted into a machine learning process using the artificial neural network (ANN). Different from previous ANN curve fitting, by this method, the conventional MoM solving process is transformed into an ANN training process. Consequently, we can utilize multiple linear regression (MLR) to train the model, which is typical training process in machine learning regime [6]. The trained model is the final solution of MOM. The proposed approach is used to solve both static parasitic capacitance extractions and dynamic electromagnetics scattering problem. 978-1-5386-3284-0/17/$31.00 �17 IEEE S1N ?1 V1 ? ][ ? ] = [ ? ] SMN ?N VM ? ? ? Our novel approach begin by reconstructing MoM into a training model, as shown in Fig. 1. That is a single layer perceptron (neural network structure). While the matrix element Smn is used as the input training data, the charge density unknown ?? is used as the weighting coefficient. The output of the neural network is the potential Vm. Thus, solving MoM becomes a new process: feeding Smn to train the model. After training, the accurate estimation of the weighting coefficient ?? can be 973 AP-S 2017 Highly developed machine learning algorithms and computation platform provide software and hardware supports to realize our novel approach. We use the multiple linear regression (MLR) to train our model to achieve the MOM results. By utilizing the following cost function, MLR intrinsically decreases the least square error: 1 2 ??? = arg ??? ?? ?=1 (?? ? ? ? ?? ) Capacitance (pF) (a) 50 100 150 200 250 Angle ? (degrees) 300 350 400 (b) ACKNOWLEDGMENT REFERENCES [1] R. F. Harrington, Field Computation by Method of Moment, Macmillan, New York, 1968. [2] C. Craey, B. Parvais, and X. Dardenne, ?MoM Simulation of Signal-toNoise Patterns in Infinite and Finite Receiving Antenna Arrays,? IEEE Trans. Antennas Propag, vol. 52, no. 12,pp. 3245-3256, Dec. 2005 [3] P. D. Mehta and S. B. Chakrabarty, ?Electrical capacitance of dielectric coated metallic parallel piped and closed cylinder isolated infree space,? J Electrostat, vol. 71, no. 71, pp. 756-762, Aug. 2013. [4] A. I. Mackenzie, S. M. Rao, M. E. Baginski, ?Electromagnetic Scattering From Arbitrarily Shaped Dielectric Bodies Using Paired Pulse Vector Basis Functions and Method of Moments,? IEEE Trans. Antennas Propag, vol. 57, no. 7, pp. 2076-2083, May. 2009 [5] T. Mitchell, Machine Learning, McGraw Hill, 1997. [6] C. M. Bishop, Pattern Recognition and Machine Learning. Springer, Aug. 2006. [7] J. Hugunin, ?The Python Matrix Object: Extending Python for Numerical Computation,? Proceedings of the Third Python Workshop, Reston, Va., Dec. 1995. [8] http://docs.aws.amazon.com/gettingstarted/latest/awsgsg-intro/gsg-awsintro.html [9] W. C. Chew and L. Jiang, "A Complete Variational Method for Capacitance Extractions," Prog Electromagn Res, vol. 56, pp. 19-32, Jan. 2006. [10] A. G. Papagiannakis, ?Application of a point-matching MoM reduced scheme to scattering from finite cylinders?. IEEE Transactions Microw. Theory Tech, vol. 45, no. 9, pp. 1545-1553, Aug. 1997. 2.54 2.52 2.50 2.48 2.46 MOM ML-MOM, M=1008 ML-MOM, M=2268 2000 0 0 This work was supported in part by the Research Grants Council of Hong Kong (GRF 17207114 and GRF 17210815), NSFC 61271158, and Hong Kong UGC AoE/P?04/08. 2.56 1500 1 a In this paper, by combining MoM and ANN, we reconstruct conventional MoM into novel machine learning based method of moment (ML-MoM), which is solved by a standard machine learning training process-MLR in ML field. The novel MLMoM is discussed by analyzing its application on two numerical examples, which indicates the feasibility of our new method. 2.58 N x 2 IV. CONCLUSION 2.60 1000 ?=? 3 (a) Power planes as shown in Fig. 2. is used as the first example to extract capacitances. Three coupling plates (Vdd, Vcc, and GND) are located at two layers. The ground plate has a dimension of 12� mm2, and the distance between two layers is 0.2mm. The conventional MoM and the newly proposed MLMoM based on MLR training are compared in the capacitance extraction. It clearly states the efficiency and feasibility of MLMoM. 2.42 ? 4 Fig. 3. ML-MoM Current density produced by scattering on metal cylinder. (a) TMz uniform plane wave incident on a circular Metal cylinder, (b) Current distribution of ML-MoM with N=360 and M=720 III. NUMERICAL EXAMPLES 2.44 5 ? z We choose Amazon Web Service (AWS) ? a cloudcomputing service suite as machine learning hardware platform, which offers a wide range of machines and softwares with ranges of configurations. The machine used is m4 instance, the latest generation of General Purpose instance offered by AWS and configured with 16 vCPUs and 64 GB memories. Extra 200 GB volume is attached, to verify validation of a large scale of matrix sizes. 500 y z In the static field, all data in equation (1) are real numbers. based on which our model can be easily trained by MLR. However, when the data in equation (1) is complex for dynamic fields, we can still train our model to achieve MOM solutions. But the real and imaginary parts of the matrix have to be processed specially to make them into logitimate new data groups before the training. 0 6 (2) 2 ? cylinder of radius a=1m with wavelength ?=a/2. The circumference is divided into 360 segments as N for both methods and 720 segments as M for ML-MoM. The results of conventional MoM and ML-MoM based on MLR training are compared in current density. With the same N, ML-MoM can offer better result. Current Density (mA/m) obtained. According to experience in ML, more training data will provide better accuracy in the final solution. Hence, when the new algorithm is used, more field points than source points are needed, which means M>N. Hence, the MoM solving process becomes a machine learning training process. 2500 (b) Fig. 2. Power-ground plane structure for capacitance extraction. (a) Powerground plane structure vertical view and plane view, (b) Capacitance comparison between MoM and ML-MoM. N is the number of charge discretization, and M is the field testing point number. The PEC cylinder as shown in Fig. 3. is used as the example for the dynamic field. A TMz uniform wave incident upon the 974

1/--страниц