Data-driven Reduced Order Modeling For Maxwell Equations

Computational electromagnetics deals with the art and science of solving Maxwell equations numerically by using modern electronic computers.Thanks to the rapidly increasing capability of computers,computational electromagnetics has become a very important and practical analysis tool in electromagnetics,radio frequency and microwave engineering.Highly accurate and efficient numerical calculation methods are the basis for the analysis of increasingly complex electromagnetic problems.In real-time analysis,Maxwell equations need to be solved repeatedly for different time and physical parameters,and repeated high-fidelity simulations lead to a great burden on CPU computing time and memory.In this paper,two data-driven reduced-order models,based on a combination of traditional numerical computation methods and machine learning methods,are devised to address the computational challenges faced in current time-domain electromagnetics.The main research contents are as follows.For the parameterized electromagnetic prolems,in the first method,a database collecting high-fidelity snapshots is built by solving the parameterized time-domain Maxwell equations for some material parameter values using a discontinuous Galerkin time-doamin(DGTD)method.Next,a set of time-and parameter-independent reduuced basis(RB)functions are extracted from the snapshot matrix by a two-step proper orthogonal decomposition(POD)method.To perform a prior dimensionality reduction,the projection coefficient vectors are obtained by compressing the snapshots.Then the projection coefficient vectors are further compressed through a convolutional autoencoder(CAE)network.Finally,the main components of the coding matrices composed of coding vectors are extracted by singular value decomposition(SVD),and the main temporal and parametric modes of these matrices are approximated by the cubic spline interpolation(CSI)method.The generation of the RB functions and the training of the CAE and CSI are accomplished in the offline stage,thus the RB solution for new time and parameter values can be recovered rapidly via outputs of the interpolation model and decoder network.The numerical experimental results illustrate the effectiveness and superiority of the CAE-CSI ROM,which is much more efficient than the traditional DGTD method while ensuring high accuracy.When the number of basis functions extracted from the snapshot matrix by the twostep POD is large,a great many interpolation models need to be constructed,which increases the online test time.To address this problem,the second method uses a canonical polyadic(CP)decomposition to decompose the projection coefficient tensor into a series of linear combinations of rank-one vectors,and then constructs the mapping between the time and parameters and the projection coefficients by the cubic spline interpolation.The accuracy and efficiency of the proposed POD-CP reduced-order model is illustrated by experiments on the scattering of plane waves by multi-layer dielectric disk as well as multi-layer dielectric sphere.