The study of the phase retrieval and uncertainty quantification based on integral equation

Abstract

The integral equation is a powerful approach for addressing electromagnetic compatibility and inference (EMC/EMI). Based on the method of moments, the far field can be obtained via the near field far field transformation. Antenna measurement involves phase retrieval and magnitude measurement, but phase retrieval is challenging due to its sensitivity to the external environment. Therefore, a phase retrieval algorithm is employed to obtain the phase information using magnitude-only or intensity-only data. The most popular approach is two-scan techniques. To increase the independent sample data, the multi-polarization phase retrieval algorithm is proposed to change the polarization of the incident field. The surface current density on the objective can be obtained via the sampling data of the scattering field at near field. Subsequently, the far field can be computed by the reconstructed current density. Furthermore, uncertainty quantification is a significant issue in electromagnetic fabrication. Three different approaches are proposed to quantify uncertainty in the electromagnetic system. One approach is interval analysis, which estimates the bounds of the range of the quantities of interest (QoIs). Two methods can be used to obtain the center point and radius of the circular interval analysis: the original derivative to obtain the increment of the QoIs and the extension of the obtained derivative from the first-order derivative to the m-th interval. However, interval analysis can only estimate the bounds of the QoIs, and other stochastic parameters, such as mean value and standard deviation, cannot be obtained. Therefore, the polynomial chaos expansion (PCE) is employed in a physical-informed neural network to obtain the coefficients of the polynomial bases. The mean value and standard deviation can then be obtained via the coefficients. The neural network is employed to address underdetermined and overdetermined matrices. Generally, the overdetermined matrix is solved via linear regression, and the underdetermined matrix is addressed via least angle regression. However, the PCE computational model is necessary, and the number of polynomial chaos is related to the number of random variables and the highest order of the polynomial basis function, resulting in an exponential increase in computational effort with the number of random variables. Additionally, the probability density function should be known in advance to construct the polynomial bases, and if the random variables are correlated, obtaining the joint function is challenging. In light of these factors, the Wasserstein generative adversarial network (WGAN) is based on data directly. It uses high-probability data to decrease the number of sample data compared with the Monte Carlo method. For WGAN, a set of data with an approximate probability density function to the measured sample data is generated, and the mean value and standard deviation can be obtained. For three approach to estimate the stochastic parameters, the interval analysis can effectively estimate the bounds of the output, PINN-based can accurately estimate the stochastic parameters with a small number of sample data. And the WGAN-based method can use the sample data directly. Those methods provide some potential to complex EM system.