Deep neural networks for electromagnetics applied to optimization and inverse problems

Licentiate thesis, 2023

This thesis explores the possibilities to complement full-wave electromagnetic solvers with fully-connected neural networks. We emphasize problems that must be solved a very large number of times in a limited parameter domain.

We present and evaluate the normalization method ForwardNorm. ForwardNorm normalizes the outputs of the hidden layers of a neural network and enables training of very deep fully-connected neural networks. To minimize the number of samples needed train the very deep neural networks, we formulate a loss function that includes the misfit in (i) the output of the neural network and (ii) the derivatives of the output of the neural network with respect to its inputs. For certain combinations of input and output, we use continuum sensitivity analysis to compute these derivatives at a low computation cost. We also develop an auto-calibration method that simultaneously determines (i) a set of unknown amplification factors and (ii) the mean permittivity of an unknown medium under test. The method assumes that we have access to a set of measurements that are made a-priori for the purpose of characterization. The method is intended for on-line applications.

We test the methods on four different test-problems. For the first three test-problems, we consider a type of microwave measurement-device intended for an inhomogeneous dielectric medium transported through a metal pipe. In the first test-problem, we train a neural network to determine the point-wise mean and variance of the permittivity of the inhomogeneous dielectric. The trained neural network is very computationally cheap to evaluate, which makes the method appealing for real-time applications. For the second test-problem, we apply the auto-calibration method to simultaneously determine (i) the mean permittivity in the pipe and (ii) a set of unknown amplification factors. For the third test-problem, we use a deep neural network to model the microwave measurement-device with a stochastic dielectric medium and estimate high-dimensional histograms. For the fourth test-problem, we train a deep neural network to model the frequency response of an H-plane waveguide filter as a function of its geometrical parameters. We then use the neural network to optimize the geometry of the filter to achieve pass-band characteristics under geometrical uncertainty.