Physics-informed neural networks for solving parametric magnetostatic problems

The optimal design of magnetic devices becomes intractable using current computational methods when the number of design parameters is high. The emerging physics-informed deep learning framework has the potential to alleviate this curse of dimensionality. The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. We derive the variational principle for 2-D parametric magnetostatic problems, and prove the existence and uniqueness of the solution that satisfies the equations of the governing physics, i.e., Maxwell's equations. We use a deep neural network (DNN) to represent the magnetic field as a function of space and a total of ten parameters that describe geometric features and operating point conditions. We train the DNN by minimizing the physics-informed loss function using a variant of stochastic gradient descent. Subsequently, we conduct systematic numerical studies using a parametric EI-core electromagnet problem. In these studies, we vary the DNN architecture trying more than one hundred different possibilities. For each study, we evaluate the accuracy of the DNN by comparing its predictions to those of finite element analysis. In an exhaustive non-parametric study, we observe that sufficiently parameterized dense networks result in relative errors of less than 1%. Residual connections always improve relative errors for the same number of training iterations. Also, we observe that Fourier encoding features aligned with the device geometry do improve the rate of convergence, albeit higher-order harmonics are not necessary. Finally, we demonstrate our approach on a ten-dimensional problem with parameterized geometry.