FE-PINNs: finite-element-based physics-informed neural networks for surrogate modeling

Abstract: We present a method whereby the finite element method is used to train physics-informed neural networks that are suitable for surrogate modeling. The method is based on a custom convolutional operation called stencil convolution which leverages the inverse isoparametric map of the finite element method. We demonstrate the performance of the method in several training and testing scenarios with linear boundary-value problems of varying geometries. The resulting neural networks show reasonable accuracy when tested on unseen geometries that are similar to those used for training. Furthermore, when the number of training geometries is increased the testing errors systematically decrease, demonstrating that the neural networks learn how to generalize as the training set becomes larger. Further extending the method to allow for variable boundary conditions, properties, and body forces will lead to a general-purpose surrogate modeling framework that can leverage existing finite element codes for training.