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

We present a method whereby the finite element (FE) method is used to train physics-informed neural networks that are suitable for surrogate modeling. The method is based on a new class of mesh-based neural network operators called stenciled operators, which leverage the inverse isoparametric map of the FE method. In contrast with graph-based methods, stenciled operators can incorporate non-local information over arbitrary spatial domains with computational costs that are length-scale invariant. Using stenciled convolutional operations as an exemplar, we demonstrate the performance of the method in several training and testing scenarios with linear-elastic 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 FE codes for training.