Interpretable physics-encoded finite element network to handle concentration features and multi-material heterogeneity in hyperelasticity

Abstract

Physics-informed neural networks (PINNs) have recently prevailed as differentiable solvers that unify forward and inverse analysis in the same formulation. However, PINNs have quite limited caliber when dealing with concentration features and discontinuous multi-material heterogeneity, hindering its application when labeled data is missing. We propose a novel physics-encoded finite element network (PEFEN) that can deal with concentration features and multi-material heterogeneity without special treatments, extra burden, or labeled data. Leveraging the interpretable discretized finite element approximation as a differentiable network in the new approach, PEFEN encodes the physics structure of multi-material heterogeneity, functional losses, and boundary conditions. We simulate three typical numerical experiments, and PEFEN is validated with a good performance of handling complex cases where conventional PINNs fail. Moreover, PEFEN entails much fewer iterations (<10%) than some published improved PINNs (namely the mixed form and domain decomposition method), and the proposed PEFEN does not employ extra variables for stresses or special treatments for subdomains. We further examine PEFEN in hyperelastic multi-layer strata with and without a pile, validating its ability for more practical applications. PEFEN is also tested for inverse analysis. In 3D experiments, transfer learning with PEFEN is validated. PEFEN need much less memory than FEM (<20%), and its training from zero initialization is faster than FEM forward analysis (>1 million dofs). It is also discussed that PEFEN may act like domain decomposition in a refined way, and a simple experiment validates that PEFEN can solve the problem with multi-scale frequency. The PEFEN, thus, proves to be a promising method and deserves further development.