Learning finite element convergence with the Multi-fidelity Graph Neural Network

Abstract

Machine learning techniques have emerged as potential alternatives to traditional physics-based modeling and partial differential equation solvers. Among these machine learning techniques, Graph Neural Networks (GNNs) simulate physics via graph models; GNNs embed relevant physical features into graph data structures, perform message passing within the graphs, and produce new attributes based on the system’s relationships. Like many machine learning frameworks, GNNs are limited by excessive data generation costs and limited generalizability outside of a narrow training domain. To address these limitations, we introduce the Multi-Fidelity Graph Neural Network (MFGNN), a supervised machine learning framework that uses low-fidelity projections to inform high-fidelity modeling across arbitrary subdomains represented by subgraphs. We implement the MFGNN for two-dimensional elastostatic problems with finite element training data. The MFGNN is trained to produce accurate analysis given low-fidelity evaluations and emulate the convergence behavior of traditional finite element analysis (FEA). Through subdomain abstraction, we also extend the MFGNN as a general model for new boundary conditions and material domains outside of the training domain.