GNN-based physics solver for time-independent PDEs
This work addresses the problem of long-range information exchange in time-independent PDEs for researchers and engineers in scientific and industrial settings, representing an incremental advancement in neural operator methods.
The paper tackles the challenge of solving time-independent PDEs with GNNs by introducing two new architectures, Edge Augmented GNN and Multi-GNN, which outperform baseline methods by a factor of 1.5 to 2 on solid mechanics problems and generalize to unseen domains, boundary conditions, and materials.
Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge - the Edge Augmented GNN and the Multi-GNN. We show that both these networks perform significantly better (by a factor of 1.5 to 2) than baseline methods when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings.