Mesh-based GNN surrogates for time-independent PDEs.

Area of Science:

• Physics-based deep learning
• Computational solid mechanics
• Graph neural networks (GNNs)

Background:

• Deep learning models excel at simulating physical systems but struggle with time-independent problems requiring extensive information exchange.
• Deeper GNNs are needed for these problems, but can slow down training.
• Existing methods like MeshGraphNets have limitations in handling complex, time-independent physical dynamics.

Purpose of the Study:

• To introduce novel GNN architectures designed to efficiently handle time-independent physics problems.
• To improve the accuracy and generalization capabilities of physics-based GNNs.
• To enable GNNs to tackle a broader range of scientific and industrial applications.

Main Methods:

• Development of two new GNN architectures: the edge augmented GNN and the multi-GNN.
• Application of these architectures to time-independent solid mechanics problems.
• Implementation of a novel coordinate transformation for rotation and translation invariance to handle variable domains.

Main Results:

• The proposed edge augmented GNN and multi-GNN architectures significantly outperform baseline methods like MeshGraphNets.
• The models demonstrate strong generalization across unseen domains, boundary conditions, and materials.
• The coordinate transformation effectively addresses variable domain challenges.

Conclusions:

• The novel GNN architectures provide an effective solution for simulating time-independent physical systems.
• These advancements broaden the applicability of neural operators based on GNNs.
• The research lays the foundation for using GNNs in complex scientific and industrial simulations.