[Paper Review] GNN-based physics solver for time-independent PDEs

Rini Jasmine Gladstone, Helia Rahmani+4 more|arXiv (Cornell University)|Mar 28, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes Edge Augmented GNN (EA-GNN) and Multi-GNN (M-GNN) architectures to improve graph neural networks for solving time-independent partial differential equations (PDEs) in solid mechanics. By enhancing long-range message passing and using a simulation coordinate system, the models achieve 1.5–2× lower error than baselines and generalize robustly to unseen domains, boundary conditions, and rotations/translations.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of long-range information exchange in graph neural networks (GNNs) for time-independent PDEs, particularly in static solid mechanics.
• Overcome the limitations of standard GNNs in capturing long-range interactions due to reliance on local, short-range message passing.
• Develop GNN architectures that generalize across unseen domains, boundary conditions, and material properties.
• Ensure invariance to rotation and translation in physical systems through a novel coordinate transformation strategy.
• Enable resolution- and geometry-invariant physics simulation using graph-based deep learning surrogates.

Proposed method

• Introduce Edge Augmented GNN (EA-GNN), which adds skip connections or enhanced edge features to improve long-range message passing in GNNs.
• Propose Multi-GNN (M-GNN), a multi-stream architecture that aggregates information from multiple GNN branches to enhance long-range connectivity.
• Apply a coordinate transformation to map physical domain coordinates into a simulation reference frame, enabling rotation and translation invariance.
• Train models in the simulation coordinate system to ensure generalization under domain transformations, including scaling, rotation, and translation.
• Use a graph-based representation where nodes correspond to mesh points and edges represent spatial connectivity, with physical parameters (e.g., material properties, loads) encoded as node and edge features.
• Leverage physics-informed loss functions that incorporate the weak form of the PDE and boundary conditions to guide training.

Experimental results

Research questions

• RQ1 Can GNN architectures be enhanced to effectively model long-range interactions required in time-independent PDEs such as static elasticity?
• RQ2 How do EA-GNN and M-GNN compare to standard GNNs in terms of accuracy and generalization for time-independent solid mechanics problems?
• RQ3 To what extent can coordinate transformation in the simulation reference frame ensure rotation and translation invariance in GNN-based physics solvers?
• RQ4 Can the proposed models generalize to unseen domains, boundary conditions, and material properties without retraining?
• RQ5 What is the impact of domain scaling and geometric transformations on the generalization performance of GNN-based PDE solvers?

Key findings

• EA-GNN and M-GNN reduce relative error in nodal deformation prediction by a factor of 1.5 to 2 compared to baseline GNNs across all test cases.
• For domain scaling (scale = 0.5), EA-GNN achieves a relative error of 0.09 ± 0.07 in u_x and 0.09 ± 0.07 in u_y, significantly outperforming the baseline B (0.87 ± 0.56 in u_x).
• For double scaling (scale = 2.0), EA-GNN maintains low error (0.22 ± 0.12 in u_x), indicating robustness despite increased domain size and unseen coordinate positions.
• The models generalize well to out-of-distribution boundary conditions (Dirichlet/Neumann on disconnected sets), with EA-GNN achieving 0.11 ± 0.08 error in u_x and 0.13 ± 0.09 in u_y.
• Rotation and translation invariance is successfully achieved when models are trained in the simulation coordinate system, with no increase in error under these transformations.
• The baseline model (B) fails to generalize under rotation and translation, showing 1.5–2× higher error than EA-GNN and M-GNN, confirming the necessity of coordinate transformation.