Multiscale graph neural networks with adaptive mesh refinement for accelerating mesh-based simulations

TL;DR

This work tackles the computational burden of simulating complex multiphysics on refined meshes by introducing a multiscale graph neural network (GNN) framework that uses adaptive mesh refinement (AMR) to mimic multigrid solvers. The approach employs Graph Transformer message-passing across multiple mesh resolutions, with downscaling and upscaling steps and skip connections to preserve information and mitigate over-smoothing, while an encoder–decoder structure predicts displacements and crack fields. Transfer learning enables rapid adaptation to new crack configurations (center, shear, right-edge) with dramatically reduced training data, achieving high accuracy (often <0.3–1.2% error) at substantially reduced computational times (e.g., SSR ~3.9 h for 30 cases vs ~43.5 h for a high-fidelity PF model). The results demonstrate robust performance across various crack geometries and loading conditions, highlighting the method's potential to accelerate a broad class of AMR-based engineering multiphysics problems.

Abstract

Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems

Figures (12)

• Figure 1: Architecture of the multiscale GNN framework for the Four-Stage Refinement architecture. The model first uses MLP networks to encode the input graph representation. The feature embedding is then passed through MP GNN blocks followed by mesh coarsening operations denoted by $GNN_{D1}$ and $GNN_{D2}$. $GNN_{n}$ involves an additional MP GNN block to operate on the coarsest mesh. The resulting coarsened embedding is then reconstructed through MP GNN blocks followed by mesh upscaling operations denoted by $GNN_{U3}$ and $GNN_{U4}$. The dashed green lines represent skip connectors. The final reconstructed embedding is passed through a decoder MLP network to predict the crack field and displacement fields at future times.
• Figure 2: Representation of the instantaneous refined mesh graphs involving a) refinement levels 0-4, b) refinement levels 0-2, and c) refinement level 0 (i.e., coarsest mesh).
• Figure 3: Problem geometry and input parameters $\{C_{P},C_{\theta},C_{L}\}$ set-up for a) initial case of left-edge cracks subjected to tension, b) center cracks subjected to tension, c) left-edge cracks subjected to shear, and c) right-edge cracks subjected to tension.
• Figure 4: Comparison of the FSR, TSR, and SSR frameworks versus the high-fidelity PF model for predicting a) crack field, $\phi$, b) x-displacement fields, and c) y-displacement fields in left-edge crack cases subjected to tension.
• Figure 5: Comparison of average $\%$ errors on left-edge crack cases for a) FSR GNN crack-field predictions, b) FSR GNN x-displacement predictions, c) FSR GNN y-displacement predictions, d) TSR GNN crack-field predictions, e) TSR GNN x-displacement predictions, f) TSR GNN y-displacement predictions, g) SSR GNN crack-field predictions, h) SSR GNN x-displacement predictions, and i) SSR GNN y-displacement predictions.