MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations

TL;DR

MAgNET introduces a graph U-Net framework for mesh-based simulations by embedding Multichannel Aggregation layers and novel graph pooling/unpooling into an encoder–decoder architecture. It demonstrates competitive predictive accuracy against CNN U-Nets on structured meshes and extends seamlessly to unstructured meshes, enabling fast surrogates for nonlinear finite element problems in hyperelastic solids. The work provides extensive FEM-based datasets, open-source code, and cross-validation results, highlighting the framework's versatility and potential for real-time surrogate modeling. While accuracy is strong for displacements, the authors discuss physics-related errors and outline pathways to enhance physics-informed training, dynamic problems, and probabilistic formulations to broaden applicability.

Abstract

In many cutting-edge applications, high-fidelity computational models prove to be too slow for practical use and are therefore replaced by much faster surrogate models. Recently, deep learning techniques have increasingly been utilized to accelerate such predictions. To enable learning on large-dimensional and complex data, specific neural network architectures have been developed, including convolutional and graph neural networks. In this work, we present a novel encoder-decoder geometric deep learning framework called MAgNET, which extends the well-known convolutional neural networks to accommodate arbitrary graph-structured data. MAgNET consists of innovative Multichannel Aggregation (MAg) layers and graph pooling/unpooling layers, forming a graph U-Net architecture that is analogous to convolutional U-Nets. We demonstrate the predictive capabilities of MAgNET in surrogate modeling for non-linear finite element simulations in the mechanics of solids.

Figures (22)

• Figure 1: A novel graph U-Net neural network surrogate model for mesh-based simulations. MAgNET accurately captures non-linear FEM responses.
• Figure 2: A schematic of Graph U-Net architecture for mesh based inputs. Colors indicate different types of layers. $c_1, c_2, \ldots, c_5$ stand for channel dimensions. Different arrows indicate different layers: the graph Multi-channel Aggregation (MAg) layer, the graph pooling/unpooling layers, and the concatenation layer.
• Figure 3: Adjacency matrices for the (a) square and (b) triangular meshes. The dashed lines in (a) represent additional edges that are added to the original mesh.
• Figure 4: Local aggregation in MAg (a) works very similar to the filter application in CNN (b). However as opposed to CNN, MAg uses different set of weights at different spatial locations with heterogeneous window size. In CNN, a constant filter slides across the channel.
• Figure 5: One arbitrary choice of non-overlapping subgraphs to create a pooled graph. Subgraphs ${\bf G}_{1}, \ldots, {\bf G}_{5}$ are represented with different colors and are generated by the Algorithm \ref{['alg:adjacency algo']}.