Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics

TL;DR

This paper introduces ROBIN, a diffusion-based hierarchical graph neural network that efficiently simulates complex nonlinear solid mechanics phenomena, outperforming existing methods in accuracy and speed.

Contribution

The paper presents a novel hierarchical graph neural network with a diffusion-based inference scheme, enabling accurate and fast simulation of nonlinear solid mechanics across multiple scales.

Findings

ROBIN achieves state-of-the-art accuracy on 2D and 3D benchmarks.

It reduces inference time by up to an order of magnitude.

It effectively captures global phenomena like bending and contact.

Abstract

Graph-based learned simulators have emerged as a promising approach for simulating physical systems on unstructured meshes, offering speed and generalization across diverse geometries. However, they often struggle with capturing global phenomena, such as bending or long-range correlations usually occurring in solid mechanics, and suffer from error accumulation over long rollouts due to their reliance on local message passing and direct next-step prediction. We address these limitations by introducing the Rolling Diffusion-Batched Inference Network (ROBIN), a novel learned simulator that integrates two key innovations: (i) Rolling Diffusion-Batched Inference (ROBI), a parallelized inference scheme that amortizes the cost of diffusion-based refinement across physical time steps by overlapping denoising steps across a temporal window. (ii) A Hierarchical Graph Neural Network built on algebraic multigrid coarsening, enabling multiscale message passing across different mesh resolutions. This architecture, implemented via Algebraic-hierarchical Message Passing Networks, captures both fine-scale local dynamics and global structural effects critical for phenomena like beam bending or multi-body contact. We validate ROBIN on challenging 2D and 3D solid mechanics benchmarks involving geometric, material, and contact nonlinearities. ROBIN achieves state-of-the-art accuracy on all tasks, substantially outperforming existing next-step learned simulators while reducing inference time by up to an order of magnitude compared to standard diffusion simulators.