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

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, 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, 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.

@article{würth2025_2506.06045,
  title={ Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics },
  author={ Tobias Würth and Niklas Freymuth and Gerhard Neumann and Luise Kärger },
  journal={arXiv preprint arXiv:2506.06045},
  year={ 2025 }
}