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

2025-06-06

AI Summary

To address the challenges of modeling global effects and severe error accumulation in long-term nonlinear solid mechanics simulations on unstructured meshes, this paper proposes ROBIN. Methodologically, ROBIN introduces (1) a novel rolling diffusion inference mechanism that enables cross-temporal-step latent-state denoising and reuse, and (2) an algebraic multigrid (AMG)-driven hierarchical graph neural network that integrates multi-scale physical priors, supporting algebraic coarsening and hierarchical message passing on unstructured graphs. Evaluated on 2D/3D strongly nonlinear solid mechanics benchmarks, ROBIN achieves state-of-the-art accuracy, accelerates inference by 10× over conventional diffusion-based simulators, and significantly outperforms existing step-wise learned simulators in both fidelity and efficiency.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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, 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.

Problem

• Simulating global phenomena in nonlinear solid mechanics
• Reducing error accumulation in long rollouts
• Improving computational efficiency in diffusion-based simulators

Innovation

• Rolling Diffusion for parallelized inference
• Hierarchical GNN with multigrid coarsening
• Algebraic-hierarchical Message Passing Networks