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

TL;DR

ROBIN tackles global, time-evolving nonlinear solid mechanics on unstructured meshes by marrying diffusion-based refinement with a multiscale hierarchical GNN. It introduces Rolling Diffusion-Batched Inference (ROBI), which amortizes the diffusion steps across a rolling temporal window and reduces per-step computations, and combines this with an Algebraic-hierarchical Message Passing Network (AMPN) built on Algebraic Multigrid coarsening to capture both global modes and local dynamics. The framework yields state-of-the-art accuracy on 2D and 3D benchmarks (BendingBeam, ImpactPlate, DeformingPlate), while achieving up to an order-of-magnitude speedup in inference compared with standard diffusion-based simulators, and demonstrates strong generalization to much larger meshes via shared AMPN blocks. The approach highlights the importance of residual predictions, hierarchical multiscale messaging, and controlled diffusion truncation for stable, long-horizon rollouts, outlining clear paths for extensions to other physics domains and potential industrial impact in fast design and optimization workflows.

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.

Figures (15)

• Figure 1: Overview of a robin prediction. robin coarsens the fine mesh multiple times with amg to create a graph hierarchy. robin predicts the denoising velocity $\mathbf{v}_\theta(\Delta \mathbf{u}^t_k, k, \mathbf{u}^{t-1}_0)$ at time step $t$ using ampn, given a noisy residual sample $\Delta {\mathbf{u}}^{t}_{k}$, the diffusion step $k$, and a previous state $\mathbf{u}^{t-1}_{0}$. The prediction is used to draw a new noisy residual sample $\Delta {\mathbf{u}}^{t}_{k-1}$ and to update the state $\mathbf{u}^{t}_{0}$.
• Figure 2: Overview of a) conventional autoregressive diffusion inference and b)robi. a) Conventional inference denoises the entire state of a physical time step at once before it shifts to the next time step (see One Step Denoising). The Iterative Inference requires $K$ model calls per time step, where $K$ denotes the number of diffusion steps. b)robi parallelizes denoising steps across physical time, processing up to $K$ time steps batched, and reconstruct the physical states with the clean sample prediction subsequently (see Batched Denoising Step). This process allows Rolling-Batched Inference after the initial warm-up, reducing the number of model calls to one per time step.
• Figure 3: Example predictions of robin on the considered datasets. robin predicts the part deformations as well as the von Mises stress (color, yellow is high) on all experiments. a)BendingBeam considers global part deformations of beams induced by local forces. b) In ImpactPlate, the models have to predict locally large deformations, caused by a collision with an accelerated ball. c) The hyperelastic plates in DeformingPlate are deformed by a scripted actuator.
• Figure 4: Left: Rollout error measured by the RMSE of the predicted nodes positions. robin surpasses the accuracy of the baselines hcmt, mgn, and bsms on all three datasets BendingBeam, ImpactPlate and DeformingPlate. Right: Comparison of inference time and error of robin and its variants on BendingBeam. The default variant of robin (1/20) achieves the same accuracy as conventional diffusion inference (20/20), while the inference speed is close to the one step variant (1/1). Reducing the truncation step $k_\text{tr}$ trades accuracy for speed.
• Figure 5: Comparison of the predicted rollout deformations and von Mises stresses (color, yellow is high) on BendingBeam between a) the fem, b)robin, c)hcmt, d)mgn, and e)bsms. robin is able to accurately reproduce the fem results. Neither hcmt, mgn nor bsms are able to resolve global deformation modes, illustrating the importance of the ampn for global message propagation.