Title: Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neural PDE Solvers

Diffusion Models

Graph Neural Networks

Learned Simulation

AI4Science
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a method for learned, GNN-based simulation targeted at solid mechanics applications. It introduces a novel rolling diffusion method and hierarchical message passing scheme which allows fast & accurate predictions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Quasistatic elasticity simulations are challenging for GNN simulators; yet the presented approach does seem to be quite effective at them\n- All the absolutely necessary ablations are there (even though comparisons to other HGNN or diffusion approaches would have been nice)\n- This is a nice demonstration that rolling diffusion can be a useful approach for learned simulators\n- Overall, the paper is well written and easy to follow\n\nWeaknesses:\n- The main issue I see is that the paper splits its attention too much between rolling diffusion, HGNNs and applications in solid mechanics. The paper does the minimum diligence in all of these areas, but doesn't deliver any fundamental insights in either that might transfer to other work.\nE.g., there are so many HGNN papers with slightly different choices of graph construction and message propagation, and this paper adds yet another option without properly evaluating it against other options. Similarly, I feel the effect of conditioning on not fully denoised previous timesteps is underexplored. There is one ablation on this with downstream performance, but it would have been much more interesting to develop an understanding to what this does to simulation rollouts; e.g. does this affect coherence of high-frequency details?\n- The choice of experimental domain is a bit odd. Quasistatic elasticity is a good domain for studying hierarchical message passing, but less so for diffusion-- these simulations are very deterministic and smooth. So diffusion a) seems much less important here but also b) the tradeoffs in conditioning on not fully resolved diffusion might look very different in a less smooth domain. So it would have been good to also include a domain with more chaotic dynamics on multiple scales (e.g. CFD, weather, fracture,...).\n- (Minor) The description of ROBI and corresponding figure could be improved; it's a bit of a notation-fest, and the figure is hard to parse. It took me a few passes to understand which step is conditioned on what exactly in diffusion & simulation time.\n\nQUESTIONS:\nNo major open questions. Overall I think this a paper worth accepting as this is an interesting approach which shows progress on a challenging problem. I would however encourage the authors to think about more thoroughly investigating the effect of some of the choices made in the paper, either as follow-up work if this paper is accepted, or a major revision of this paper if it is not.\nE.g. I'd love to see a proper review of different ways to perform HGNN, and how this affects different physical domains, from smooth simulations with long-range effects such as the ones studied here, to local chaotic CFD simulations. Similarly, a deep dive into the various choices in diffusion & denoising and the effect this has on consistency and frequency distribution could be very interesting.\n\nLIMITATIONS:\nyes, except for the points mentioned above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes an auto-regressive hierarchical graph neural network and diffusion model based framework for modelling the dynamics of systems in solid-mechanics from a Lagrangian frame. The paper extends the work of the paper PDE-Refiner Lippe et al. 2023 with several key extensions. They extend onto problems on unstructured meshes with Lagrangian dynamics using a hierarchical GNN. The mesh hierarchy utilises a novel AMG based approach from classical FE meshing, which creates visually more balanced and appealing coarse meshes compared to recent ML baselines. The authors also introduce a strategy Rolling Diffusion Batched Inference (ROBI), that allows accelerated roll-out at inference time by initialising the denoising process of future states conditioned by \"early-denoising\" of preceding states. Experiments are performed on 3 dynamical problems from solid mechanics, baselined against 2 SOTA models MGN and HCMT. A sensitivity analysis of the ROBI procedure wrt to $m$ the denoising stride and $K$ the denoising steps and a model ablation study of key architecture components are also performed.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper would be a straight accept for me if the quality of the write up was better but I had to spend quite some time to extract the details. In particular the clarity of the write up in sections 3.1 and 3.2 regarding ROBI and the HGNN could be improved, I list my concerns below to see they can be improved.\t\t\n\t\t\nIn section 3.1:\t\t\nRegarding the v-prediction target, the explanation was not clear. My eventual understanding is when k=K (first denoising steps) this is where we sample pure noise which is input to the model but also the target is closest to the 1 step direct target. Conversely when k~0 (latter denoising steps) the target is the remaining noise on the sample. So when the SNR is the highest the target is noise and when the SNR is the lowest the target is the signal. I found this quite counter intuitive and perhaps it can be clarified/reworded. Also, \"One step models\" are not defined, this confused me for a long time until I read PDE-Refiner paper.\n\n- There is some abiguity if there are also neural nets used to predict $\\mu$ and $\\Sigma$. But are these different to the main denoiser that predicts the denoising velocity and if so how are they trained?\n- There are no definitions of t_B/k_B/t_b/k_b\t\n- It was not immeadiate to derive the formula on line 148, it could be pointed out this comes from substituting $\\epsilon$ from the noising step into 1.\n- Clarifications on figure 3, what is meant by \"pooling\", as pooling is usually associated as a step within coarsening. In the caption does \"1 (top)\" refer to line c) ?\t\t\n\nIn the ROBI section, the writing would benefit from clarifying/reasserting that the network has 2 inputs, the t-1 condition and the current t sample at denoise level k, as this is non-standard in diffusion models, AR is a niche case. I now understand this is because the model takes in 2 arguments, previous step as condition and current noisey sample. It is the \"early-denoising\" of the preceding t-1 conditioning that gives a \"trade-off\" against accuracy when m<<K as it is utilised from a less developed state. Similarly, The sentence that starts \"Using these properties, we begin\", I thought for a long time the N(0,1) was a typo, given the prior sentence is motivating \"sufficient to condition the next step\".\n\nIn section 3.2:\t\t\nThe description of root-node AMG coarsening is also unclear, I would not be able to implement it from the description. Neither the rote-node or smoothing aspects from \"algebraic root-node-based smooth aggregation\" are explained. The description of upsampling/downsampling are not well explained \"Those matrices are constructed by smoothing the sparse initial aggregation mapping using the adjecency matrix\" I think I understand there is some kind of graph diffusion rewiring but providing more details would again strengthen the paper.\t\t\n\nI have some small evaluation concerns. No numerical figures are reported just bar charts that are hard to read and interpret. For the rollout in Figure 5b, I understand we compare this is a sensitivity study for different $m$ and $K$ but it is a slight red flag that ML baseline and even classic data generation times are not provided. Especially since this is a diffusion model where inference time is a known challenge.\n\nQUESTIONS:\nNo\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nGraph based simulators are widely used for the mesh based data modalities, however these models struggle with long range dependencies and correlations. Furthermore autoregressive rollouts leads to error accumulation which introduces another source of error. This paper introduces ROBIN a diffusion based learned simulator to address these issues. A rolling diffusion denoising scheme is introduced to accelerate the overall generation process by parallelizing the diffusion across time. Furthermore a hierarchical graph neural network built on multigrid coarsening is introduced to allow for multiscale message passing.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n1) The concept of a rollout diffusion inference and autoregressive diffusion is rapidly growing popularity for world models and other scenarios where actions and inputs are continuously introduced to the model. The rollout diffusion idea of the paper is interesting to see in the context of solid mechanics. \n\n2) The multi-resolution message passing opens doors to mixing simulations at different scales as well. This is something that can be explored by other researchers to have a generations informed from variety of scales. \n\n\n**Weakness**\n1) A major concern is the time parallel denoising scheme for inference. Authors mention that for generation of T frames (timesteps) with K denoising steps the time complexity will be O(KT). However the first diffusion models introduced for spatiotemporal data were not trained or used at inference with rollouts. These models (and most current video generative models) denoise all frames (timesteps) in parallel with an inflated UNet so the actual complexity would be O(K). It seems the paper uses the argument of efficiency throughout to develop the the rollout style denoising scheme to achieve O(K-m + mT).\n\n2) Paper provides the arguments to how they get to O(K-m + mT) in \"Rolling Diffusion-Batched Inference (ROBI)\" section however their claim in introduction is that they reduce the inference time from O(KT) to O(T)\n\n3) Novelty of the ROBI is unclear. How does ROBI differ from the current autoregressive diffusion frameworks. \n\n4) Provided experiments are limited. Given the claim of a new denoising scheme is introduced authors potentially should provide comparisons with other autoregressive diffusion frameworks (mostly in image or video generation). Otherwise there is a wide range of PDE problems that are more commonly used for benchmarking that can be used to compare methods with baselines. \n\n5) Baselines are old and and limited. There has been major researches focusing on PDE generations in recent years however the paper only compares to two baselines one of which is older than 5 years. MGN is commonly is not used as a standalone baseline and it helps complete an overall picture when also having other baselines.\n\nQUESTIONS:\n1) Can authors provide a clearer picture of the novelty and comparison with existing autoregressive diffusion methods? It is not clear how the rollout denoising differs from others!\n\n2) If it is possible to put the dependence of the denoising between steps into equation it would become much more clear to the reader. Can authors formulate such equation (Even if it is a small inline equation)?\n\n3) Can authors clarify why a parallel generation is not possible in this concept? This way the complexity simply becomes O(T)!\n\n4) Why did the authors focus on a very limited area? Is it possible to expand this to other PDEs? This will help understanding the generalizability.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
The authors proposed a novel method, Rolling Diffusion Batched Inference (ROBI), that can accelerate inference in DDPM-based simulation, and then they combined it with a Hierarchical Graph Neural Network that combines multiscale message passing with ROBI to provide fast, accurate diffusion-based simulations.

STRENGTHS AND WEAKNESSES:
The strengths of this paper are:
1. The paper has a clear model structure and mathematical formula which reinforce the theoretical soundness and reliability of the proposed model.
2. The model's novelty is good, although many people are using the Diffusion model, the authors improve it here.
3. The paper features a detailed model structure that is easy to follow and understand, and the experiment is well-designed, clear, and straightforward.

QUESTIONS:
1. How is the computation cost compared to HGNN? Does it take longer or shorter to train？
2. How does ROBI learn the parameter compared to conventional diffusion?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-07