EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions

Q1: Differentiability for $G_{l>1}$.

EvoMesh avoids using fully connected graphs for ${G}_{l>1}$. Instead, it ensures differentiability through:

1. Node selection: Differentiable via Gumbel-Softmax (L183-197).
2. Edge construction: Edges are formed based on the selected nodes and the connectivity of the original graph, further enhanced by $K$-hops (L199-219).
3. Edge feature generation: A differentiable process that encodes distance vectors and norms between connected nodes into high-dimensional edge features, serving as inputs for edge and node updates in message passing.

As a result, the prediction error can be backpropagated to both the generated edges and selected node features, allowing the node selection module to be optimized.

Q2: Novelty beyond AMP.

We appreciate the reviewer’s focus on evaluating the rationale behind AMP. Meanwhile, we'd like to emphasize that the primary novelty of EvoMesh lies in its learnable dynamic graph structures, with AMP being one component for enabling this capability. We hope the reviewer will consider the broader innovation of our work as a whole.

• Core Contribution - Learnable Dynamic Graph hierarchies: Unlike traditional PDE solvers that rely on fixed/predefined graphs, EvoMesh introduces a differentiable method to learn time-evolving graph structures that adapts to physical system dynamics.
• Role of AMP: AMP enables anisotropic intra-level propagation to determine node selection; and its predicted importance weights facilitate inter-level feature propagation.

Q3: Efficacy of AMP.

The efficacy of AMP is demonstrated in two ways:

• Fig 4 (Inter-level): By comparing the model with M1-M3 baselines, it suggests that AMP's predicted importance weights are crucial for inter-level information propagation.
• Anisotropic vs. Isotropic (Intra-level): We here use isotropic summation for intra-level feature aggregation, while retaining the AMP importance weights for inter-level propagation. The results are shown below:

Q4: Number of levels/downsampling ratio.

Number of hierarchy levels: We use the same number of layers as the static bi-stride hierarchy method. Please refer to Appendix C.4 for the impact of varying hierarchy numbers.

Downsampling ratio: Below, we present the average downsampling ratios for EvoMesh and the static bi-stride hierarchies. We visualize static and EvoMesh hierarchies at https://sites.google.com/view/evomesh/.

Q5: vs. Uniform node-sampling.

We present the required comparisons below. The baseline model uses an equal number of uniformly sampled nodes at each level.

Q6: MGN reports lower RMSE for CylinderFlow/Airfoil/Plate.

Most models converge after 1 million steps, while MGN trains for 10 million steps (10 days for CylinderFlow, 20 for Airfoil). To fix this, we use 500 trajectories for the Plate dataset, whereas MGN paper uses 1000. For Airfoil, our sequence length is 100 steps, while MGN uses 600 (detailed in Appendix). We now include MGN results in our response to Reviewer Mnuf Q4, showing EvoMesh outperforms MGN.

Q7: AMP vs. GAT.

Please see our response to Reviewer Mnuf Q3.

Q8: Related work -- DiffPool.

We'll include a discussion of these methods in the revised paper -- Most differentiable graph pooling methods like DiffPool are designed for graph classification, where unpooling is not typically needed. In contrast, EvoMesh focuses on mesh-based physics simulation, requiring dynamic modeling across multiple scales and integration of global and local details. Dynamic hierarchy construction remains largely under-explored in this field.

Q9: Ratios of retained points.

We provided sampling ratios of EvoMesh in our previous response to Q4, which shows variations across different graph layers. While static methods follow predefined reduction steps, EvoMesh adaptively retains nodes based on the optimization objective, which is solely aimed at improving prediction accuracy. We do not impose constraints on sampling ratios.

Is it possible for the model to retain all nodes? Theoretically, yes. However, in practice, the model prioritizes the most relevant nodes for the simulation, those that contribute the most to the final prediction. This behavior emerges naturally during training, suggesting that dense high-level graph structures may not be necessary for this task.

We thank the reviewer for the insightful comments. Below, we address each comment point-by-point.

Q1: Missing reference -- LayersNet.

Thank you for bringing LayersNet to our attention. We will include it in our related work and provide detailed comparisons in the revised manuscript. Below is a brief discussion of the key differences:

• Distinct application domains: While LayersNet focuses on garment animation influenced by external forces, EvoMesh is a general mesh-based simulation method for both Eulerian (e.g., fluid and structure simulations) and Lagrangian (e.g., cloth simulation) systems.
• Different hierarchical structures: LayersNet employs a static, patch-based two-level hierarchy, where garments are represented as particle patches to simplify interaction modeling. In contrast, EvoMesh introduces a learning-based, time-evolving graph structure that dynamically adapts to physical system dynamics, with the entire structure being learned in a fully differentiable manner.
• Attention vs. AMP mechanisms: LayersNet employs specialized attention mechanisms to handle rotational symmetry in garment dynamics. In EvoMesh, AMP serves as the core mechanism for dynamic graph construction, where the differentiable node selection is guided by AMP features. Additionally, AMP’s predicted importance weights facilitate inter-level feature propagation. We also discuss the impact of replacing AMP with GATConv, which results in degraded performance, as detailed in our response to Reviewer Mnuf Q3.

Q2: More visualized results.

We provide additional visualizations in https://sites.google.com/view/evomesh/, including prediction showcases of FoldingPaper, constructed comparisons for static-based bi-stride hierarchy and dynamics hierarchies of EvoMesh. We hope this addresses your concern.

Q3: Is the proposed method bring extra computational overhead?

We have provided a comprehensive efficiency comparison in Appendix Table 13, covering training time, inference time, and model size.

Admittedly, the dynamic hierarchy construction in EvoMesh does introduce some additional computational overhead. However, as shown in Table 13, this cost is manageable as the overall training time remains comparable to that of static hierarchy methods. This efficiency is primarily attributed to the significant reduction in the number of edges at each hierarchy level, as detailed in our response to Reviewer TDvH Q7.

We greatly appreciate the reviewer’s valuable comments.

Q1: Details of DiffSELECT.

We here revise Eq. (6) to clarify the node selection process. Specifically, we employ Gumbel-Softmax independently for each node based on a 2-dimensional logits vector predicted by model $\phi^v$. This enables stochastic sampling of node retention in the downsampled graph while ensuring differentiability for gradient-based optimization.

The formulation is as follows:

$$z\_i^l=\text{Gumbel-Softmax}(\mathbf{l}\_i^l)=\frac{\exp\left((\log p\_{i,0}^l+g\_{i,0}^l) / \tau \right)}{\sum_{k=0}^1 \exp\left((\log p_{i,k}^l + g_{i,k}^l) / \tau \right)},$$

where:

• $\mathbf{l}\_i^l=(\log p\_{i,0}^l, \log p\_{i,1}^l)$ represents the logits for node $v_i$ at layer $l$,
• $g_{i,k}^l$ is the Gumbel noise sampled independently for each node and category,
• $\tau$ is the temperature parameter controlling the smoothness of the sampling process.

This process allows for a differentiable approximation of discrete node selection. We will include this explanation in the revised manuscript.

Q2: On FeatureMixing.

FeatureMixing is to integrate features from finer levels with those from coarser levels, to better capture both local and global information in the U-Net architecture. While the EXPAND operation upsamples coarser-level features (e.g., level $L$) to match the resolution of the current level ($L−1$), naive upsampling can introduce noise or lead to uneven upsampling. FeatureMixing refines and aligns coarser-level features before fusion. The whole upsampling process works as follows:

1. EXPAND: Coarser-level features ($\mathbf{F}_L$) are upsampled to the resolution of level $L-1$.
2. Refinement: A message-passing step is applied to refine the upsampled features ($\text{EXPAND}(\mathbf{F}_L)$):
   $\mathbf{\tilde{F}}_L = \text{MessagePassing}(\text{EXPAND}(\mathbf{F}_L))$
3. Fusion: The refined coarser-level features ($\mathbf{\tilde{F}}\_L$) are combined with the current level’s intra-level features ($\mathbf{F}\_{L-1}$):
   $\mathbf{F}\_{L-1}^{\text{mixed}} = \mathbf{F}_{L-1} + \mathbf{\tilde{F}}_L$

We will improve the clarity of FeatureMixing in the revision.

Q3: Multi-step rollouts on OOD Mesh Task.

We have supplemented the rollout predictions with 50 steps and results of additional baselines on the Cylinder and Airfoil datasets in the rebuttal period. Results are presented in the reply to Reviewer Mnuf Q4. As shown, EvoMesh outperforms other methods in both the 1-step and 50-step rollout predictions, demonstrating superior generalization.

Q4: Qualitative showcases of paper-folding.

Visualizations of EvoMesh and MGN rollouts are available in https://sites.google.com/view/evomesh/. EvoMesh better captures fine details and deformation, adapting to intricate folds and complex geometries. We hope this addresses your concern.

Q5: Gaussian input noise.

The values for Gaussian input noise are reported in Appendix A, Table 7. We adopted the same noise scale used in the MGN and BSMS-GNN papers for each dataset, without task-specific tuning. The MGN paper and its referenced GNS paper[1] offer a detailed discussion on the selection of the Gaussian noise scale, and subsequent work generally adhered to these established values. The same scale is applied in the OOD task, regardless of mesh resolution.

[1] Learning to Simulate Complex Physics with Graph Networks.

Q6: Typo in Figure 2 and Figure 8a.

"DHMP" indeed refers to EvoMesh, and we'll correct this in the revised manuscript.

Q7: Why similar runtime to static hierarchy models? Any optimizations to improve efficiency?

EvoMesh maintains a runtime comparable to static hierarchy models because the computational overhead of hierarchy construction is balanced by a substantial reduction in the number of edges at each hierarchical level. As shown below, EvoMesh consistently yields fewer edges per layer compared to the static bi-stride hierarchy. Since message passing complexity is primarily driven by the number of edges rather than nodes, this reduction substantially lowers computational costs.

Notably, EvoMesh achieves this without explicit efficiency-driven modifications. Instead, this efficiency emerges naturally during training, as EvoMesh autonomously identifies key nodes, which we observe to be relatively sparse. This partially suggests that the dense high-level graph structures may not be necessary for this task.

Q8: Code release.

We'll release both the code and datasets upon acceptance.

We thank the reviewer for the insightful comments.

Q1: Novelty compared to BSMS-GNN.

While both EvoMesh and BSMS-GNN adopt U-Net-based hierarchical structures for multi-scale modeling, EvoMesh introduces critical innovations in end-to-end joint learning of graph hierarchies and physics dynamics that adapt to changing physics states:

Core Innovation: Adaptive vs. Static (preprocessed) Hierarchies

• EvoMesh integrates graph structure learning and dynamics learning into a fully differentiable framework, enabling time-evolving graph hierarchies that adapt dynamically to changing physical conditions. Specifically, this is implemented with anisotropic intra-level propagation, where edge weights are modulated by local physical states to enable directionally sensitive message passing, and learnable inter-level propagation, which adaptively learns interactions between hierarchical levels to optimize information flow across scales.
• BSMS-GNN separates hierarchy construction (preprocessing) from dynamics learning. It employs a fixed, pre-defined graph hierarchy with static connectivity throughout the simulation, limiting its ability to refine its structure in temporally evolving systems.

Q2: EvoMesh vs. EAGLE (Janny et al.).

In methodology,

• EAGLE uses a two-scale hierarchical message-passing approach, downscaling mesh resolution via geometric clustering of mesh positions. The fixed-size clustering is precomputed with a modified k-means algorithm, independent of dynamics modeling.
• EvoMesh constructs a multi-scale hierarchy that enables the joint learning of adapative high-level graph structures and physical dynamics within an end-to-end differentiable framework.

The table below compares the performance of EAGLE and EvoMesh. The results show that EAGLE’s purely geometric two-scale hierarchy results in slightly lower predictive performance, while EvoMesh benefits from its context-aware dynamic hierarchy.

Q3: AMP vs. GAT

The key distinction between AMP and GAT is that AMP enables differentiable dynamic hierarchy construction, a capability not supported by GAT. While AMP shares similarities with attention-based methods like GAT in computing importance weights, the detailed implementation differences are as follows:

• GAT applies computed weights to aggregate node features through weighted summation;
• Ours: (a) directly applies predicted weights to edge features, (b) leverages weights for inter-level feature propagation.

This dual mechanism enables AMP to dynamically learn graph hierarchies adaptable to evolving physical systems while jointly modeling physics dynamics through gradient-based learning, rather than relying on static hierarchical structures.

To further demonstrate the effectiveness of AMP, we replaced it with GATConv in EvoMesh, using a single attention head. The results below show that AMP layers used in EvoMesh outperform GATConv. Full comparisons between AMP and GAT will be included in the revised manuscript.

Q4: There are inconsistencies in the choice of methods used for comparison in different experimental setups.

Table 2: Here, we present the results of MGN, showing that EvoMesh achieves superior performance.

Table 3: For the Paper Folding dataset, fixed hierarchy-based approaches (e.g., BSMS-GNN and HCMT) are excluded because their hierarchies, derived from the initial graph input, cannot be applied across the entire sequence as the mesh structure evolves.

Table 4: We have supplemented this table with additional baseline models in the rebuttal period. Additionally, we present results from 50-step rollouts to offer further insights into the performance of extended simulations. The results demonstrate the generalizability of EvoMesh to OOD mesh structures through dynamic hierarchy construction. Full results will be provided in the revised paper.