EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Authors: Huayu Deng
Xiangming Zhu
Yunbo Wang
Xiaokang Yang
Venue: ICML 2025 poster (2025)
Keywords: Physical Simulation
Message Passing Networks
Primary area: deep_learning->graph_neural_networks
TL;DR: EvoMesh is a fully differentiable framework that jointly learns graph hierarchies and physical dynamics.

Abstract: Graph neural networks have been a powerful tool for mesh-based physical simulation. To efficiently model large-scale systems, existing methods mainly employ hierarchical graph structures to capture multi-scale node relations. However, these graph hierarchies are typically manually designed and fixed, limiting their ability to adapt to the evolving dynamics of complex physical systems. We propose EvoMesh, a fully differentiable framework that jointly learns graph hierarchies and physical dynamics, adaptively guided by physical inputs. EvoMesh introduces anisotropic message passing, which enables direction-specific aggregation of dynamic features between nodes within each hierarchy, while simultaneously learning node selection probabilities for the next hierarchical level based on physical context. This design creates more flexible message shortcuts and enhances the model's capacity to capture long-range dependencies. Extensive experiments on five benchmark physical simulation datasets show that EvoMesh outperforms recent fixed-hierarchy message passing networks by large margins. The project page is available at https://hbell99.github.io/evo-mesh/.

EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Huayu Deng' Xiangming Zhu
Yunbo Wang' Xiaokang Yang'
Abstract
Table I. Comparison of mesh-based physical simulation mod- els. Dynamic hierurchy refers to hierarchical graph structures that evolve over time, Adaptive indicates that the graph structures are determined by physical inputs. Prop. denotes feature propagation.
Graph neural networks have been powerful tool for mesh-based physical simulation. To effi- ciently model large-scale systems, existing meth- ods mainly employ hierarchical graph structures to capture multi-scale node relations. However, these graph hierarchies are typically manually de- signed and fixed, limiting their ability adapt the evolving dynamics of complex physical sys- tems. We propose EvoMesh, fully differentiable framework that jointly learns graph hierarchies and physical dynamics, adaptively guided by physical inputs. EvoMesh introduces anisotropic message passing, which enables cnrrection-speciiie aggregation of dynamic features between nodes within each hierarchy, while simultaneously learn- ing node selection probabilities for the next hi- erarchical level based on physical context. This design creates more flexible message shortcuts and enhances the model's capacity to capture long-range dependencies. Extensive experiments on five benchmark physical simulation datasets show that EvoMesh outperforms recent fixed- hierarchy message passing networks by large mar- gins. The project page is available at https: //hbel/ 199. g thub. seeevoomesst
Dynamic Hierarchy
Adaptive Hierarchy
Anisotropic
Learnable
Model
Intra-level Prop Inver-level Prop
MGN (2021)
X
Lino et al. (2022)
X
X
x
BSMS (2023)
X
X
x
x
Eagle (2023)
X
X
V
V
HCMT (2024)
V
X
EvoMesh
V
broadcast from neighboring nodes (Sanchez-Gonzzlez et al., 2020; Pfaff et al., 2021; Allen et al., 2023). Existing meth- ods generally rely on repeated local message passing to propagate influence over long distances, which becomes extremely costly for large-scale mesh graphs. A common solution involves using multi-scale graph structures to create direct information shortcuts between distant nodes. (Lino et al., 2022; Cao et al., 2023; Yu et al., 2024; Han et al., 2022; Fortunato et al., 2022).
However, as shown in Table 1, previous methods commonly rely on heuristic node selection to create predefined (data- independent) coarser message passing graphs (Cao et al., 2023; Yu et al., 2024). These predefined graphs limit the model's adaptation ability in two key ways. First, the fixed graph hierarchies, applied to the entire input sequence, do not account for the variety of physical contexts. In practical systems like turbulence, even with identical boundary con- ditions, small changes in initial conditions lead to sig- nificant differences in subsequent dynamics. Second, since the spatial correlations in a physical process can evolve over time, static graph hierarchies are insufficient for capturing the time-varying node interactions.
1. Introduction
Simulating physical systems with deep neural networks has achieved remarkable success due to their efficiency com- pared with traditional numerical solvers. Graph Neural Networks (GNNs) have been validated as a powerful tool for mesh-based simulation, such as for fluids and rigid colli- sions (Wu et al., 2020). The primary mechanism driving the GNN-based models is message passing, where time-varying physical quantities are encoded within the mesh structure and are temporally updated by aggregating information
To tackle this challenge, we propose a novel neural network approach named EvoMesh, which constructs addapivee and time-evolving graph hierarchies based on the input phys- ical quantities. The key insight is to develop differentiable node selection method that allows for flexible correlation of long-range, dynamic node interactions. This is techni- cally supported by an anisotropic message passing (AMP) mechanism, which (i) aggregates neighboring features with non-uniform, learnable importance weights within each hi-
'MoE Key Lab of Artificial Intelligence, Al Institute, Shanghai Jiao Tong University. Correspondence to: Yunbo Wang <yun- bow @sw.u.edu.cn........
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
erarchical level, (ii) predicts the probabilities of the node being retained for the next hierarchy based on physical con- text, and (iii) adaptively learns cross-hierarchy interactions to optimize information flow across scales. To enable differ- entiability in the node selection process, we approximate the discrete downsampling decisions using Gumbel-Softman
updates the node states with aggregated messages from its neighbors. In existing GNN-based mesh simulation meth- ods, multi-layer perceptrons (MLPs) with residual connec- tions are commonly employed for d-(-) and Φ" ("(-), with the par-meraiiiiic aggregation function eh(-) being defined as the sum of edge features. Notably, since the aggregation function treats all neighbors equally, the contributions from neighboring nodes may be averaged out, and the repeated mnessgge-passingg process can further dilute distinctive node features. This issue is exacerbated in dynamic physical systems, where transferring directed patterns is crucial, Attention-based methods address this issue by reweight- ing neighbor features, either locally or globally (Veličković et al., 2018; Yu et al., 2024; Han et al., 2022; Yun et al., 2019). While effective for directional aggregation, most weighting remains limited intra-level features and does not support dynamic graph hierarchy construction.
Another advantage of the AMP mechanism is its ability to enable features to transfer between nodes with varying importance, aligning with the directionally non-uniform nature of the dynamic patterns, as observed in scenarios such as CylinderFlow, AirFoil, and Flying Flag simulations. It applies to both intra-level and inter-level feature prop- agation. In contrast, as shown in Table 1, most previous GNN-based mesh simulation methods perform isotropic feature aggregation within the intra-level transition and rely on unlearnable importance weights to transfer inter-level information across hierarchical levels, assuming equal con- tributions from neighboring nodes.
Hierarchical MPNNs. To facilitate long-range modeling, hierarchical MPNNs process information at L scales by cre- ating a graph for each level and propagating information between them (Lino al., 2022; Fortunato et al., 2022; Cao et al., 2023; Yu et al., 2024). Let 91 = (V1, E rep- resent the graph structure at the finest level, i.e., the input mesh. The lower-resolution graphs G2 Gs..., Gr, with |i > [V2] >...> > [zl,, contain fewer nodes and edges, which allows for more efficient feature propagation over longer physical distances with certain propagation steps. The typical process for constructing multi-scale structures primarily involves downsampling and upsampling between adjacent graph hierarchies. Downsampling reduces the num- ber of nodes while upsampling transfers information from a lower-resolution graph to a higher-resolution one. The downsampling operation includes two steps:
Overall, our contributions are summarized as follows:
We present EvoMesh, which generates dynamic graph hierarchies through differentiable node selection, enabling adaptive modeling of multi-scale physical relations.
EvoMesh employs anisotropic message passing to enable directionally varied feature propagation both within and across graph hierarchies.
On average, EvoMesh outperforms fixed-hieraree hierarchy mod- els by around 20% across range of standard benchmarks. It also demonstrates strong generalization to test cases with time-varying mesh structures, novel resolutions, and out-of-distribution dynamics.
2. Preliminaries
SELECT: Nodes are selected from the current graph structure G/ to create a new, coarser graph 9441. Various strategies have been proposed to construct V4+1, including hand-crafted designs (Lino et al., 2022; Cao et al., 2023; Yu et al., 2024), geometric clustering (Han et al., 2022; Janny et al., 2023), and differentiable pooling methods that predict cluster assignments or select top-ranked infor- mative nodes (Ying et al., 2018; Gao & 2019; Lee et al., 2019; Ranjan et al., 2020). The edges E1 in G4 are constructed by connecting the selected nodes based on the original edges & However, this process can sometimes lead to loss of connectivity and introduce partitions (Gao & Ji, 2019; Lee et al., 2019; Cao et al., 2023). To mitigate this, connectivity in e can be strengthened by adding K K-hop edges.
Message passing. We consider simulating mesh-based physical systems, where the task is to predict the dynamic quantities of the mesh at future timesteps given the current mesh configuration. A mesh-based system is represented as a bi-directed graph g = (V,8)', where V and È denote the set of nodes and edges, respectively. Message passing neu- ral networks (MPNNs) compute the node representations by stacking multiple message passing layers of the form:
Edge update: ē.j- = (e Vt. Vy);
(1)
Node update: V, d² (v., di ((ev) Vj. E E))),,
(2)
where v, is the feature of node 2s V and &e denotes non-parmatric aggregation function. The function df up- dates the features of edges based on the endpoints, while d"
REDUCE:
The features of the nodes in V1+1 are aggre- gated from their corresponding neighborhood features in the finer graph Gt-
Bi-directed means each original undirected edge is represented
twice G: if there is an edge between i and j, it is represented as two directed edges + j and j + i. Each node has a self-loop.
The upsampling process is represented by EXPAND, which
2

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
i
+
Anisotropic Message Passing (AMP)
E N
0.5
U
0.1
0.05
9
1XAMP
9
9
1xAMP
9
v
Vis
DiffSELECT
JEM
EXPAND
REDUCE
Differentiable Pooling
9
1XAMP
92
G
1xAMP
G2
0.2
0.00
0.5
DiffSELECT REDUCE
EXPAND
0.1
0.05
G
1XAMP
G
Forward
DiFfSELECT node
REDUCE
Gradient
EXPAND
Figure 1. The architecture of EvoMesh. Physical dynamics is modeled multiple graph resolutions with adaptive structures, 91, .2...., Gr., and are processed using their respective AMP layers. The Di DifELEECCT operation performs differentiable pooling to create coarser graphs with learnable downsampling probabilities. REDUCE and EXPAND integrate inter-level information using learned feature aggregation weights over the neighhoring nodes. EvoMesh is trained end-to-end with one-step supervision.
the inverse of the REDUCE function and transfers informa- tion from the coarser level back to the finer level. Most previ- ous work generates coarser graphs either by using numerical software or by downsampling the imput mesh through heuris- tie pooling strategies (Cao al., 2023; Lino et al., 2022; Yu al., 2024; Janny et al., 2023). This process performed during the data preprocessing stage. The preprocessed hier- archy with the same input mesh topology is reused across different initial conditions and time steps.
Section 3.4, we outline the implementation details.
3.1. Anisotropic Message Passing
We introduce the AMP layer, which facilitates information propagation both within and between graph hierarchies, enabling EvoMesh to effectively capture local and long- range dependencies simultaneously.
As shown in Eq. (2), a common non non- parameeriiicc aggregation in GNN-based mesh simulation is to use the summation for node update: v, o" Vi, Eny ENs, éis) where Uj No denotes a neighboring node of e in the graph.
3. Method
In this section, we introduce EvoMesh, a fully differentiable model that adaptively generates time-evolving graph hierar- chies over the sequence, while simultaneously simulating the physical system over these learned hierarchical graphs. Figure demonstrates an overview of the proposed model, pipeline. The
To differentiate the contributions of neighboring nodes, the AMP layer employs learnable parameters d" to predict the anisotropic importance weight of edge feature é;ÿ with respect node u;. These weights are then normalized across the neighborhood of U, using softmax function:
which operates in an sncode-process-decco encoder first maps the input field to a latent feature space = (v.lv, E Vi} } at the original mesh resolution. Subse- quently, we model the physical dynamics across the learned multi-scale graph hierarchies with adaptive graph structures.
(xis))))
Wij = 30 Vo, Vy), = ΣLENS exp (wa+)
The normalized coefficients are used to compute a linear combination of the corresponding edge features. This linear combination serves as the final input for the node update function 6" given node feature v;
In Section 3.1, we present the details of the AMP layer. In Section 3.2, we discuss the approach for learning context- aware graph hierarchies. In Section 3.3, we describe the inter-level downsampling and upsampling processes that incorporate AMP-based feature propagation. Finally, in
vi 6" (vo, Σ anjess) B,NN,,
(4)
3

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
The proposed AMP layer enables the implicit assignment of varying contribution weights to the updated edge fea- tures within the same neighborhood. Analyzing the learned direction-specific weights in AMP further enhances inter- pretability. We adopt an MI.P implementation for o² while alternative designs, such as graph attention (Veličković et al., 2018) or cross-attention (Vaswani et al., 2017), are also fea- sible. detailed comparison is provided in Appendix C.2.
we enhance the connectivity in ε by incorporating the K- K-hop edges during edge selection, defined follows:
)/ El {es B0k Uk2 ................................ UKK-I Vi
s.t. ei,b : A21.
Ckk 1 E }1}.
(7)
In essence, Eij E & if there exists a sequence of interme- diate nodes {Uk, Uly...., u consecutively connected by edges in & or eij E EG.. The edges in & are defined as:
&
3.2. Differentiable Multi-Scale Graph Construction
E1 = {ess Bey, by E V/41 s.t. Cij E & {
With the AMP layer functioning within each graph level, local dependencies are effectively propagated throughout the high-resolution graphs, guiding the selection of nodes to be discarded in the next hierarchy for improved long-range modeling. We now delve into the details of the differen- tiable node selection method (DifESELECT) for hierarchi- cal graph construction.
(8)
E consists of edges from the enhanced edge set E {) that connect nodes in Vi+1- As K increases, nodes in & can be connected through additional intermediate nodes, thereby improving long-ram oong-range connectivity. In practice, the most effective value of K is found to be 2. We include further discussions in Appendix C.3.
In the Di DSSEEETTT operation, we train the node update module ø² based on anisotropic aggregated edge features to produce a 2-dimensional probability vector π. for each node u. This vector -ं (#i.o: T(.1) represents the probabilities of discarding or retaining node 2; in the next-level coarser
The graph construction process is fully differentiable, al- lowing for seamless integration into differentiable physical simulators. By flexibly adapting graph hierarchies based on simulation states, it paves the way for more accurate predic- tions of the spatiotemporal patterns in complex systems.
graph Gi+1. We rewrite Eq. (4) as follows:
(vb Eajel
8// - o²
V/ Σ
alje!y
(5)
3.3. Inter-Level Feature Propagation with AMP
OyENo
During the downsampling process from Gi the generated coarser graph Gi+1, as illustrated in Figure 1, the REDUCE operation aggregates information to each node in Vi+1 from its corresponding neighbors in V. Conversely, the EXPAND operation unpools the reduced graph back to a finer resolu- tion, delivering the information of the pooled nodes to their neighbors at the finer level.
In the next step, we apply Gumbel-Softmax sampling (Jang et al., 2017) independently each node, using the log- probabilities (log o : log nf,1) as logits. This produces soft one-hot vector z (z),0= 2
for each node:
k = Gumbel-Softmai (log Dlo nh,0- log lo1
Prior work employed non- non-parametric aggregation in inter- level propagation, convolving features based on the nor- malized node degree. It simplifies intricate relationships between nodes and neglects the directional aspects of infor- mation flow. To address this, EvoMesh is designed to learns inter-level aggregation weights that are both data-specific and time-varying. Specifically, the importance weight at computed by the AMP layer inherently captures the rele- vance of node By" 's features to node U; at the graph level I. These weights can be directly reused for the REDUCE and EXP AND operations in the downsampling and upsampling processes. We provide details of these operations as follows:
exp (logTT4 + s/T
(6)
Ek exp (JogTi,N' + )/T
where gf is Gumbel noise sampled independently for each node, and is the temperature parameter controlling the smoothness of the sampling. In this way, the node set V4+1 is adaptively constructed based on node features from the finer graph level. The straight-through Guumhe--Softmax estimator provides a differentiable approximation to hard sampling, thereby facilitating end-to-end training. We im- plement the Gumbel-Softmas with temperature annealing to stabilize training, initially encouraging the exploration of hierarchies and gradually refining the selection process.
REDUCE: Let i; be the node at the coarser graph level. The downsampling process aggregates the information of the current neighbors M, by reusing the weight aly:
The edges ε141 in the coarser graph Gi+1 are constructed by connecting the selected nodes using the original graph's edges E,. However, this process may result in disconnected partitions (see Appendix Figure 6). To address this issue,
1+1 <- RED
:= =EEN, a.y v
EXPAND: We first unpool the node features from the coarser graph G0+1 back to the finer level G. To achieve

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table Quantitative comparison one-step and long-term prediction errors. We report the mean results over 3 random seeds,
with corresponding standard deviations detailed in Appendix C.7. Promotion denotes the improvement over the second-best model.
RMSE-1 (x10 3
RMSE-All (x 10 x00")
Model
Cylinder
Airfoil
Flag
Plate
Cylinder
Airfoil
Flag
Plate
MGN (2021)
0.3046
77.38
0.3459
0.0579
59.78
2816
115.3
3.982
Lino et ol. (2022)
3,9352
85.66
0.9993
0.0291
27.60
2080
118.2
2.090
BSMS-GNN (2023)
0.2263
71.69
0,5080
0.0632
16.98
2493
168.1
1.811
Eagle (2023)
0.1733
51.55
0,3805
0.0392
20.05
2344
127.7
7.797
HCMT (2024)
0.9190
48.62
0,4013
0,0295
23.59
3238
90.32
2,468
EvoMesh
0.1568
41.41
0.3049
0.0282
6.571
2002
76.16
1.296
Promotion
9.53%
14.8%
11.9%
3.10%
61.3%
3.75%
15.7%
28.5% %
this, we record the nodes selected during the downsam- pling process and use this information to place the nodes back in their original positions in the graph. Then, we reuse the previously computed importance weights aty to assign weighted features from Gi+1 back to Gi- The EXPAND operation is formally defined as: v EXPAND ((v) (v))1 .ig (ENN)))):: :- EJEN; V v/ aly.
CylinderFlow: Simulation of incompressible flow around cylinder based on 2D Eulerian meshes.
Airfoil: Aerodynamic simulation around airfoil cross- sections based on 2D Eulerian meshes.
FlyingFlag: Simulation of flag dynamies in the wind based on Lagrangian meshes with fixed topology.
DeformingPlate: Deformation of hyper-elastic plates based on Lagrange tetrahedral meshes.
ReatureMixing: While the EXPAND operation up-
samples coarser-level features of G1+1 to match the res-
olution of the current level Gt, naively upsampled fea-
FoldingPaper: Deformation of paper sheets with evolving
tures may suffer from artifacts misalignment. To meshes driven by varying forces at the four corners.
mitigate this, we introduce FeatureMixinn to re-
We mainly compare EvoMesh with the following methods:
fine and fuse coarse-level features with intra-level in-
formation at the current resolution. Specifically, we
MGN (Pfaff et al., 2021), which performs multiple times of message passing on the original graph.
apply an additional anisotropic message passing step the upsampled features V and then integrate these features with the original intra-level features vi in
Lino er al. (Lino et al., 2022), which also trains MPNNs on manually-set multi-scale mesh graphs
Gi (prior to downsampling) using a skip connection: vi, (ely)eene) == vl + AMP (V., (V) (elybex))
! + FeatureMi
BSMS-GNN (Cao et al., 2023), which generates static hierarchies using bi-stride pooling and performs message passing on predefined meshes.
3.4. Implementation Details
Eagle (Janny al., 2023), which constructs two-scale hierarchy using precomputed geometric clustering and performs message passing at both levels.
We train EvoMesh using the one-step supervision that mea- sures the Le loss between the ground truth and the next-step predictions. We include detailed descriptions of the imple- mentation of encoder, decoder, node update function and edge update function Appendix
HCMT (Yu et al., 2024), which generates static hicrar- chies by applying Delaunay triangulation to the bi-stride pooled nodes, and enables directed feature propagation with the attention mechanism.
4. Experiments
All models are trained using the Adam optimizer for 1 1M steps, with an exponential learning rate decay from 10-4 to 10-6 over the first 500K steps. We provide further details on the architecture and hyperparameters of the compared models in Appendix D.
In this section, we present the key results of the proposed method. Additional analyses can be found in Appendix C.
4.1. Experimental Setup
We evaluate EvoMesh on five mesh-based benchmarks from previous work (Pfaff et al., 2021; Cao et al., 2023; Wu et al., 2023; Narain et al., 2012). For detailed descriptions, includ- ing the input physical quantities, please refer Appendix A.
4.2. Main Results
Standard benchmarks. Table 2 presents the root mean squared error (RMSE) of one-step prediction (RMSB-1) and
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Lino et al.
BSMS-GNN
EvoMesh
Ground Truth
50
100
t= t=200
t=400
400
High Ervor
Low Error
Figure 2. Prediction showcases 400 future steps CylinderFlow. From the displayed error maps, it is evident that EvoMesh
effectively captures long-term dynamics, providing predictions that closely align with the ground truth.
100
t=400
Figure 3. A demonstration of how the learned hierarchies adapt to evolving physical dynamics. Top: true data, Bottom: the temporal difference of the velocity fields between adjacent time steps alongside the constructed coarser-level mesh graph (G==4). The highlighted areas demonstrate notable experimental phenomenon: the mesh dynamically evolves with the data context and aligns with the critical areas of change in data.
the velocity field from the
long-term rollouts for 100-600 future time steps (RMSE- all). EvoMesh consistently outperforms the compared mod- els across all benchmarks. This demonstrates the effective- ness of building context-aware, time-evolving hierarchies with learnable, directionally non-uniform feature propaga- tion both within and across graph levels. Figure 2 presents long-term predictions on CylinderFlow, based solely on the system's 's initial conditions at the first step. As see, EvoMesh captures the complex, time-varying fluid flow around the cylinder obstacle more successfully, with its pre- dictions closely matching the ground truth evolution. More results are shown in Appendix C.9.
highlighted by the temporal differences in the true data. We have two observations here: First, the constructed hi- erarchy evolves as the data context changes. Second, the time-evolving graph structures align with the high-intensity regions, either velocity fields (top) or in their temporal variations (bottom). These results highlight the effectiveness of our approach in capturing significant dynamic patterns.
Paper simulation with changing meshes. We evaluate EvoMesh in more challenging setting with time-varying meshes for paper folding simulation, generated using the ARCSim solver (Narain et al., 2012; Wu et al., 2023), and compare EvoMesh with MGN (Pfaff et al., 2021). Lino el al., BSMS-GNN, Eagle, and HCMT rely on pre-computed hierarchies during preprocessing. which limits their applica- bility in scenarios with dynamically changing mesh topolo- gies. Therefore, do not include them in this evaluation.
Can the learned hierarchies adapt to evolving data dy- namics? Figure 3, we visualize the time-evolving hi- erarchies constructed by EvoMesh at different time steps, where coarser-level nodes tend to concentrate in regions
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table Simulation results 2D paper folding with time- varying imput meshes. We here compare EvoMesh with MGN, as BSMS and HCMT require pre-computed hierarchies, which are unsuitable for scenarios involving continuously changing meshes.
propagation, its use of uniform node sampling results in sub- optimal performance, suggesting the necessity of adaptive and data-aware hierarchy construction.
Furthermore, in Figure 5, we visualize the variance of pre- dicted anisotropic edge weights and compare with areas where physical quantities present substantial variations over time. The results reveal strong correlation between the anisotropic learning mechanism and the rapidly changing dynamics of the physical system.
Model
RMSE-1 I (x10-3)
RMSE-AN All (x10 ?)
MGN (2021)
0.0618
24.08
EvoMesh
0.0544
7,412
Promotion
12.0%
69.2%
We assess the models using ground-truth remeshing nodes provided by the ARCSim Adaptive Remeshing component, following the setup from (Pfaff et al., 2021). As shown in Table 3, EvoMesh achieves superior short-term and long- term accuracy compared to MGN, indicating that the time- evolving graph hierarchies in our approach can better fit physical systems with significant geometric variations, represented by the time-varying input mesh structures.
4.4. Generalization Analyses
Generalization to out-of-distribution mesh resolutions. Nearly all existing machine learning models for mesh-based simulations are not resolution-free and may fail when eval- uated on unseen mesh resolutions. We assess the gener- alization performance of EvoMesh by training it on low- resolution meshes and testing it on high-resolution meshes, The average number of nodes in the test data is twice that of the training data, and the number of edges is three times greater. Table 4 reports one-step and 50-step rollout errors on -uito--distribution (OOD) mesh resolutions. EvoMesh shows strong generalization on the CylinderFlow and Air- foil datasets, highlighting its zero-shot capability to handle refined mesh structures. This performance gain is largely attributed to EvoMesh's ability to construct hierarchical graphs adaptively, allowing it to scale effectively with in- creased resolution. On the other hand, MGN achieves lower errors on the FlyingFlag and DeformingPlate datasets, in- dicating that mesh-based architectures remain effective for systems with more regular structures and smoother deforma- tion dynamics. While our method does not yet achieve full generalization across arbitrary resolutions, truly resolution- free modeling remains an open challenge that calls for more advanced architectural design. Nevertheless, this holds sig- nificant value in practical applications and has the potential to greatly reduce the time overhead of numerical simulation processes for preparing the large-scale mesh data required for model training.
Model stability under variable graph structures. Due to the stochasticity of Gumbel-Softtax sampling in OifFEEE,,,, we evaluate the stability of trained EvoMesh by conducting three independent runs the test set. The mean and standard deviations of the prediction errors reveal minimal discrepancies across different runs, as shown in Table 11 in Appendix C.6. These findings demon- strate that once trained, EvoMesh generates consistent graph hierarchies based on the same inputs.
4.3. Ablation Studies
EvoMesh has three key components: (i) evolving graph hierarchy, (ii) anisotropic intra-level propagation, (iii) learn- able inter-level propagation. To evaluate the contribution of each component, we implement several ablated vari- ants of EvoMesh, including: Static((i-strrieee Aniiorop)-- Unlearnable (MI), (M2), wwwir-nnnniiooonaaaaeee (M3), and Dynamic- Anisotropic-Unlleanaahe Unlearnable (M4), and compare them against the BSMS-GNT baseline, which uses static hierarchies, isotropic intra-level summation, and unlearnable inter-level propagation. Both M/ and M2 adopt the same static bi-stride hierarchy via preprocessing as BSMS-GNN. M3 constructs the hierarchy via uniform node sampling in each hierarchy, while M4 applies dynamic hierarchy construction without learnable inter-level updates.
Generalization to physical variations. We evaluate EvoMesh under strong distribution shifts in the input physi- cal quantities. Table 5 presents data statistics and the RMSE results on the CylinderFlow and Airfoil datasets. EvoMesh consistently outperforms the compared models in both short- term and long-term simulations. This advantage mainly comes from its ability to learn evolving hierarchies and model intra-level and inter-level interactions based on phys- ical context. When the fluid dynamics in the test set become more complex- caaplle--charaterrieeeee by increased variance in the velocity field over time-the dynamics patterns propagate more rapidly in space. EvoMesh adaptively constructs graph hierarchies and more effectively captures long-range node interactions in response to evolving physical contexts.
Figure demonstrates the effectiveness of direction-aware message propagation both intra- and inter-levels, as well as the benefit of learning dynamic graph hierarchies. Com- paring MI with BSMS-GNN shows that integrating AMP improves performance even under a static hierarchy. Fur- thermore, the comparison between EvoMesh and M4, as well as between M/ with M2, highlights the importance of learnable inter-level propagation in capturing hierarchical signal flow. In addition, although M3 benefits from learnable
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Static-Aniso Unllammlllee
Alatoo-nniooLLLamabble
Uniform-Ariso-Learrnall
Dynamic-Ariiso-Unleemaaa
EvoMesh
0.2
15
150
RMSE-AII
0.4
RMSE-AII
RMSE-1
RMSE-1
10
100
0.1
0.2
50
0.0
0.0
0
M1 M2 M3 M4 Ours
M1 M2 M3 M4 Ours
M1
M2 M3
M4 Ours
M1 M2 M3 M4 Ours
(a) CylinderFlow
(b) Flag
Figure 4. Ablation studies. We provide analyses of time-evolving hierarchies, anisotropic intra-level propagation, and learnable inter-level
feature propagation. The red dashed lines represent results from BSMS-GNI (Cao et al., 2023). Lower values indicate better performance.
Importance Weight Variance
Physical Quantity Variance
Figure demonstration of how the predicted anisotropic edge weights respond dramatic changes in physical quantities over time. Top: Visualizations of the variance in the generated anisotropic weights, calculated on adjacent edges. Bottom: Variance in physical quantities over time. The strong correlation between them highlights the AMP's ability detect significant patterns data.
5. Related Work
et al., 201 (),8, soft-body deformation and fluids (Li et al., 2019; Mrowca et al., 2018; Sanchez-Gonzalez et al., 2020; Rubanova et al., 2022: Wu et al., 2023), rigid body dynam- ics (Battaglia et al., 2016; Li et al., 2019; Mrowca et al., 2018; Bear al., 2021: Rubanova et al., 2022), and aero- dynamics (Belbute- et al., 2020; Hines Bekemeyer, 2023; Pfaff et al., 2021; Fortunato et al., 2022; Cao al., 2023). Among these, MeshGraphNets (Pfaff et al., 202 1) serves as representative, introducing general scheme for representing meshes as graphs and learning mesh-based dy- namics, inspiring subsequent works that focus on improving modeling capacity and computational efficiency.
Barning-basedd and GNN-based physical simulation. Recent advances have demonstrated that lcarning-basee approaches can efficiently tackle complex and high- dimensional physical simulation tasks, including fluid dy- namics (Zhu et al., 2024), structural analysis (Kavvas et al., 2018; Thai, 2022), and climate modeling (Kurth et al., 2018; Rasp et al., 2018; Rolnick al., 2022; Lam et al., 2023). These methods can be broadly categorized by their data representations: partial differential equations (Raissi al., 2017; 2019; Lu et al., 2019; Li al., 2021; Wang et al., 2021), particle-based systems (Li et al.. 2019: Sanchez- Gonzalez et al., 2020; Ummenhofer et al., 2020; Prantl et al., 2022), and mesh-based systems (Pfaff et al., 2021; Lino et al., 2022; Fortunato et al., 2022; Cao et al., 2023). The rapid inference and differentiable nature of these mod- els have facilitated range of downstream applications, such as inverse design (Wang& & Zhang, 2021; Goodrich et al., 2021; Allen et al., 2022; Janny et al., 2023). In particu- lar. Graph Neural Networks (GNNs) have emerged as a powerful tool for modeling physical systems across various domains, including articulated bodies (Sanchez-Gonzalez
Hierarchical GNNs for physical simulation. Hierarchi- cal GNNs leverage multi-scale graph structures (Lino et al., 2022: Han al., 2022; Fortunato al., 2022: Allen et al., 2023; Janny et al., 2023; Cao al., 2023; Yu al., 2024) reduce computational overhead by operating coarser rep- resentations and to facilitate long-range information propa- gation. For example, Sean-TTranffomrr-Gi. GMUUS (Han et al., 2022) employs uniform sampling for pooling, while Ea- gle (Janny et al., 2023) adopts a two-scale message pass- ing scheme with geometric clustering by preprocessing the
8

---

EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table One-step (RMSE-1) and 50-step (RMSE-50) rollout prediction errors on out-of-distribution (OOD) mesh resolutions. The average number of nodes in the test data is twice that of the training data, and the number of edges is three times greater.
RMSE-1 1 (x 10 B)
RMSE- 50 (x 10 3)
Model
Cylinder
Airfoil
Flag
Plate
Cylinder
Airfoil
Flag
Plate
MGN (2021)
1.0596
169.6
0.4215
0.0359
7.833
1829
55.96
0.2467
Lino et al. (2022)
25.893
144.4
0.8906
0.0475
65.21
1391
93.68
3,9845
BSMS-GNN (2023)
0.9177
202.3
0.6486
0.0474
2.097
1677
59.18
0.2554
HCMT (2024)
1.3864
205.5
1.0634
0.0354
7.541
2569
86.87
0,2957
EvoMesh
0.4855
126.7
0.5536
0.0368
1.077
812.5
58.29
0.3780
Table Generalization results across domains with various scales of input velocities. The domain gap is presented by the variance and norm of data training/test splits. Increase denotes the relative increase of the test data compared to the training data.
tions, such as DiftPool (Ying et al., 2018), TopKPool (Gao & Ji, 2019), SAGPool (Lee et al., 2019), and ASAPool- ing (Ranjan et al., 2020). These methods construct hier- archical representations of graphs by either learning soft cluster assignments or selecting top-ranked nodes based on learned importance scores. However, most of these meth- ods are primarily designed for static graphs and focus on global graph-level tasks, where unpooling or reconstruction of the original graph structure is not required. In contrast, mesh-based physical simulation demands fine- fine-grained local information and remporally-evolvine hierarchies for accu- rate modeling of physical dynamics. Our approach differs by constructing context-aware, time-varying graph hierar- chies tailored for mesh-based simulation, enabling flexible integration of global and local features and supporting dy- namic adaptation to changing physical conditions.
Cylinder
Airfoil
Split
Var
Norm
Var
Norm
Train
7.92
579.6
288.3
173.4
Test
13.43
826.3
827.4
180.6
Increase
64.5%
42.5%
186.9%
4.20%
Cylinder
Airfoil
Model
RMSE-1
RMSE-AII
RMSE-1
RMSE-AII
MGN
4.99x10-3
1,020
1.193
88.23
Lino et al.
5,60x10-3
1,415
3.226
410.5
BSMS-GNN
2058x10-3
0.251
1.035
30.32
Eagle
2.49x10-*
0.273
0.931
51.83
HCMT
7.35
1.047
1.697
63.18
EvoMesh
2.14x10-3
0.091
0.665
22.57
6. Conclusions and Limitations
Promotion
14.1 %
63.7%
28.6%
25.6%
In this paper, we introduced EvoMesh, a neural network that significantly advances the state-of-the-rt in mesh-based simulation. Our key innovation is adaptively creating the lemporally-evolving and context-aware graph structures of hierarchical GNNs through a differentiable node selection process. To this end, we proposed an anisotropic message passing mechanism to enhance the propagation of long-tern dependencies between distant nodes, aligning with the di- rected nature of significant dynamic patterns. Extensive experiments show that EvoMesh outperforms existing mod- els, especially those with fixed graph hierarchies, in both short-term and long-term predictions.
mesh structure. LayersNet (Shao et al., 2023) introduces a static, patch-based hierarchy for garment animation, sim- plifying interaction modeling via particle patches. More recent works (Lino al., 2022; Cao et al., 2023; Yu al., 2024; Garnier et al., 2024; Hy & Kondor, 2023) integrate hierarchical GNNs with U-Net architectures (Ronneherger al., 2015), using static multi-level structures for message passing. For instance, Lino et al. (2022) relies on manually defined grid resolutions, BSMS-GNT (Cao et al., 2023) proposes a bi-stride pooling strategy with enhanced edge connectivity, and HCMT (Yu et al., 2024) further refines the hierarchy using Delaunay triangulation and replaces message passing with graph attention. However, these meth- ods typically employ precomputed, static hierarchies and uniform feature aggregation, limiting their adaptability to dynamic physical environments. In contrast, our approach constructs context-aware, temporally evolving graph hier- archies with learnable anisotropic feature propagation, en- abling robust adaptation to diverse initial conditions and rapidly changing dynamics.
A potential limitation of this work is the need to improve the interpretability of the learned hierarchies. Additionally, we would consider incorporating physical priors into EvoMesh to enhance the model's robustness and generalizability, par- ticularly in resolution-free problem settings, which have been less explored in existing meshhbassed approaches.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (62250062), the Smart Grid National Science and Technology Major Project (2024
Differentiable graph pooling. A variety of differentiable and learnable graph pooling methods have been proposed to enable end-to-end training of hierarchical graph representa-

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
the Shanghai Municipal Science and Technology Major Project (2021SHZID2X01022, and the Fundamental Re- search Funds for the Central Universities.
Fortunato, M., Pfaff, T., Wimsherger. P., Pritzel, A., and Battaglia, P. Multiscale meshgraphnets. In ICML 2022 2nd Al for Science Workshop, 2022. URL https:// openreview net. forum/orumPP wet.tooooriid---.T......... 3d--G3RRI
Impact Statement
Gao, H. and Ji, S. Graph u-nets. In international conference on machine learning, pp. 2083-2092. PMLR, 2019.
In this work, we adhere to the highest ethical standards throughout all stages of research. No human subjects were involved, and no personal data used, ensuring full com- pliance with privacy and security protocols. All datasets employed are publicly available, minimizing concerns re- garding sensitive information exposure.
Garnier, P., Viquerat, J., and Hachem, E. Multi-grit graph neural networks with self-attention for computational mechanics. arXiv preprint arXiv:2409. /899, 2024.
Goodrich, C. P., King, E. M., Schoenholz, S. S., Cubuk, E. D., and Brenner, M. P. Designing assfmbnns ki- netics with differentiable statistical physics models. Pro- ceedings of the National Academy of Sciences, 118(10): e20240831 18, 2021.
We recognize that while physical simulation models can drive significant advancements, they also have the potential for misuse if applied inresponsibly. Thus, we emphasize the importance of thoughtful consideration regarding the con- text and ethical implications when deploying these models, particularly in high- applications such as healthcare, engineering, and environmental management.
Han, X., Gao, H., Pfaff, T., Wang, J.-X., and Liu, L. Predict- ing physics in mesh-reduced space with temporal atten- tion. In International Conference on Learning Represen- tations, 2022. URL https: openreview .ntt/ forum? (errm?dd=xctLdNN Www.ddNddCCC
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Appendix
A. Datasets
employ four established datasets from MGN (Pfaff et al., 2021); CylinderFlow, Airfoil, Flag, and Deforming
The CylinderFlow case examines the transient incompressible flow field around fixed cylinder positioned at different
locations, with varying inflow velocities.
The Airfoil case explores the transient compressible flow field at varying Mach numbers around the airfoil, with different angles of attack.
The Flag case involves a flag blowing in the wind on a fixed Lagrangian mesh.
The DeformingPlate case involves hyperelastic plates being compressed by moving obstacles.
The Cylinder Flow, Airfoil, and Flag datasets are each split into 1,000 training sequences, 100 validation sequences, and 100 testing sequences. The Deforming dataset split into 500 training sequences, 100 validation sequences, and 100 testing sequences.
We also consider more challenging dataset, Poper, where varying forces the four corners deform paper with time-varying Lagrangian mesh graphs, generated using the ARCSim solver (Narain et al., 2012; Wu et al., 2023). This dataset is divided into 500 training sequences, 100 validation sequences, and 100 testing sequences.
We present statistical details of all five datasets in Table and the input physical quantities in Table 7.
Table 6. Statistics of the CylinderFlow, Airfoil, Flag, DeformingPlate. and FoldingPaper datasets.
Dataset
Average # nodes
Average # edges
Mesh type
# Hierarchies
# Steps
CylinderFlow
1886
5424
triangle, 2D
600
Airfoil
5233
15449
triangle, 2D
100
Flag
1579
9212
triangle, 2D
400
DeformingPlate
1271
4611
tetrahedron, triangle, 2D
FoldingPaper
110
724
3
325
Table 7. Comparisons of the edge offsets and node inputs of different physical systems.
Dataset
Type
Edge offset ezj
Node Input
Outputs
Noise Scale
Fllw
Eulerian
Xzj, [Xay
Vs,
ú;
2e 2
Airfoil
Eulerian
Xıj. [Xg]
pe, Up, Th
p.,
P.
2c
lel
Flag
Lagrangian
Xiy-
( [ig
4
#; 3c
DeformingPlate
Lagrangian
X1ys [X [igl
&e n4
4.
2; 3e 3
FoldingPaper
Lagrangian
Xay, [x
d.
3; 3e
B. Model Implementation
We present model configurations of different physical systems below:
Edge offsets. X and stand for the mesh-space and world-space position. For an Eulerian system, only mesh position used for eij. while for Lagrangian system, both mesh-space and world-space positions are used. The edge offsets directly used as low- low-dimensional input the edge update function d" These features concatenated and fed into de without any transformation through an MIP or other encoding processes to generate a higher-dimensiooal representation.
Input and target of the physical term of node U,. e is the velocity, p is the density, P the absolute pressure, and the dot à a;-1
stands for temporal change for variable stands for the node type of t;. Random Gaussian noise is added to the node input features to enhance robustness during training (Pfaff et al., 2021; Fanchez-Gonzaler et al., 2020; Cao et al., 2023). All the variables involved are normalized to zero-mean and unit variance via preprocessing. The preprocessed physical term is fed to the encoder to transform into a high- figeh-immnsionall representation.
The encoder, decoder, node update function o², and edge update function 6f all utilize two-layer MLPs with ReLU activation and a hidden size of 128. Similarly, the importance weight network 6" in AMP implemented using a two-layer ML.P.
13

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table 8. Qualitative results of model variants of EvoMesh and the baseline model.
RMSE-1 (x10-3)
RMSE-All (x 10
Model
Cylinder
Flag
Cylinder
Flag
BSMS- (Cao et al., 2023)
0.2263
0,5080
16.98
168.1
Unlearnable (Mf)
0.1995
0.4804
9.621
121.1
carnable (M2)
0.1695
0.4666
8.317
109.9
Uniform- catamm-Annistropee-LLaamee (M3)
0.2019
0.3999
9.357
145.27
Dynamiee-Anisottpii--Uueeee Unleemmabe (M4)
0.1631
0.3538
7.793
82.65
EvoMesh
0.1568
0.3049
6.571
76.16
Figure 6. Mesh visualization Flag Dataset. Original mesh (left), sub-level graph after differentiable node selection with K-hop enhancement with K = 2 (middle), and sub-level graph after node selection without K hhop enhancement (right).
LayerNorm is applied to the MLP outputs, except for the decoder and the importance weight network. We set for edge enhancement, which is aligned with the setting of BSMS-GNT (Cao et al., 2023). In the Gumbel-Softmax for differentiable node selection, temperature annealing decreases the temperature from
0.1 using a decay factor of y 0.999, encouraging exploration of hierarchies while gradually refining their selection to ensure stability. EvoMesh is trained with Adam optimizer, using exponential learning rate decay from 10-4 to 10-6, All experiments are conducted using 4 Nvidia RTX 3090. We mainly build EvoMesh based on the released code of BSMS-GNN (Cao et al., 2023).
C. Additional Results
C.1. Ablation Study
In Sec. 4.3, we compare different variants of our EvoMesh model against the baseline, to evaluate the effectiveness of (i) dynamic hierarchy construction based the input mesh topology and physical quantities, (ii) anisotropic intra-level feature propagation, (iii) learnable inter-level feature propagation. The variants investigate include:
Saitc:B-ssrkkk--iissttiiieeeeeed (MI).
StaiccBissrde/AhiioonppieLLaaa (M2),
Unijomm-Anisotropi-LLLamaaee (M3),
Dyaamic-nnisorppi--UUleeae Unleraaaaaeeeeeeeebbblbeeeeb (M4).
In this ablation study, we utilize static graph hierarchy preprocessed using bi-stride pooling described in the BSMS-GNN paper (Cao et al., 2023) for static hierarchy variants, along with non-parametric intra-level aggregation function from previous works (Pfaff et al., 2021; Cao al., 2023). Additionally, BSMS- GNN employs unlearnable node degree metries to generate lnter-level aggregation weights, which convolve features based on the normalized node degree for inter-level propagation. We show the quantitative RMSE values of Figure 4 in Table 8.
C.2. Alternative Implementation of AMP
We evaluate the effectiveness of the AMP module by comparing it with alternative attention-based designs (Veličković et al., 2018; Vaswani et al., 2017).
Unlike standard attention mechanisms such as Graph Attention Networks(GAT) (Veličković et al., 2018), AMP enables differentiable dynamic hierarchy construction- an essential capability for modeling evolving physical systems. While
14

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table 9. Results of different implementations of AMP module.
Cylinder
Flag
AMP Implementation
RMSE-1 (x10-3)
RMSE-All (x10-3)
RMSE-I IMSE-I(x100)
RMSE-AIl (x10
EvoMesh-GATTonn
0.2025
10.253
0.3009
106.2
EvoMesh- CooMesshCrossAttentioo
0.1881
8.356
0.3650
75.12
EvoMesh
0.1568
6.571
0.3049
76.16
AMP and GAT both compute importance weights, their usage differs substantially: GAT uses these weights to aggregate node features, whereas AMP directly applies the predicted weights to edge features and reuses them for inter-level feature propagation. This dual mechanism allows AMP to jointly support dynamic hicrarchy learning and physics modeling via end-to-end training. To assess the benefits of AMP over GAT, we replace AMP with GATConv (using a single attention head) in EvoMesh. As shown Table 9, the A.MP-based model consistently outperforms the GAT-based variant across datasets, highlighting the importance of dynamic hierarchy construction.
We further explore the flexibility of AMP by implementing it with cross-attention mechanism (Vaswani et al., 2017) in place of the default MLP. Specifically, edge features eij are refined by attending to node features through scaled dot-product attention:
Cager CrossAttennionn = Vi. K e.g. = ēit),
and the resulting attention scores are reused for inter-level propagation. Experimental results Table 9 show that while the cross-attention implementation competitive, the original MLP-based AMP achieves better overall accuracy. This suggests that the edge features é.j already encode relevant node information, making the explicit inclusion of v; through attention redundant and potentially less efficient.
C.3. Edge Enhancement
When constructing the lower-level graph Gi+1 based the selected nodes, the edges ε1ạ1 are formed by connecting these nodes using the original edges E, from the previous graph. However, this approach may lead disconnected partitions, as observed in previous work (Lee et al., 2019; Cao et al., 2023; Gao & Ji, 2019), and illustrated in Figure 6. To address this issue, we enhance the connectivity of & by incorporating K-hop edges during the edge construction process. We investigate the impact of different K values, specifically K 2, 2,3,4, 4, on the Flag dataset. The results are presented in Table 10,along with comparisons of the computational efficiency.
Notably, K yields the lowest RMSE across all conditions (RMSE-1, RMSE-50, and RMSE-all), indicating superior performance compared to higher K values. Despite the performance decline observed with K and K 4, they still outperform the baseline results, indicating the effectiveness of dynamic hierarchical modeling and anisotropy message passing.
Table 10. Results for different values of K in edge enhancement. Here, K 1 denotes directly using edges of selected nodes from previous graph levels. Training time and memory usage are measured with a batch size of 32, while inference time and memory are evaluated with batch size of 1.
Training
Infer
RMSE-1 1 (x
RMSE-AIl (x
Time (ms)
vRAM (GBs)
Time (ms)
vRAM (GBs)
=
0.3296
100.1
31.57
14.75
23.60
1.17
K 2
0.3049
76.16
33.67
16.53
26.33
1.24
K
0.3380
86.84
34.67
18.49
33.21
1.25
K 0.3510
105.4
35.27
18.76
32.25
1.28
C.4. Hyperparameter Analyses on Number of Hierarchies
We conduct an ablation study to assess the impact of varying numbers of hicrarchies on model performance. The results from the CylinderFlow dataset, illustrated in Figure 7, demonstrate that EvoMesh consistently outperforms BSMS-GN across all tested numbers graph hierarchies. Both models show improved performance with increased hierarchy depth up to 7, indicating that deeper levels help capture more complex interactions and thus enhance accuracy. However, slight
15

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
BSMS- GNN
EvoMesh
40
0.35
35
0.30
RMSE-1 (x11-11
0.25
RMSE-AII (x10-1
30
25
0.20
20
0.15
15
0.10
10
0.00
#Hierarchy
#Hierarchy
Figure 7. Model comparisons different numbers of graph hierarchies CylinderFlow dataset.
BSMS SSMS-GNN
EvoMesh
Bi-tride Pooling
Di#fSELBCT
High error
50%
40%
30%
10%
Low
0%
1//
Error distribution for BSMS-GNN and EvoMesh
Ratios of challenging nodes
Figure 8. (a) Error maps, where nodes with the top 10% of errors in each model's predictions are marked in yellow and referred to "challenging nodes". (b) EvoMesh retains more challenging nodes coarser graph hierarchies to capture multi-scale dependencies.
performance decline is observed at level 9, which may suggest the onset of overfitting. Overall, the dynamically learned hierarchies in EvoMesh are shown to be more effective compared to the predefined static hierarchies used in BSMS-GNN,
C.5. Effectiveness of Evolving Hierarchies
From Figure 4, by comparing EvoMesh vs. M2 and M4 vs. MI, we observe the advantages of learning adaptive and temporally evolving hierarchical graph structures. These results highlight the significance of adaptively modeling interactions in -eeeext-dependent graphs. To better understand how EvoMesh constructs dynamic hierarchies, we visualize the distribution of nodes with the top 10% prediction errors Figure 8(a). Accordingly Figure 8(b), we observe that EvoMesh retains a higher proportion of "ccaallengnn" nodes in the coarser message passing levels, enabling our model to capture multi-scale dependencies more effectively, especially in areas where finer message passing levels struggle. In contrast, the predefined static hierarchies in the Bi-stride pooling baseline are data-independent and may inevitably overlook modeling long-range relations surrounding these pivotal nodes, even though they typically present higher errors than those in EvoMesh.
C.6. Stability Analysis
Given the inherent randomness introduced by the Gumbel-Softmai sampling process in DiFfSELECT, evaluated the stability of EvoMesh by running the trained model on the test set in three independent trials. We report the mean and standard deviation of the prediction errors in Table 11. Despite the stochastic nature of the node selection process, the results show a very small standard deviation, demonstrating that EvoMesh reliably constructs stable and consistent dynamic hierarchies. This stability can be attributed to the SESELEECT operation, where the node update module d" generates probabilities for retaining nodes the next-level graph based on anisotropic aggregated edge features. The Gumbel-Softme
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table 11. Evaluation of EvoMesh with three independent tests.
Cylinder
Airfoil
Flag
Plate
R (x10-3)
0.1506 +3.6B4
36.27 +5.76-4
0.2741 +2,4B-2
0.0263 45.6E-6
RMSE-AIl (x10-3)
6.317 20.33
2018 4130
68.66 =2.9
1.327 20,02
Table 12. Full quantitative results over three training seeds.
Cylinder
Airfoil
Flag
Plate
RMSE-1 (x10-²)
MGN (2021)
0.3046 +1.08E
77.38 +1.34E+1
0.3459 26.34B-2
0.0579 #2.64BB3
Lino et al. (2022)
3,9352 +11.3E22
85.66 #0.351+1
0,9993 +244B.2
0.0291 #0.19B.3
BSMS- GNNS (2023)
0,2263 #4.39E-2
71.69 #1.41++1
0,5080 +0.48B-2
0.0632 #14,3B-3
Eagle (2023)
0.1733 +3.02B-2
0.385 =1.03+1
51.55 =0.87B-2
0,0392 20.52B-3
HCMT (2024)
0.9190 261.2E-2
48.62 20.51E+1
0.4013 =1.76B-2
0.0295 #3.45B-3
EvoMesh
0.1568 =0.94R-2
41.41
#0.664+1
0.3049 #6.34E-2
0.0282 #2.65-3
RMSE-AII (x10-3)
MGN (2021)
59.78 #2.00++
2816 11.9E+2
115.3 =1.30++1
3.982 11.14B-2
Lino et al. (2022)
27.60 80.86+1
2080 20.39
118.2 20.58++1
2.090 213.26-2
BSMS-GNN (2023)
16.98 40. 12E+1
2493 21.70
168.1 20.65++1
1,811 20,425-2
Eagle (2023)
20.05 +0.67E+1
2344 +2.1 IB+2
127.7 #0.88++1
7.797 #2.356-2
HCMT (2024)
23.59 +1.38+1
3238 +3.62E42
90.32 =0.50E+1
2.468 +42,4E-2
EvoMesh
6.571 #0.0GE+1
2002 #1.02E+2
76.16 #1.30++1
1.296 #1.14--2
technique, coupled with temperature annealing, enables differentiable and stable node selection across hierarchy levels. As result, the dynamic hierarchies are constructed in a manner that is not only consistent but also optimized for long-range dependencies. Morcover, the prediction errors from EvoMesh are significantly smaller than those of the baseline models, underscoring the robustness and reliability of the model, even with its dynamic node selection mechanism.
C.7. Full Results over Multiple Training Seeds
In Table in the main manuscript, we report the mean results calculated over three random seeds. In Table 12, we provide full comparisons between our model and the baseline models, including standard deviations.
C.8. Computation Efficiency
We evaluate computational efficiency based on four criteria: training hours, inference time per step, and the total number of model parameters. The results are presented in Table 13.
C.9. Rollout Showcases
Figures 9 showcase rollout error maps for the Airfoil, Flag, and DeformingPlate datasets. EvoMesh exhibits much lower rollout errors than the baseline models.
C.10. Constructed Dynamic Hierarchies
We visualize the constructed context-aware and temporally evolving hierarchies in Figure 10. We can see that the constructed hierarchies evolve as the input context changes and the evolving graph structures align with high-intensity regions. More visualizations of the evolution of the graph structure across the sequence are included in the supplementary material.
D. Baseline Details
We compare EvoMesh with following competitive basclines: (1) MGN (Pfaff et al., 2021) which performs multiple message passing on the input high-resolution mesh topology: (2) Lino et al.(Lino et al., 2022), which uses manually set grid resolutions and spatial proximity for graph pooling; (3) BSMS DSMS-GNN (Cao et al., 2023), which uses predefined bi-stride pooling prior as preprocessing to generate static hierarchies on same mesh topology; (4) Eagle (Yu et al., 2024), which uses
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
Table 13.
The detailed measurements of computation efficiency for EvoMesh and baseline models.
Measurements
Dataset
BSMS-GNN
HCMT
EvoMesh
Cylinder
37.11
80.60
35.96
Airfoil
79.09
114.32
75.45
Training cost (hrs)
Flag
18.27
66.70
17.14
Plate
39.80
99.84
41.85
Cylinder
16.55
79.52
21.79
Airfoil
38.04
106.34
58.84
Infer time/step (ms)
Flag
17.18
85.87
26.33
Plate
28.44
100.78
47.45
Cylinder
2.05M
2.03M
2.66M
Airfoil
2.58M
2.03M
2.27M
# Parameter
Flag
2.06M
2.03M
2.67M
Plate
2.87M
2.03M
3.20M
a two-scale hierarchical message-passian paasgin- approach, downscaling mesh resolution via geometric clustering of mesh positions; (5) HCMT (Yu et al., 2024), which uses Delauny triangulation based on bi-stride nodes and adopt attention mechanism to enable non-uniform feature propagation. The architecture details of the compared models follows:
MGN. In MGN, we use 15 message passing steps in all datasets. The encoder, decoder, node update function, and edge update function are configured the same way as in our model.
Lino et al.
We use the four-scale GNN structure proposed in the work of Lino et al. (2022). The edge length of the smallest cell for each dataset is 1/10 of the average scene size, with each lower scale doubling in size, We follow its original paper to use 4 message passing steps at the top and bottom levels and two for the others.
BSMS-GNN. We use the same number of graph hierarchies in EvoMesh and as in BSMS BSMS-GNN. We the minimum average distance as the seeding heuristic for the BFS search recommended in its original paper. The multi-level building is processed in one pass. The inter-level propagation uses the normalized node degree to convolve features from neighbors to central nodes. The encoder, decoder, node update function, and edge update function are set up the same way as in our model. We perform one message passing step at each graph level.
Eagle. We use the same-size KMeans algorithm in the preprocessing step with cluster size of 10. The encoder consists of stacked GNN layers. The graph pooling module comprises a single-layer gated recurrent unit followed by a single-layer MLP. The hidden dimension is set to 128 across all datasets. The attention module includes sequential attention blocks, each with single attention head. A final layer normalization is applied after the last attention block. The decoder shares the same architecture that used in EvoMesh.
The hidden dimension and the number of attention heads in the HCMT block are set to 128 and 4, respectively, We use the same number hierarchies as in EvoMesh. For the Cylinder and Airfoil datasets, due to the presence of hollow sections the mesh, we do not apply Delaunay triangulation for remeshing. Instead, we use edge connections generated through bi-stride pooling. Like EvoMesh, we use a single message passing step at each graph level.
HCMT.
Notably, the node encoder, decoder, node update function, and edge update function of MGN, Lino et al., BSMS GNN, HCMT and Eagle have the same network architecture as those in EvoMesh. To reduce the number network parameters, we avoid separately encoding the edge offset eij. Instead, we concatenate with the node latents and use this combined input for the edge update function to compute ëij-
All models are trained using the Adam optimizer for 1M steps, with an exponential learning rate decay from 10-4 to 10-6 and decay rate of 0.79 from the first 500K steps. The batch size is set to 32.
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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
EISMS GINN
High emorr
EvoMesh
(a) Airfoil
BSMS-GNN
High error
EvoMesh
Low error
(b) Flag
BSMS-GNN
High error
EvoMesh
Low error
(c) DeformingPlate
Figure Showcases of rollout prediction maps Airfoil, Flag and DeformingPlate dataset.
19

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EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions
t=1
t= 400
91-2
91=3
-1=4
G1=5
G1-6
Figure 10. Row 1: The velocity field from the true data on the CylinderFlow dataset. Row 2-6: The temporal difference of the velocity fields between adjacent time steps alongside the constructed coarser-level graphs.
20


Reviews:
[{"review_id": "ViGAurruSw", "reviewer": "Reviewer_Mnuf", "summary": "The paper presents EvoMesh, a graph neural network for mesh-based physical simulations that dynamically learns evolving graph hierarchies instead of relying on fixed structures. Using anisotropic message passing, it adaptively selects nodes based on physical inputs, improving long-range dependency modeling. Experiments show improvement over fixed-hierarchy GNNs across multiple simulation datasets.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the insightful comments. \n\n> Q1: Novelty compared to BSMS-GNN.\n\nWhile 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**:\n\n**Core Innovation: Adaptive vs. Static (preprocessed) Hierarchies** \n- *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. \n- *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. \n    \n>Q2: EvoMesh vs. EAGLE (Janny et al.).\n\nIn methodology,\n- *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.**\n- *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.\n\nThe 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.\n\n| | RMSE-1 ($\\times 10^{-2}$)| | RMSE-All ($\\times 10^{-2}$) | |\n|-|-|-|-|-|\n|| Cylinder| Flag | Cylinder| Flag |\n|EAGLE| 0.1733|0.3805| 20.05|127.7|\n| EvoMesh | 0.1568| 0.3049 | 6.571| 76.16 |\n\n> Q3: AMP vs. GAT\n\nThe 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:\n- GAT applies computed weights to aggregate node features through weighted summation;\n- Ours: (a) directly applies predicted weights to edge features, (b) leverages weights for inter-level feature propagation.\n\nThis 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.\n\nTo 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. \n\n| | RMSE-1 ($\\times 10^{-2}$)| |RMSE-All ($\\times 10^{-2}$) ||\n|-|-|-|-|-|\n||Cylinder| Flag |Cylinder| Flag|\n| EvoMesh-GATConv|0.2025|0.3009|10.253| 106.2|\n| EvoMesh |0.1568|0.3049|6.571|76.16|\n\n> Q4: There are inconsistencies in the choice of methods used for comparison in different experimental setups. \n\n**Table 2:** Here, we present the results of MGN, showing that EvoMesh achieves superior performance. \n\n|| RMSE-1 ($\\times 10^{-2}$)||||RMSE-All ($\\times 10^{-2}$) ||||\n|-|-|-|-|-|-|:-:|:-:|:-:|\n|| Cylinder | Airfoil |  Flag  | Plate| Cylinder |Airfoil|Flag|Plate|\n|MGN|  0.3046|77.38| 0.3459 | 0.0579 |59.78| 2816|115.3|3.982|\n|EvoMesh|0.1568| 41.41| 0.3049 | 0.0282 |6.571 |2002|76.16| 1.296 |\n\n**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. \n\n**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.\n\n||RMSE-1 ($\\times 10^{-2}$)|| RMSE-50 ($\\times 10^{-2}$) | |\n|-|-|-|-|-|\n| |Cylinder|Airfoil|Cylinder|Airfoil|\n| MGN (**New**) | 1.0596|169.577|7.8332|1829.1|\n| Lino et al. (**New**) |25.8930|144.35|65.207|1299.4|\n| BSMS-GNN|  0.9177  |202.30 |2.0971|1677.3|\n| EvoMesh |  **0.4855**  | **126.70**|**1.0771**|**812.47**|", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Make sense.", "theoretical_claims": "Yes, seems correct", "experimental_designs_or_analyses": "Yes, seems reasonable.", "relation_to_broader_scientific_literature": "The paper advances GNN-based physical simulations by replacing fixed graph hierarchies with adaptive, time-evolving structures via differentiable node selection.", "essential_references_not_discussed": "The related work seems include sufficiency references.", "other_strengths_and_weaknesses": "Overall, the proposed approach appears reasonable, and the results show improvement over baseline methods. However, a primary concern is that the novelty of the method seems limited, as it closely resembles the BSMS-GNN in overall structure.", "other_comments_or_suggestions": "Please find comments in the question section.", "questions_for_authors": "1. Could the authors provide more detail about the differences or novel aspects of the proposed approach compared to BSMS-GNN?\n2. BSMS-GNN’s main contribution is its computational efficiency, which may come at the expense of prediction accuracy. In contrast, methods like EAGLE (Janny et al.) emphasize multi-scale message-passing algorithms and typically achieve better predictive performance.\nThe authors might consider comparing their approach to such methods.\n3. The anisotropic message-passing layer appears similar to a variant of GAT. Could the authors explain this resemblance in more detail?\n4. There are inconsistencies in the choice of methods used for comparison in different experimental setups. For example, why is MGN not included in Table 2 but only in Table 3? Likewise, why are methods like MGN and Lino et al. (2022) excluded from Table 4 comparisons?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y67rVWwB3h", "reviewer": "Reviewer_TDvH", "summary": "The paper presents a novel hierarchical graph network architecture in which the hierarchy is determined in a data-driven manner. Additionally, it introduces an anisotropic message-passing step that incorporates an attention mechanism into the aggregation process. With these innovations, the proposed EvoMesh architecture outperforms various hierarchical baselines across multiple challenging tasks by a significant margin.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We greatly appreciate the reviewer’s valuable comments.\n\n> Q1: Details of DiffSELECT. \n\nWe 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.\n\nThe formulation is as follows:\n\n$$\nz\\_i^l=\\text{Gumbel-Softmax}(\\mathbf{l}\\_i^l)=\\frac{\\exp\\left((\\log p\\_{i,0}^l+g\\_{i,0}^l) / \\tau \\right)}{\\sum_{k=0}^1 \\exp\\left((\\log p_{i,k}^l + g_{i,k}^l) / \\tau \\right)},\n$$\n\nwhere:  \n- $\\mathbf{l}\\_i^l=(\\log p\\_{i,0}^l, \\log p\\_{i,1}^l)$ represents the logits for node $v_i$ at layer $l$,  \n- $g_{i,k}^l$ is the Gumbel noise sampled independently for each node and category,  \n- $\\tau$ is the temperature parameter controlling the smoothness of the sampling process.  \n\nThis process allows for a **differentiable approximation of discrete node selection**. We will include this explanation in the revised manuscript.\n> Q2: On FeatureMixing.\n\nFeatureMixing 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.\nThe whole upsampling process works as follows:\n1. **EXPAND**: Coarser-level features ($\\mathbf{F}_L$) are upsampled to the resolution of level $L-1$.  \n2. **Refinement**: A message-passing step is applied to refine the upsampled features ($\\text{EXPAND}(\\mathbf{F}_L)$):  \n     $\\mathbf{\\tilde{F}}_L = \\text{MessagePassing}(\\text{EXPAND}(\\mathbf{F}_L))$\n3. **Fusion**: The refined coarser-level features ($\\mathbf{\\tilde{F}}\\_L$) are combined with the current level’s intra-level features ($\\mathbf{F}\\_{L-1}$):  \n   $\\mathbf{F}\\_{L-1}^{\\text{mixed}} = \\mathbf{F}_{L-1} + \\mathbf{\\tilde{F}}_L$\n\nWe will improve the clarity of FeatureMixing in the revision.\n> Q3: Multi-step rollouts on OOD Mesh Task. \n\nWe 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. \n> Q4: Qualitative showcases of paper-folding.\n\nVisualizations 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.\n> Q5: Gaussian input noise.\n\nThe 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.\n\n[1] Learning to Simulate Complex Physics with Graph Networks.\n> Q6: Typo in Figure 2 and Figure 8a.\n\n\"DHMP\" indeed refers to EvoMesh, and we'll correct this in the revised manuscript.\n\n> Q7: Why similar runtime to static hierarchy models? Any optimizations to improve efficiency?\n\nEvoMesh 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.\n\nNotably, 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.\n\n|| CylinderFlow (#nodes, #edges)||(#nodes, #edges)||\n|-|-|-|-|-|\n|Layer|Static |EvoMesh| Static|EvoMesh|\n|1|(949, 8128)|(851, 5322)|(791, 7130)|(696, 4347)|\n|2|(481, 6333)|(433, 3150)|(402, 5728)|(438, 3975)|\n|3|(247, 5308)|(229, 2021)|(208, 5210)|(230, 2681)|\n|4|(133, 4645)|(115, 1172)|(118, 5654)|(168, 2178)|\n|5|(81, 4285)|(50, 484)|(77, 4923)|(44, 588)|\n|6|(41, 1654)|(31, 358)|(40, 1537)|(29, 174)|\n\n> Q8: Code release.\n\nWe'll release both the code and datasets upon acceptance.", "claims_and_evidence": "All claims are backed up by evidence.", "methods_and_evaluation_criteria": "## Method\nI did not fully understand the details of the DiffSELECT operation. The model needs to sample nodes during training, and through reparameterization, the gradient somehow propagates through this process. However, the exact mathematical formulation of Eq. (6) remains unclear from reading only the current paper. Providing a precise formulation—perhaps in the appendix—would improve the paper, as this is a crucial component enabling the data-driven hierarchical selection.\n\nSimilarly, the motivation behind FeatureMixing is not entirely clear. Is it intended to combine features from a coarser level with those from the current level? A more detailed explanation would help clarify this aspect of the architecture.\n\n## Evaluation\nThe evaluation appears thorough, covering a range of ablations and baselines. However, I missed a complete rollout from the OOD mesh resolutions—or, if a full rollout is too noisy, a multi-step rollout could still be informative. Additionally, a qualitative showcase of the paper-folding task would be valuable. Since, to my understanding, this is a new task and dataset, visualizing the results would help illustrate the model's performance more effectively.", "theoretical_claims": "No theoretical claims were made.", "experimental_designs_or_analyses": "The selection of only three seeds seems somewhat low. However, the standard deviations suggest that significant variations between different training runs are unlikely.", "relation_to_broader_scientific_literature": "This work builds upon hierarchical graph network simulators, introducing novel aspects in both anisotropic message passing and data-driven hierarchy construction. While Attentive Graph Neural Networks share some similarities with the anisotropic message passing steps, their structure differs significantly, making this a clear contribution.", "essential_references_not_discussed": "Not the case.", "other_strengths_and_weaknesses": "The introduction effectively communicates the problem, the proposed method, and its broader context. Additionally, the analysis and evaluation are thorough. I enjoyed reading the paper. However, as mentioned earlier, some details in the method description—particularly regarding the DiffSELECT operation and the FeatureMixing method—are missing. Expanding on these aspects would improve clarity.", "other_comments_or_suggestions": "- In Figure 2, the authors speak from “DHMP”, I suspect that this is an older acronym of the EvoMesh method? \n- Same goes for Figure 8a.", "questions_for_authors": "- According to the appendix, the runtime was similar to that of static hierarchy models. Why is this the case? Given that hierarchy construction is typically costly, did the authors employ any optimizations or tricks to improve efficiency? Clarifying this would be helpful.\n- Will the authors release their code and datasets? Given the complexity of the data-driven hierarchy implementation, making the code available would help increase adoption of the method. Additionally, I would appreciate the opportunity to compare it more easily in my own work. I consider raising my score to Strong Accept if that is the case.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Pp4zZA7Qvi", "reviewer": "Reviewer_2MCy", "summary": "This paper proposes anisotropic message passing (AMP) with hierarchical structure for mesh-based simulation. Specifically, the AMP enables\tGNN to predict the edges’ weights before aggregation. The hierarchical graphs are constructed dynamically given the predicted importance. Experiments on five different domains show that the proposed method outperforms baselines.\n\n## Update after rebuttal\nThe author has addressed my concerns, so I remain positive about this paper and will keep my original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the insightful comments. Below, we address each comment point-by-point.\n\n> Q1: Missing reference -- LayersNet.\n\nThank 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:\n- *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.\n- *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.\n- *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**.\n\n> Q2: More visualized results.\n\nWe 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.\n\n> Q3: Is the proposed method bring extra computational overhead?\n\nWe have provided a comprehensive efficiency comparison in **Appendix Table 13**, covering training time, inference time, and model size.\n\nAdmittedly, 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**.", "claims_and_evidence": "The claims are clear.", "methods_and_evaluation_criteria": "Make sense.", "theoretical_claims": "N/A. The claims are generally correct.", "experimental_designs_or_analyses": "The experiments are sufficient. Visualized results on all domains are provided either in the main text or appendix. Both the mean and standard deviation are reported in the quantitative results.", "relation_to_broader_scientific_literature": "Please refer to \"Essential References Not Discussed\".", "essential_references_not_discussed": "The paper should discuss LayersNet [1], which also deals with learning-based simulation. The relation and similarity should be discussed. For example, LayersNet also applies hierarchical structure for mesh-based simulation; the attention mechanics and the AMP's selection share similarity.\n\n[1] Shao, et al. Towards Multi-Layered 3D Garments Animation, ICCV2023.", "other_strengths_and_weaknesses": "The experiments are sufficient and convincing. It would be better if the author could provide more visualized results, such as videos with more frames.", "other_comments_or_suggestions": "Please refer to \"Questions For Authors\".", "questions_for_authors": "Here I summarize and list my questions:\n \n1. Is the proposed method bring extra computational overhead? What is the efficiency of this method?\n2. Missing reference in \"Essential References Not Discussed\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LMl7W7a50M", "reviewer": "Reviewer_oo3z", "summary": "This paper presents a hierarchical GNN method for learning PDEs. In contrast to many other such methods, it uses differentiable graph construction, so the message passing hierarchy is learned end-to-end.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1: Differentiability for $G_{l>1}$.\n\nEvoMesh avoids using fully connected graphs for ${G}_{l>1}$. Instead, it ensures differentiability through:\n1. Node selection: Differentiable via Gumbel-Softmax (L183-197).\n2. Edge construction: Edges are formed based on the selected nodes and the connectivity of the original graph, further enhanced by $K$-hops (L199-219).\n3. 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.\n\nAs 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.\n\n> Q2: Novelty beyond AMP.\n\nWe 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.\n* **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.\n* **Role of AMP**: AMP enables anisotropic intra-level propagation to determine node selection; and its predicted importance weights facilitate inter-level feature propagation.\n\n> Q3: Efficacy of AMP.\n\nThe efficacy of AMP is demonstrated in two ways: \n- 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.\n- 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:\n \n||RMSE-1($10^{-2}$)||RMSE-All($10^{-2}$)||\n|-|-|-|-|-|\n||Cylinder|Flag|Cylinder|Flag|\n|Isotropic|0.4521|0.3752|40.59|118.9|\n|EvoMesh|0.1568|0.3049|6.571|76.16|\n\n> Q4: Number of levels/downsampling ratio.\n\nNumber 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.\n\nDownsampling 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/.\n\n|Node retention|Cylinder||Flag||\n|-|-|-|-|-|\n|Layer|Static|EvoMesh|Static|EvoMesh|\n| 1| 0.5029| 0.4510| 0.5009| 0.4409|\n| 2| 0.5069| 0.5090| 0.5082| 0.6288|\n| 3| 0.5147| 0.5294| 0.5174| 0.5251|\n| 4| 0.5356| 0.5031| 0.5654| 0.7323|\n| 5| 0.5131| 0.4306| 0.5582| 0.2626|\n| 6| 0.5039| 0.6331| 0.5129| 0.7182|\n\n> Q5: vs. Uniform node-sampling.\n\nWe present the required comparisons below. The baseline model uses an equal number of uniformly sampled nodes at each level.\n \n||RMSE-1($10^{-2}$)||RMSE-All($10^{-2}$)||\n|-|-|-|-|-|\n||Cylinder|Flag|Cylinder|Flag|\n|Uniform sample|0.3019|0.3999|9.357|145.27|\n|EvoMesh|0.1568|0.3049|6.571|76.16|\n\n> Q6: MGN reports lower RMSE for CylinderFlow/Airfoil/Plate.\n\nMost 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. \n\n> Q7: AMP vs. GAT. \n\nPlease see our response to **Reviewer Mnuf Q3**.\n\n> Q8: Related work -- DiffPool.\n\nWe'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.\n\n> Q9: Ratios of retained points.\n\nWe 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. \n\nIs 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.", "claims_and_evidence": "Yes. The main claim is better accuracy compared to multi-scale GNNs with fixed hierarchy, and the paper does provide evidence for this.", "methods_and_evaluation_criteria": "Benchmark datasets are fine for the task. Although I probably would have included a dataset with fast-acting/long-range physics, for which single-level GNNs perform poorly and hierarchies are necessary.\n\nI'm not really convinced by the motivation for AMP. Oversmoothing is particularly a problem for long-range information retrieval tasks, and much less of an issues for learned PDE solvers. Those generally provide geometry information in edge features, plus by definition in PDEs information does spread locally. Summation aggregation is also actually much closer to the underlying computations in physics than attention: see force superposition in Lagrangian and differential kernel computation in the Eulerian view.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "There's a few important comparisons & baselines missing.\n\n1. I didn't see any test for the efficacy of AMP-- there should be an ablation of AMP against 'isotropic' summarization\n2. It would be very nice to analyze/discuss number of levels/downsampling ratio for EvoMesh & the baselines. How different are the hierarchies used by the baselines or the 'static' ablation? Is spatial location important, i.e. how does EvoMesh compare against a hierarchy with same amount of nodes per level, but sampled uniformly across the domain? Etc.\n3. There should be a comparison against a single-level GNN (probably MGN, since most of the datasets are taken from that paper). I took a glance at the RMSE number in EvoMesh and the MGN paper, and RMSE reported for MGN seem actually lower for CylinderFlow, Airfoil, Plate? (it's possible there are differences in how those are computed though).", "relation_to_broader_scientific_literature": "AMP looks _very_ similar, if not identical, to Graph Attention Networks (Velickovic et al). So either this should be removed as a contribution, or if the authors believe that it is sufficiently different, GAT needs to be discussed & compared to, and notable differences pointed out.", "essential_references_not_discussed": "GAT (see above).\nThere is also a bunch of literature on differentiable graph pooling (e.g. Ying et al's DiffPool), which should probably be discussed in related work.", "other_strengths_and_weaknesses": "Strengths: \nI like the premise of the paper-- there's clear evidence that for GNNs, multi-scale hierarchies are important for many problems in physics. But the optimal way to construct these hierarchies is still TBD, and being able to perform differentiable end-to-end graph construction would be very impactful.\n\nWeaknesses:\n1. As mentioned above, part of the contribution may not be novel.\n2. In a way, node selection is the easy part. The paper needs to explain how it gets around the issue of differentiable edge graph construction -- fully connected graphs for gl>1 are very expensive, which is often cited as the reason that fully differentiable end-to-end graph construction is not viable.\n3. The paper needs a bit more thorough evaluation (see missing experiments above)", "other_comments_or_suggestions": "### After rebuttall ###\n\nThe additional experiments (especially vs MGN, GAT) strengthen the paper, and hence I have improved my score. A paper with all the additional experimental results and updated explanations might meet the bar for ICML.\n\nBut there's still a bunch of unanswered questions around differentiability: differentiable sampling with GS for graph construction seems to be rarely done, yet somehow this paper seems to make it work without any major new innovation. It was my understanding that with GS sampling gradient quality wouldn't be sufficient, and most papers in this space I'm aware of use soft sampling with all edges active.\nAlso I'm not sure the role of AMP is properly addressed-- in the rebuttal the authors pose the theory that while it might not necessarily help with physics it helps with node selection. It would be nice to properly investigate this (and the relation to GAT).\n\nSo while I wouldn't be opposed to having the paper published, I do feel that with taking a step back, a performing a major revision, this could be a much better paper, which provides much more value to the ML community.\n\n###", "questions_for_authors": "- From fig 10, it looks like the downsampling ratio is around ~2 per level. Is there anything in the model that encourages a particular ratio, or put pressure to use less points? What's stopping the model from retaining every single point?\n\n- Differentiable node sampling makes sense; however, what about edge construction? Differentiable pooling often uses fully connected graphs, with down-weighting for inactive edges, to make this process differentiable. But it sounds like this paper doesn't use fully connected graphs for G_l>1-- how is differentiability guaranteed?\n\nThese points also need to be described more explicitly in the paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces a GNN model for mesh-based physical simulations based on dynamic graph hierarchies. The paper has merits that have been highlighted by the reviewers. These include the formulation of the problem and the good performance on standard benchmarks. However, the reviewers have also indicated limitations mainly related to the following points: (i) novelty of the methodology and relation to previous approaches (the AMP component); (ii) efficiency of the proposed methodology. Considering the author's responses, I am in favor of accepting the paper. I encourage the authors to take these comments into consideration and revise the manuscript accordingly to better position their work in relation to previously proposed approaches.
Decision: Accept (poster)