Learning Deformable Body Interactions With Adaptive Spatial Tokenization
Authors: Hao Wang
Yu Liu
Daniel Biggs
Haoru Wang
Jiandong Yu
Ping Huang
Venue: Accepted by TMLR ()

Abstract: Simulating interactions between deformable bodies is vital in fields like material science, mechanical design, and robotics. While learning-based methods with Graph Neural Networks (GNNs) are effective at solving complex physical systems, they encounter scalability issues when modeling deformable body interactions. To model interactions between objects, pairwise global edges have to be created dynamically, which is computationally intensive and impractical for large-scale meshes. To overcome these challenges, drawing on insights from geometric representations, we propose an Adaptive Spatial Tokenization (AST) method for efficient representation of physical states. By dividing the simulation space into a grid of cells and mapping unstructured meshes onto this structured grid, our approach naturally groups adjacent mesh nodes. We then apply a cross-attention module to map the sparse cells into a compact, fixed-length embedding, serving as tokens for the entire physical state. Self-attention modules are employed to predict the next state over these tokens in latent space. This framework leverages the efficiency of tokenization and the expressive power of attention mechanisms to achieve accurate and scalable simulation results. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art approaches in modeling deformable body interactions. Notably, it remains effective on large-scale simulations with meshes exceeding 100,000 nodes, where existing methods are hindered by computational limitations. Additionally, we contribute a novel large-scale dataset encompassing a wide range of deformable body interactions to support future research in this area.

Published in Transactions on Machine Learning Research (11/2025)
Learning Deformable Body Interactions With Adaptive Spa- tial Tokenization
Hao Wang"
Yu Liu"
Daniel Biggs
Apple devin. wwwghao@apple..com com
Apple yliu66eapple.com com
Apple
dbi abigga@apple.com
Haoru Wang
Jiandong Yu Apple
Ping Huang Apple
Apple
haoru_ voooupuanggapple.comm com
jiandong_rudapple.com
huang_pingdapple.com
Reviewed on OpenReview:
Abstract
Simulating interactions between deformable bodies vital fields like material science, mechanical design, and robotics. While learning- methods with Graph Neural Networks (GNNs) effective at solving complex physical systems, they encounter scalability issues when modeling deformable body interactions. To model interactions between objects, pairwise global edges have to be created dynamically, which is computationally intensive and impractical for large-scale meshes. To overcome these challenges, drawing on insights from geometric representations, we propose an Adaptive Spatial Tokenization (AST) method for efficient reprosentation of physical states. By dividing the simulation space into grid of cells and mapping unstruetuued meshes onto this structured grid, our approach naturally groups adjacent mesh nodes. We then apply cross-attention module to map the sparse cells into compact, fixed-length embedding, serving as tokens for the entire physical state. Self-attention modules are employed to predict the next state over these tokens in latent space. This framework leverages the efficiency of tokenization and the expressive power of attention mechanisms to achieve accurate and scalable simulation results. Extensive experiments demonstrate that our method significantly outperforms stato-of-the-art methods modeling deformable body interactions. Notably, it remains effective on large-scale simulations with meshes exceeding 100,000 nodes, where existing methods hindered by computational limitations. Additionally, contribute a novel large-scale dataset encompassing wide range of deformable body interactions to support future research in this area.
1 Introduction
Solving interactions between deformable bodies plays a vital role in a wide range of applications, including mechanical design (Thompson & Sung 1986,
material science (David Müzel et al. 2020 Barbero 2023),
Zienkiewicz & Taylor 2005), and robotics (Collins et al.
2021 The finite element method (FEM) widely
used numerical approach for addressing such problems Courant al. 1994]. However, FEM typically incurs high computational costs and requires significant manual effort from engineers to ensure solver stability and
convergence OOcl 2000 Hughos 2003 Cook et al. 2007 EIrefaic et al. 2024). Recently, there has been growing interest in leveraging learning-based methods to address deformable body simulation. Nesh (MGN) and related approaches (Pfaff et al.) 20200: Sanchez-Gonzale et al. 2020 Fortunato et al. 2022 represent unstructured meshes as graphs and employ stackod message-passing blocks to propagate physical information across the mesh. Subsequent variants have introduced various graph pooling operations and
"These authors contributed equally.

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U-Net-like architectures (Lino et al., 2021 Deshpande et al. 2024 Cao et al. 2023 to solve the multi-scale challenges in different. simulation tasks. To model interactions wetween distinct deformable bodies, these methods typically construct dynamic edges based on the proximity of mesh nodes (Pffff et al. 2020 Yu et al.
2024). However, maintaining global pairwise edges across all mesh nodes becomes a significant computational bottleneck, limiting the scalability of such models to larger or more complex scenes. Rubanova el al. (0024)
proposed to use 3D implicit representations like Signed Distance Function (SDF) for more efficient collision detection, but it has be built on the solid body prior. Thuerey et al. (2018) interpolated the input 2D field onto a 128x x 128 grid and applied convolutional U-Net to process the grid imput for CFD simulation. O Pinheiro et al. 2023) proposed representing atoms as continuous densities and molecules as discretizations of 3D space on voxel grids. Similar to image tasks, they applied score-based generative model to generate molecules by denoising noisy voxelized representations. Beyond regular grids suitable for convolutional operations, later methods such as GraphCast (Lam et al. 2023) introduced a background grid to which the mesh is attached, enabling more efficient message passing in graph neural networks. However, this effectively converts an unstructured mesh into a structured one, which may work well for simple, regular geometries but. can suffer from reduced accuracy and efficiency when applied to complex shapes.
Methods such as GraphCast (Lam et al. 2023 propose creating a background grid to which the mesh is attached, enabling more efficient message passing, However, this effectively converts an unstructured mesh into a structured one, which can work well for simple, regular geometries such as spheres. When applied to complex shapes, however, this approach may suffer from reduced accuracy and efficiency. Transformer-basee models such as HCMT (Yu et al., (024)) propose using two sets of attention blocks to model contacts and collisions between deformable objects. However, the attention matrices are constructed over mesh nodes and global edges, rosulting in high memory consumption. This makes the approach computationally prohibitive when scaling to larger meshes.
Recent advances in computer vision Dosovitskiy et al. 2020 Radford et al. 2021 Li et al. 2023 and computer geometry (Zhang et al. 2023b Li ct al. 2025 have shown that applying learnable tokenization to raw input signals is critical for downistream understanding and generation tasks. In computer vision (Dosovilskiy et al.) 2020 Radford et al. 2021). input images are typically divided into patches, which are then mapped via linear layers to patch embeddings. These embeddings, along with added positional information, are processod further using token-wise operations. Similarly, in geometric tasks, 3D reppessnted meshes, point clouds, or signed distance functions (SDFs)-ad are typically embedded into compact latent representations before being passed to downstream components (Zhang et al.] 2023a Li et al., 2025). These approaches inspired us to design an effective tokenization strategy for physical states in simulation, where the state can be viewed as a set of vector fields defined over a given 3D geometry.
In this work, we introduce Adaptive Spatial Tokenization (AST), a novel method for encoding diverse physical states into fixed-length embeddings. We begin by quantizing the spatial domain into grid of cells, effectively mapping the unstructured mesh onto structured spatial partition. To manage this representation efficiently, we also provide the option to use an octree-like hicrarchical indexing system for scalable storage and fast lookup. As showed in Figure [ in our proposed representation, adjacent mesh nodes are naturally grouped into shared cells, enabling structured local interactions. We then apply sparse convolution to propagate information between cells, allowing inter-object interactions to emerge naturally. This approach climinates the need for explicitly constructing costly pairwise global edges, which is a common bottlenock in prior graph-based methods. Inspired by the use of cross-attention in Transformer architectures and its success in 3D shape representation (Zaaaa et al. 2023b Li et al., 2025). we design cross-attention module that queries features from the quantized sparse cells using compact, fixed-length vectors-on our adaptive spatial tokens. These tokens are then processed in latent space using a Transformer-style network composed of a series of self-attention layers, enabling next-step prediction of the physical state. We conduct extensive experiments across various scenarios involving deformable body interactions. Our method consistently outperforms state-of-the-art basclines, particularly in modeling inter-object interactions. Furthermore, in large-scale simulations involving meshes with over 100,000
most existing methods fail due to memory constraints-ou approach continues to produce accurate and stable predictions.
Our primary contributions can be summarized as follows:
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Mesh Nodes for
Object V/j
Mesh Nodes for
Object 2 1/2
Mesh to mesh edge intra object E
Dynamic world edge inter objects E, Sparse spatial cubes
Figure 1: Spatial Cells for interactions.
We propose Adaptive Spatial Tokenization (AST) to efficiently represent physical states in simulation by mapping unstructured meshes into structured grid cells and compressing them into a compact, fixed-length representation via cross-attcntion.
2. We develop attention-based model that operates spatial tokens in latent space to simulate interactions between deformable bodies, without relying on explicit global connectivity.
Our method achioves significant improvements over state-of-the-art baselines, particularly in modeling complex object interactions.
We introduce ABCD-XL, novel large-scale dataset featuring diverse deformable body interactions for benchmarking.
2 Related Work
In recent years, solving physies simulation with learning based methods have been an active research area. One of the most representative work MGN (Pfaff et al. 2020), treated as graph. adopts PrcooddProoces-Deee Pecode Decode architecture
which mesh data with physical states is (Sanchez Gonzalez GSanalzzzGonzalez et al. 2020). stacking multiple message passing layers as the core processing module to propagate the physics information through the graph. Several variants, such as BSMS (Cao et al.
2023 MAgNET (Deshpande et al. 2024 and
GNN-U-Net [Gladstone et al. 2023), introduce enhancements including virtual edges, graph pooling, and U-Net structures to improve computational efficiency and handle long-range interactions. To simulate interactions between objects, these methods typically construct dynamic world edges by connecting spatially close mesh nodes across objects at each time step. FIGNet (Allen et al. 2023) extends this idca by defining multiple edge types (face-to-lace, mesh-lo-mesh, and object-lo-mesh) to betler capture rigid body dynamics. More recently, SDF-Sim (Rubanova al. 2024) introduces implicit representations such as signed distance functions (SDFs) for efficient collision detection between rigid bodies. However, both methods focus rigid body interactions and do not offer scalable solutions for deformable body simulations. HCMT (Yu et al. 2024h explores the use of Transformer-style attention blocks to model the dynamics between deformable bodies. While promising, it still relies on constructing world edges and requires computing dense node-wise attention matrices, which limits its scalability to large moshes.
Recently, learnable tokenizers have been widely adopted in both computer vision (Oord et al. 2017 Dosovitskiy et al. 2020: Radford et al. 20211) and computer geometry (Wang) 2023 Zhang al. 2023b Li et al.] 2025)
to improve efficiency and scalability, leading to state-of-the-art performance in downstream tasks such understanding and generation. In vision tasks, methods like VIT (Dosovitskiy et al. 2020 first split the input image into fixed-size patches. These patches are mapped to embeddings using lincar layers, which, when combined with positional embeddings, form patch-level image tokens. VQ-VAE (Oord et al., 2017) uses a variational autoencoder to learn reconstructable tokens for image patches and applies vector quantization to map them into discrete codebook. In computer geometry, hicrarchical structures like the Octroe (Meagher 1982) are designed for efficient storage of 3D shapes and their properties (Wang et al. 2017)). In Octformer (Wang 2023), following the approach of ViT, sparse convolutions applied to structured
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point clouds to tokenize the imput and map it into latent space for downstream tasks. Shape2VecSet (Zhang et a.. 2023a) and Tripos (Li et al. 2025) use cross-attention modules to map 3D shapes to fixed-length vectors, treating them as latent tokens. Diffusion models are then trained over these learned tokens for generation tasks.
Building on insights from previous works, we treat the physical state simulations as vector fields defined over a given 3D shape, and propose Adaptive Spatial Tokenization to push the boundaries. Inspired by approaches in graphies and vision tasks, pipeline divide vectors in the space to cell patches, encode cells to compact, fixed-length tokens, and apply attention- modules to complete next-step prediction in the simulation. We conduct series of experiments and ablation studies to demonstrate how Adaptive Spatial Tokenization enhances both the efficiency and performance of simulations involving deformable bodies with interactions.
3 Preliminaries
In this section, we briefly introduce the preliminary techniques used in our method and refer interested readers to the original sources for further details.
3.1 Graph Operations
Graphs offer flexible structure for representing complex data. We leverage message-passing operations to aggregate features on input graphs before transforming them into discrete tokens.
Message-Passee Passing Message passing refers to feature aggregation operalions over graphs. While numerous variants exist in the literature, we adopt the formulation from Pfaff et al. (2020) and extend it to heterogeneous graphs. Given an edge set E,, edge attributes }ei} defined on each edge, and the corresponding sender and receiver node features { and {o]}, the message passing operation <- MMessage-Passingee Passing(e, v", v²)
proceeds in two steps:
eyy (" of, of" = S" Σ e(g)
(1)
where Ie is the edge update function and f is the node update function, typically implemented as multilayer perceptrons (MLPs). The implementation details of the MLPs can be found in Section A.5 The sum taken over all sender nodes i connected to receiver node j. Message passing operations can also be applied in the absence of explicil edge features, resulting in edge-free formulation: v" <- Message-Passiene Passing(v" v²),, where the edge update function simplifies to: -
(2)
3.2 Spatial Operations
Although graph operations are flexible and generalizable to various data structures, they can be inefficient and memory-intensive due to explicit edge reprosentations. To address this, we introduce spatial techniques from the 3D vision literature originally designed for large-scale point clouds to enable more efficient. computation.
Sparse Convolution Sparse convolution is powerful (echnique widely used in 3D shape analysis and synthesis (Graham 2015 Yan et al.) 2018 Wang et al. 2017p. It enables efficient operations sparse 3D representations, such as point clouds and volumetric grids, by leveraging octree structures to compress spatial information without loss. In the context of mesh-based physical simulation, we briefly outline how sparse convolution is applied, and refer readers to Wang et. al.] (2017) for more comprehensive details.
Starting from unit cell (the level-O cell defined over the space [-1, 1]3) that encompasses the entire 3D object, an octree is constructed by recursively subdividing each cell into eight child cells whenever the parent cell contains least one mesh node. This process continues until maximum predefined level-T. is reached,
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and the cells at level-L have side length 2!-1. Each cell is assigned coordinate P. = (mz, yt, 21) that indicates its position in the uniform grid at level-1, derived by uniformly dividing the unit cell into 2 segments per axis.
A sparse convolution from level-(1+) + s) to level-l is applied at each non-empty level-l cell dt using the following rule:
K
c = Σ R=1
dta wo Saeeaay +b', ie [1,
(3)
where of denotes the feature d, N is the number of non-empty cells at level-I and v²t = [of o K is the mumber of neighbors involved in the convolution, Wy and are learnable convolution weights and bias, and ((l + s, s, k) returns the k-th neighboring cell relative to dt at level-(l + s). More precisely, based on the cell at position Pi+. = floor (P/2),, where Pt is the position of c, {N(1 + s,i, designates a group of relative positions representing the convolution kernel. For example, a 3 x 3 x 3 convolution kernel corresponds to {I(I + 8, i, i, pointing to the cells at positions:
((T++s + M + B, 2l+ + +)) 6.8,,E1-1,0,11
(4)
If no cell is present at the corresponding position, the features are treated as all zeros, similar to a padding operation. Recursively applying sparse convolution layers forms an OCNN operation, defined as
Neour OCNN(c²) SparseCony (oo. (...
55)
where Neone denotes the number of convolutional layers, and each SparseCony' operation have distinct configurations based on specific design choices.
Farthest Point Sampling The Farthest Point Sampling (FPS, (Eldar et al.) 1997 Qi ct al.
2017)
algorithm selects a representative subset of points (or cells, in our context) based on spatial distribution. is defined as
FPS(c,p"), where e is the input feature set with corresponding spatial positions pf, and C e is the sampled subset.
(6)
4 Method
4.1
Problem Setup
We consider the evolution of the physies-based system discretized on mesh, which could be directly represented by heterogeneous graph G, (1V)" VE }, {Em12m Egm2e _gem)) }). V.t and V represent the node sets associated with physical properties (c.g., material properties, strain, stress) defined on mesh nodes and element nodes, respectively. The edge sets Em2mm Em and & capture physical relationships between mesh-to-mesh, mesh-to-element, and element-to-mesh pairs, respectively.
We choose to include the element nodes besides the mesh nodes form a heterograph, as we found that explicitly modeling elements is critical for realistic physical simulations. Many physical quantities- such strain and stress are defined via integration over entire elements rather than at individual nodes. Thus, representing such properties at the element level aligns more naturally with formulations found in classical PDE solvers.
At each time step f, certain node positions or physical properties may be externally specified. These are collectively referred to as the boundary condition B., For example, in a quasi-static scenario where a deformable object being compressed by a rigid body, the movement of the rigid body must be provided; otherwise, the resulting deformation cannot be inferred solely from the current state.
The objective is to model the evolution of the vector field by learning transformation F:
G =[B,, GG, Gi-1, Gr-2..... Gr-b41),
(7)
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Figure 2: Model structure overview. Graph-based physical states are encoded into latent tokens via Adaptive Spatial Tokenization (AST), processed with attention- based mechanism, and decoded back for next-step prediction. The green cells are selected by the FPS algorithm and serve as query tokens in the cross-attention shown in the top
context
Tokenization (Sec. 4,2.4)
Spetial Cells
query
Mosh Graph (
Spatia Calls Cz fq
C2
CrossAtti
FPS
Tokens
Embedoing (Sec. 4.2.2)
SparseCamv
Mesh To Coll Aggregation (Sec. 4.2.3)
query
context
Mesh
Spalial Celle
CrossAttn
SeltAttn
Processing (Sec. 4.2.5)
S Decoding (Sec. 42255
Mesh Aggregation (Section 4.2.3
Cell Tokenization (Section
4.2.4
where h is the number of historical steps considered.
4.2
Adaptive Spatial Tokenization
4.2.1 Overall Architecture
Existing methods typically employ message-passing or graph pooling operations to retain the expressive power graph representations. However, graph-based structures do not scale well large-scale meshes, and their information aggregation inherently slow due to the localized nature of propagation. To overcome these limitations, our novel -nnoach-Adaptivee Spatial Tokenization (AST) aggregates features defined graphs into compact latent tokens through the following steps:
1. Encode the raw features on G3 into an embedded feature graph G, (Section 4.2.2)
2. Aggregate the mesh node features v from G. into cell features v defined on spatial cells CL. (Section 4.2.3
3. Project the cell features vf into a fixed-length set of latent tokens h, (Section 4.2.4).
4. After transformer- based processing on h,, decode the tokens back to spatial cells CL.ss and then reconstruct the output graph G.+1 (Section 4.2.5
Notably, the spatial cells are constructed on a per-frame basis to capture instantaneous spatial interactions at cach time stop. The overall model architecture is illustrated Figure 2
4.2.2 Graph Embedding Encoder
The input heterograph G, will first be transformed G, = (().m V.), ) {Em2m EE ge2m Em2e7) via the graph E
embedding encoders. Specifically, node features v/n, vf and edge features e, mm2m e m2n cf2m are projected
into latent. space using MI.Ps, resulting in vi, Vi, @m2mm emeee and 2 An F2M (element-to-mesh)

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message-passingg operation is then applied to aggregate features from element nodes to mesh nodes:
m + Messagg-Passsig((,"""" Vi v.).
(8)
Optionally, when element-leve features are available, an M2M (mesh-to-mesh) message-passinng operation can be performed to encode positional and structural information via Emn2m;
TTC
( v.))
(9)
These message-passing operations encode the input graph structure and features into the mesh node repre- sentation V²"
4.2.3 Mesh To Cell Aggregation
Although graphs offer high flexibility and generalization across diverse data structures, message-passe operations typically require explicitly materializing v² and v' along graph edges, which leads to significant memory overhead on large-scale graphs. Our method overcomes this limitation by partitioning space into a regular grid and mapping mesh nodes onto it.
We construct an octree of depth L based on the mesh node positions p.". and the non-empty leaf cells are referred as Cr.3- A visualization is shown in Figure [ We then establish edge sets between mesh nodes and spatial cells denoted Em=2s and (12mmbased on spatial inclusion, i.e., whether a mesh node falls within given cell. The positions of the cells pf are defined their center coordinates, and the cell features V; are obtained by an average of the connected mesh nodes.
The number of equivalentn,, the octree level L-is a design parameter akin to the world edge radius: mesh nodes falling within the same cell are considered to interact. For large-scale graphs or dense moshes, we split the space with finer resolution cells CL+LocNN: stored in a (L + LocNx)-level octree, and then apply sparse convolutions to downscale the features to the I, level. Details for the sparse convolution refers
section 3.2
We discussed the impact of spatial cells resolution in section A.5
It's worth noticing that although we do not explicitly model interactions between separate graphs, by aggregating the mesh nodes into spatial cells, it can capture such interactions through cells that encompass nodes from different graphs. Experiments in section 5.3 showed the effectiveness of modeling the interactions by spatial cells.
M2C (mesh-to-cell) message passing performed aggregate mesh node features to the corresponding cell features:
VC
Messigg-PassingggV V.T).
(10)
Later, reconstruct features back onto the original graph from the cell representations, a C2M (cell-to-mesh) cressge-passsing operation can be performed:
V
Messaage-asssnnggTee vi).
(11)
4.2.4
Cell Tokenization
Spatial cells efficiently and effectively capture the structure and information present in the original graph. While OCNN message-passing operations can be applied to further aggregate the features within each interaction cell, these methods are inherently Docal-propagating information incrementally through neighborhood connections. This introduces an inductive locality bias into the learned representations. In contrast, attention mechanisms (Vaswani et al. 2017) are designed to overcome such limitations by enabling global feature aggregation a single step, without relying on local connectivity. This makes them particularly well-suited for capturing long-range dependencies and holistic patterns in complex graphs. For brief review of our attention module design, see Section A.4
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FPS
query
context
query y
context
CrossAttn
CrossAttn
Figure 3: The encoder (Ieft) and decoder (right) cross-attention blocks. We use v to denote feature vectors on the sparse grid (i.e., the previously defined sparse cells), and h to denote the compact latent tokens. Our attention module design are detailed in Section A.4
We apply cross-attention mechanisms to transform features embedded in spatial cells C₁ into compact latent tokens h, with selected dimension droken; as illustrated in Figure
V PosEmmbv{..p3
(12)
h CrosAAttn(PPS(V pf), vi),
where PosEmb and CrossAttn are defined in Section A.4
4.2.5 Processor and Decoder
The latent tokens h, are further processed through LSa layers of self-attention modules to condense and integrate global information. During decoding, reconstruct features the spatial cells Ct, from the processed latent tokens, the positionally embedded features as queries and the latent tokens context for an cross-attention operation, namely:
V - Ctoss h1).
(13)
The output mesh graph features are first obtained via Equation 11 and the predicted mesh features v are then produced by applying an MLP. Similarly, the output element features V441 are computed through M2E (mesh-to-clement) messago-passing followed by an MLP.
5
Experiments
5.1
Experiment Setup
Datasets We adopt two representative public datasets from GraphMeshNets Pfaff et al. 2020 that involve object interactions: 1) DeformingPlate: A deformable object is compressed by rigid body, with ~1.3k mesh/ 4k element nodes per mesh; 2) SphereSimple A piece of cloth interacts with kinematic sphere, with ~2k mesh/4k cloment nodes per mesh. To further validate our method on large-scale physical simulation tasks an area where existing literature limited we introduce two new datasets: ACCB ABCD stands for A Big CAD Deformation, where two deformable objects squish each other, with -4k mesh/ 12k element nodes per mosh; 4) ABCD-XL: follows the same setup the ABCD dataset, except it uses significantly denser meshes, with ~100k mesh/3008 element nodes per mesh.
ABCD and ABCD-XL To our knowledge, there were no large-scale physical simulation datascts a wide variely of geometries available. We constructed a larger and more generalized dataset to fill this gap. The goal this dataset was to have a wide variety of geometric shapes that are deformed after coming into contact with each other. We used the ABC dataset (Koch al.] 2019), which is a CAD model dataset used for geometric deep learning, get a wide sample of parts and shapes deform. Within the ABC dataset, we selected 400 single-part CAD models with relatively higher CAD quality. To generate simulation, first.
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Randomized solid body selection
Auto-meshed parts
Automated assembly and applied boundary conditions
ABC Dataset ABC
Figure 4: Randomized FEA simulation dataset using geometry from ABC dataset.
Figure 5: The FEA simulation results using ABC CAD dataset highlight various deformation modes, including compression with associated tension around hole, well as plate and beam bending.
randomly select two of these CAD geometries, then auto-mesh them with the meshing tool Shabaka (Hafez & Rashid 2023). We then align the two meshed parts in 3D space and apply compressive boundary conditions to simulate the parts coming into contact. Figure illustrates the workflow of the dataset construction process. Due to the wide variety of CAD geometries used in the simulations, the resulting deformation exhibit equally wide variety of deformation modes and stress states within each element. Figure shows several example simulations and the modes of deformation achieved through contact. Using this simulation workflow, we generated two datasets with different mesh resolutions. The ABCD dataset has mesh size of 4k mesh nodes, and the ABCD-XL dataset has mesh size that is 25 times larger of 100k mesh nodes.
Dataset
# Mesh Nodes
Mesh Type
# Steps
System Type
#Samples
DEFORMINGPLATE
1271
Tetrahedral 3D)
400
Quasi-static
1000 100 100
SPHERESIMPLE
1731
Triangle 3D
400
Newtonian
1000 : 100 100
ABCD
4408
Tetrahedral 3D
20
Quasi-static
6000 300 300
ABCD-XL
99997
Tetrahedral 3D
20
Quasi-static
1000 100 100
Table Details of the datasets used in this work. System Type refers the underlying PDEs. All simulations use lincar elements. For DEFORMINGPL.NT and SPHERESIMPLE, the dataset providers report using AreSim (Narain et al. 2012) and COMSOL (De Bézenac et al. 2019), respectively. For ABCD and ABCD-XT, we generated simulations using ABAQUS (Barbero) 2023 #Samples indicates the number of simulations generated for training, validation, and testing, respectively.
Basclines We compare our method against several strong basclincs across all datasets. 1) McshGraph- Nets(MGN): A state-of-the-arl message- message-passing- passing-based graph neural network; 2) Bi-stride Multi-scale GBSSS: Extends MeshGraphNets with bi-stride pooling to construct U-Net structure for improved scalability; 3) Hicrarchical Contact Mesh Transformer(HCM) specifically designed to model interaction problems using contact-aware mesh transformer blocks. The details of our training settings be found in Section A.2
A Transformer-basee architecture

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5.2 Results
We evaluate all methods the benchmark datasets by selecting the checkpoints with the lowest validation loss and report their rollout inference accuracy the test set in Table All experiments DEFORMINGILATTE, SPHERESIMPLE, and ABCD are conducted on single machinc equipped with 4 V100 GPUs. For the large-scale ABCD-XL dalaset, experiments are run on a machine with 8 V100 GPUs. Other training details can be found in Section A.1 and Section A.2 Our method demonstrates superior performance compared to prior methods, achieving a substantial improvement.
Dataset
MGN
BSMS
HCMT
Ours
5.5 + 0.2
5.4 0.5
2.9 + 0.2
1.1 0.1
DEFORMING
6891 + 89
10719 + 544
7272 + 45
4842 + 174
SPHERESIMPLE
19.0 + 4.9
15.0 + 0.8
Diverge
14.4 0.8
ABCD
u
0.641 + 0.007
0.736 + 0.017
0.541 + 0.006
0.505 + 0.002
0.480 + 0.002 2.11 t 0.82
ABCD-XL
o OOM
OOM
OOM
Table 2: RMSE (rollout-all, x10-3 for displacement) evaluation results. u X Xg denotes displacement and a denotes stress. OOM stands for out-of-memory. RMSE is computed as described in Section A.1 with results shown as mean standard deviation over three independent runs.
5.3 Ablation and Parameter Analysis
In this section, we first visualize the spatial cells to demonstrate their effectiveness capturing interactions, and then present a parameter analysis of key design choices, including the spatial cell length and the number of latent tokens.
Spatial Cell for Interactions Existing graph-based methods typically rely on world edges to model interactions. However, computing world edges requires evaluating pairwise distances between mesh nodes, leading to an O(n²) complexity that limits scalability on large-scale meshes. In contrast, our method leverages spatial quantization reduce this complexity to O(n) by aggregating nodes into structured cells. We visualize these cells in Figure D demonstrating their ability to effectively capture interactions. While it is possible to compute world edges using similar spatial quantization techniques, our approach goes further-by encoding graphs into compact latent tokens, our model combines the expressive power of graph representations with the conaalational efficiency and global context aggregation capabilities of token-based processing.
Quantization Cell Length We run model on the DEFORMINGPLATE dataset with different Leeli values and report the validation loss in Figure 7 When Teell 7, all mesh nodes are assigned to single cell at the initial frame, tokens then fail to capture interactions among neighboring nodes, degrading accuracy despite the finer resolution. These results verify that grouping a reasonable number of mesh nodes into interaction cells plays a vital role in effectively learning deformable body interactions.
Number of Latent Tokens DEFORMINGPLATE. dataset, reporting both the validation loss and the training epoch time in Figure 8 As shown, too few latent tokens degrade model accuracy, while an excessive number increases training time. Owing to the parallelism of GPUs, model efficiency remains stable up to a certain threshold, beyond which additional tokens improve performance the cost of runtime. In general, a typical choice for datascts fewer than 10k mesh nodes is 512 latent tokens, which provides sufficient capacity while maintaining comparable training and inference efficiency to smaller values.
We conduct parameter search on the number of latent tokens using the
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Figure 6: We visualize the spatial cells the ABCD dataset (left) and DEFORMINGPLATTE dataset (right). The figure displays one representative feature channel across the cells. Warmer colors correspond to higher feature norms. Their concentration at the collision regions indicates that the spatial cells effectively capture the critical interactions between the disconnected meshes.
Choices the quantization level
Choices of the number of latents
25
1.40
3.5
1.35
2600
dbppeccmmen eeceementt RASE (X00-)
traninn epoch brmc
1.25
24400
2300
1.15
2200
1.05
23.00
1024
Quantization
Figure 7: Validation results on the DRFORMING- PLATE dataset with different Leel scttings. The nodes/cells ratio denotes the average number of mesh nodes per cell at the initial frame.
Figure Validation loss and training epoch time on the DEFORMINGPLATE dataset for varying numbers of latent tokens. Accuracy generally improves with more latent tokens, while training time remains stable until increasing sharply beyond threshold.
5.4 Computational Efficiency and Scaling Capability
The training and inference times are reported in Table 3 All experiments are conducted on a machine equipped with V100 GPUs. While increasing the batch size significantly improves training efficiency (c.g., 2.8x for MGN, 4.8x for BSMS, and 3.1x for ours when using batch size of 48 on SPHERESIMPLE), we ensure fair comparison by fixing the batch size to 4 (i.e., per GPU) across all methods. Our method
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demonstrates comparable training and inference efficiency to state-of-the-art graph-based approaches small-scale mesh size.
Model
DEFORMINGPLATE
SPHERESIMPLE
ABCD
ABCD-XL
Train
Val
Train
Val
Train
Val
Train
Val
MGN
9393
316
8744
361
3425
215
BSMS
17700
410
12754
508
5442
332
HCMT
16450
470
12913
510
5862
328
Ours
9333
392
7282
394
2742
188
16435
8123
Table 3: Training and inference epoch time (seconds) evaluation. The reported epoch time refers to the total time taken for a single pass over the entire dataset during training or inference. For a fair comparison, all models are evaluated on 4-GPU node with total batch size of
Scaling (Training)
Scaling Capability
100003
100200
RSMS
NCMT
HENT
20003
80300
Time [5]
Time
20003
20000
Mash Size
Mesh Size (K)
Figure 9: Training time per epoch across different mesh sizes. MGN, BSMS, and HCMT run out of memory at mesh sizes beyond 40K, 8K, and 6K, respectively.
Figure 10: Inference time per epoch across different mesh sizes.
To further evaluate scalability, we conducted experiments the ABCD-XL dataset by generating subgraphs with varying mesh sizes. We compared the training and inference time of different methods on machine equipped with V100 GPUs, and we used the training set to evaluate both training and inference efficiency. shown in Figure g 10 all methods exhibit similar performance in the small-scale regime. However, our method demonstrates significantly better scalability as the element size exceeds 20k.
Although the BSMS method demonstrated good scalability on surface meshes in its original paper, we observed that it scales poorly volume meshes due to the increased number of bi-stride edges introduced during pooling. On surface meshes, bi-stride edges consistently downsample upper-layer edges. However, in the case of volume meshes, the edge count can grow significantly. For example, in volume mesh graph with 21k mesh nodes, the number of nodes and edges across a 6-layer BSMS model are: 21k/210k, 11k 11k/266k, 5. 5.7k/534k 3.2k/1.8M, 1.8k/.66,, 0.9k/483k. 483k.
5.5 Visualizations
We present prediction result on the ABCD dataset in Figure and DEFORMINGPLATE dataset on Figure 12 Further visualizations provided in Section A.3
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GT
MGN
BSMS
HCMT
Ours
Figure 11: Visualization results on the ABCD dataset. Displacement is visualized using color warmth, with warmer tones indicating greater displacement magnitude. Our method yields predictions that are closer to the ground truth at the collision region.
GT
MGN
BSMS
HCMT
Ours
Figure 12: Visualization results on the DEFORMINGPLLATE dataset. Stress is visualized using color warmth, with warmer tones indicating greater stress magnitude. Our method produces shapes that more closely match the ground truth, indicating more accurate displacement predictions, while the red-colored cells highlight correspondingly closer stress predictions.
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6 Limitation and Future Work
Despite its effectiveness, the method has certain limitations. Specifically, the octree structure enforces fixed set of cell lengths across levels, making it difficult to identify an optimal configuration. As shown in Figure 7 the performance curve remains relatively flat near the optimal Leett but a more flexible and finer grid control could yield further improvements. Besides, we employed FPS and CA to construct an efficiency-orennted bottleneck, yet recent work such as AnchorFormer [Shan et al. (2025) introduces anchor-based approximations for full altention, offering a promising alternative that can preserve efficiency while potentially improving accuracy. Furthermore, adding additional constraints on physical laws such the conservation of energy, enforcing boundary conditions, or satisfying the underlying PDEs could also further improve the models performance and generalization.
7 Conclusion
In this paper, we proposed new method that encodes graphs into compact tokens by leveraging sparse 3D operations, followed by transformer-based processing for expressive learning. This strategy combines the structural richness of graphs with the scalability and efficiency of 3D computation, enabling our model to scale to large inputs without compromising accuracy.
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A Technical Appendices and Supplementary Material
A.1 Dataset Details
Dataset Scttings
In Table we list details of all the dataset used in the experiments.
Dataset
System
Solver
Mesh type
stops
At
TW
SPHERESIMPLE
cloth
AreSim
triangle 3D
500
0.01
0.05
DEFORMINGPLAVE PLATE hyper-el.
COMSOL
textrahedral 3D
400
0.03
BRCD/ABCD-XL
hyper-el.
Abaqus
textrahedral 3D
21
0.003
Table 4: Dataset details, rw denotes the world edge radius, meaning that nodes within a distance of rw are considered connected by world odges. Note that world edges are only computed for MGN, BSMS, and HCMT.
Our method uses slightly different input features with Pfaff al. (2020), world edges. The input and output features used in our method are summarized in Table P Specifically, v and vf represent the input features on mesh and clement nodes, respectively, while vo and vo denote the corresponding oulput features. em and 2 are input edge features for mesh-lo-mesh (m2m) and element-to- to-mesh (e2m) edges.
do not explicitly compute
We use n and m to denote node type and material type, respectively. The node type indicates whether a mesh node is a boundary node such nodes have predefined next-frame values and therefore do not require updates. denotes stress.
For MGN, BSMS, and HCMT, the input and output features follow the exact definitions from Pfaff et al. (2020) on SPHERESIMPLE and DEPORMINGPLATE. We note that in the BSMS paper, the authors included velocity as an input feature for DEFORMINGPLATE. In contrast, we follow the original MeshGraphNets (MGN) implementation, which uses node type and (relative) positions as input. On ABCD, the definitions of v em and v are consistent with that in Table and the world edges are associated with edge features p ||
Dataset V v
em22m
e22m
ve
history
SPHERESIMPLE
P:
Pr
P
P
P
P
P.
DEFORMPLATE
Ur
U,
Po
P
UJ
P
P
Pr,
0e+1
ABCD
n, m, U2
UI
Po
[
P
P
P.
ABCD-XL
n. U,
m. Oe: u,
PO
P
p.
Ø++l
Table 5: Input and output fcatures for method.
World Edge Calculation We adopt the world-edge construction method proposed in the MGN paper, with modifications accommodate the dense meshes in the ABCD dataset. Compared to DEFORMINGPLATE and SPHERESIMPLE, the meshes in ABCD computation to sometimes produce world edges even larger than mesh edges. To address this, we retain only the 1000 world edges with the smallest pairwise distances.
significantly denser, which causes the original world-edge
RMSE Calculation The Root Mean Square Error (RMSE) is computed in a per-sequence manner: we first calculate the mean squared error for each sequence, then take the square root, and finally average the RMSE across all sequences in the dataset.
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A.2 Training Settings
For all datasets, adopt pairwise training strategy where graph is randomly selected from a sequence as the imput, and its subsequent graph is used as the target. We follow the same training noise strategy as proposed in Pfaff et al. (2000). The noise scale is set to 0.003 for both the ABCD and ABCD-XL datascts. The imput and target features are all normalized to zero mean and unit variance based on the statistics of the training set. All experiments are conducted on machine equipped with four V100-32GB GPUs, unless otherwise specified.
We did some modifications to the training process for improved performance:
Batch Size We increased the training batch size from 1 or 2 to 48 (12 per GPU on 4-GPU node) for MGN and BSMS which showed a much faster training procedure. HCMT inherently not applicable on batched graphs, so we kept a batch size with per GPU.
Learning Rate We adopt square rool scaling for the learning rale with respect to batch size. Starting with a base learning rate of 0.0001 for a batch size of 2, the final learning rate LR for a batch size of 48 is computed as 0.0001 x /48/2 = 0.00049. The learning rate is lincarly warmed up from 0.0001 to LR over the first 2000 steps, followed by cosine decay to zero at the 101st epoch (training stops at the 100th epoch).
For purely graph-based methods namely MGN, HCMT, and BSMS we found that the learning rate scheduling strategy facilitates faster convergence, while the square root scaling strategy had a negative effect. Therefore, we retain the scheduling strategy and use a fixed learning rate of 0.0001 across all datasets.
Training Iterations We extend the training itcrations from 5M steps (approximatcly 25 epochs) to 100 epochs.
Loss
We mcan squared crror (MSE) loss across all experiments. For the DEFORMINGPLATE and ABCD-XL datasets, we adopt multi-head outputs to jointly predict displacement and strss,, assigning loss weights of and 0.01, respectively.
A.3 Further Discussion
MGN MeshGraphNetss (MGN) is strong baseline for mesh-based physical simulations due to the expres- siveness of its graph-based reprosentation. However, this expressivcness also introduces several challenges:
Edge overhead: The computational burden in graph models often arises from the edge set, which can be several limes larger than the node set. This issue is exacerbated on large-scale meshes, where edge-based feature aggregation results in significant computational and memory overhead.
Message-passing passing operations in graphs are inherently local, requiring many iterations to propagate information across distant nodes. For meshes with over 100K nodes, hundreds of messagc-passing steps may be needed to fully capture global interactions.
Limited global context: Message-
Scalability of world edges: On dense meshes, the number of world edges can grow prohibitively large. This not only increases computation but also makes it difficult to distinguish meaningful interactions from spurious ones.
BSMS BSMS introduces the bi-stride pooling mechanism to address some of the limitations of MGN. By recursively down-scaling the graph graph-halving the mesh size at each stage- the method reduces both the number of nodes and edges, allowing for faster propagation of global information. While this strategy proves effective for small to medium-scale meshes, it still inherits the structural limitations of graph-based methods when applied to large-scale problems. In industrial FEA simulations where mesh sizes can exceed 100K elements, the graph structure remains bottleneck. Morcover, as discussed in Scction 5.4 the bi-stride pooling algorithm fails to generalize effectively volume meshes, limiting its applicability to 3D deformable body problems. Furthermore, although pooling edges can be precomputed -thereby accelerating training teeninggthe precomputation
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process involves matrix multiplications whose complexity scales with the number of mesh nodes. This becomes prohibitively expensive for large meshes, limiting the applicability of BSMS in high-resolution simulation settings.
HCMT HCMT incorporates attention mechanisms to address one the key limitations of MGN its inefficiency aggregating global information. However, the attention computations HCMT are performed directly on each mesh node. While they modify the original attention formulation to avoid the O complexity associated with standard attention matrices, this comes al the cost of reduced theoretical soundness. Moreover, despite these modifications, the method still scales poorly to large-scale meshes, limiting its practicality in high-resolution simulation tasks.
On the SPHERESIMPLE dataset, HCMT performs well during the initial 50 frames but gradually diverges thereafter. This suggests that, while HCMT effective on the quasi-static DEFORMINGPLATE dataset, it may face difficulty generalizing to dynamic problems like SPHERESIMPLE.
A.4 Transformer Operations
Following the design in Zhang et al. (2023a), originally developed for point clouds, we exlend this framework handle vector fields.
Positional Embedding Given feature set with associated positions p", first inject spatial information into the features via positional embedding:
e' c PosEm((p")
(14)
where PosEmb: Rd + Rd column-wit wise embedding function that maps input positions p" (with d (2,33)
to the feature space of dimension de, matching the dimensionality of
Attention An altention operation is defined to aggregate three feature vectors q RR,, h E R.xed U E R2 N.x,, where NR, Ng, e R are sequence lengths and dy de, dy E IR are feature dimensions,
9h"
Attention(q. k. v) = softmax
ddz )u.
(15)
A Multi-head Attention (MHA) operation is defined as
MHA(q, k v) Concat(head),,, head, ) W head, Attention(gW?,. &W,,, vW.").
(16)
Building on the multi-head atlention mechanism and adopting pre-norm structure, we construct two types of Transformer blocks, as illustrated in Figure
Transformer Blocks
CrossAttn: Given a set of query features Cquery and context features econtext, the operation
h = CrrossAttn(Cquersy (Cquery: Ccontext)
(17)
aggregates information from ccontext into Cquery. The output maintains the same length as the query, making this block particularly useful for compressing or decompressing representations of the context features.
SelfAttn: This is the standard self-attention mechanism where the query and context features are identical, i.e.,
SelfAttn(c).
(18)
FFN stands for Feed-For Forward Network. In all our CrossAttn and SelfAttn blocks, we use an FEN module with GEGLU activation as described in Shazeer (2020)
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Query
Context
Inputs
LayerNorm
LayerNorm
LayerNorm
g.k.v
k,v
MHA
MHA
LayerNorm
LayerNorm
FFN
FFN
CrossAttn
SelfAttn
Outputs
Outputs
Figure 13: Transformer blocks. The CrossAttn is displayed on the left, while the SelfAttn is displayed on the right.
A.5 Model Details
The MLPs used in input, message-passing and output layers are two-layer MLPs with ReLU activations with output size of 128. The hidden layer size of possage-passingg and input MLPs are 128, while the output MLPs are 32. The outputs of the message-passingg MLPs are further normalized using LayerNorm. To enhance their effectiveness, we adopt Random Weight Factorization (RWF, (Wang al. 2022
MLP
Transformer We use attention layers with heads, each with feature dimension of 64. The query, key, and value projections implemented using bias-free lincar layers, while the output projection includes bias term. We adopt pre-norm setup, applying LayerNorm before both the attention and FFN layers. The FFN follows the GEGLU formulation as described in Shazeer (2020), and has hidden dimension of 512. Dropout with a rate of 0.1 is applied within both the attention and FFN modules.
Model Architecture The input and output features on graph nodes and edges are normalized using statistics computed from 400 randomly selected graph pairs from the dataset. Our model begins with input encoding MLP that maps node and edge features into hidden space of size 128. This is followed by a one-step message-passnng operation either m2m or e2m to aggregate information onto the mesh nodes.
Datasct
Leell
loenn
dtoken
LSA
SPHERESIMPLE DEFORMINGPLATE
5
256
12
5
0
256
12
ABCD
8
0
512
12
ABCD-XL
12
512
12
Figure 14: Model hyperparameter.
Mesh node positions are then quantized using an Loat-layer octree, where Loetl is a hyperparameter. A one-step message- passsig is used to aggregate mesh node features into each cell. Optionally, an OCNN module with lormn layers (also hyperparaneter) is applied to downscale the features from the Leeut-t octree layer to the (Leell - -thh layer. Each OCNN layer consists of a sparse convolution with a 3 x 3 x 3 kernel and a stride of 2, which effectively moves features up one level in the octree hierarchy.
To obtain compact latent representation, cross-attention layer applied map features on the sparse cells a fixed set of tokens of dimension dioken. These tokens are then processed using L5A self-attention transformer blocks.
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The decoding process mirrors the encoder. We first apply cross-attention mechanism to decode the latent tokens back to the octree features. These features are then upscaled to the original Lear-layer resolution using transposed sparse convolution-base messago-passinne passong operation performed to decode the cell features back to the mesh nodes. The resulting uralures are concalenated with the original mesh node Teatures from the input encoding MLP to produce v'z, which then passed through an output MLP generate the final prodictions. If output features are also required on element nodes, an additional one-step srassage-pssinng without edge features is performed from mesh to element nodes, followed by separate output MLP.
OCNN with the same number of layers as in the encoder. A one-step
The fore-mentioned hyperparameter for cach dataset are listed in Table 14
A.6 Farthest Point Sampling Implementation
We outline the FPS algorithm in Algorithm 1 while noting that readers may directly use existing open-source implementations.
Algorithm Farthest Point Sampling (FPS)
Require: Point set P= (pi, -01.....PPy.. ,PN) sample size K Ensure: Sample indices S of size K
1: Initialize S <- (-
2: Randomly choose initial index in-
3: S (.o)
for each i E (1....,N) \ S do
dii] + dist(p:, Pia) end for
for t 2 to K do
8:
j t- arg max d(i) ig3
Su 10: for each i & S do
11: dli] min min (dli), dist(p..P)) P:))
12: end for
13: end for
14: return S
B Additional Experiments
B.1 MAPE Evaluation
For completeness, we also report the Mean Absolute Percentage Error (MAPE) on the DEFORMINGPLATE, SPHERESIMPLE, and ABCD datasets in Table MAPE measures the relative prediction error as percentage and is scale-independent, but it is known to be unstable when ground-truth values close to zero. This issue is particularly relevant for DEFORMINGPLATE and ABCD, where simulated objects are centered around zero. To mitigate, we mask out ground-truth values with l|vel2 < 10-5, The MAPE used in this paper is defined as:
T
[10e Vell2
MASK
(19)
T
l/vell2
where yr and de denote the ground-truth and predicted values at time t, and MASK(-) discards terms with llvell2 10-5, We emphasize that these results are provided only as reference; RMSE remains the primary and more reliable metric for comparison across methods.
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Published in Transactions on Machine Learning Research (11/2025)
Dataset
MGN
BSMS
HCMT
Ours
DEFORMINGPLATE
0.293 + 0.006
2.099 + 0.0175
0.391 + 0.023
0.132 1 0.008
SPHERESIMPLE
u 0.059 0.001
0.031 + 0.001
Diverge
0.030 + 0.001
ABCD
u
0.626 + 0.012
1.732 + 0.033
0.673 + 0.001 0.691 + 0.004
Table 6: The Mean Absolute Percentage Error (MAPE) (rollout-all) evaluation results.
B.2 Rollout Visulization
We select one representative case from DEPORMINGPLATTE, SPHERESIMPLE, and ABCD, and show their visualizations in Figures 15 17 The per-frame RMSE of the selected cases, along with the average over the test datasets, reported in Fig Figures
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Published in Transactions on Machine Learning Research (11/2025)
Frame 250
Frame 360
Frame 390
GT
MGN
BSMS
HCMT
Ours
Figure 15: Rollout visualization results on the DEFORMIGGPLATE dataset.
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Published in Transactions on Machine Learning Research (11/2025)
Frame 88
Frame 172
Frame 500
GT
MGN
BSMS
Ours
Figure 16: Rollout visualization results on the SPHERESIMPLE datasct.
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Published in Transactions on Machine Learning Research (11/2025)
ا Figure 17: Rollout visualization results on the ABCD dataset.
Frame
Frame 10
Frame 20
GT
MGN
BSMS
HCMT
Ours
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Published in Transactions on Machine Learning Research (11/2025)
Prodiction Variance Over Time
Frudictio Varian ce Over Time (Aver age)
Dur
0.007
1ich
0.002
0.030
0.035
0.038
3.022
0.001
0.000
0.002
Figure 18: Prediction variance over time on the DEFORMINGPLATE dataset. The deformation increases and reaches its maximum at the 360th frame across all cases. Our method exhibits a smaller growth rate with respect to deformation and successfully recovers to the initial state once the deformation ends.
Prediction Variation Over Time
Prediction Varistion Over Time (Averag
0.030
Dum
OuT
Mich
0.222
0.01
0.012
0.005
Figure 19: Prediction variance over time the SPHERESIMPLE dataset. The gray region marks the interaction period between the ball and the cloth. After separation, method maintains consistent predictions.
Prediction
Prediction Varlar (Average)
0.6022
MCh
0.0008
HCMT
8595
KONS
RENT
0.000
00004
0.001
0.000
0.0000
0.007
Figure 20: Prediction variance over time on the ABCD dataset. The maximum deformation occurs at the final frame.
27


Reviews:
[{"review_id": "jcevWbMwew", "reviewer": "Reviewer_k1UN", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposed a novel Adaptive Spatial Tokenization method for efficient representation of physical states, overcoming the challenges of computationally intensive and impractical large-scale meshes required by pairwise global edges. It then developed a transformer-based model with both cross-attention and self-attention that operates on these spatial tokens in the latent space. The proposed method has been demonstrated to outperform state-of-the-art baselines on both two public datasets and a newly constructed dataset, not only in accuracy but also in efficiency and scalability.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Comparing with three baselines proposed in 2020, 2023, and 2024 respectively, which are claimed to be state-of-the-art, the proposed algorithm achieves the best performance on displacement and stress. Furthermore, its computational efficiency and scaling capability are also impressive. These points provide evidence supporting my recommendation, though I am a non-expert.\n\nHowever, there are also some concerns:\n\n1. The novelty is not entirely clear. Although AST is presented as a novel method, it is unclear which specific component is novel. Moreover, how the learnable tokenization achieves higher accuracy while also overcoming computational intensity is not analyzed in detail.\n\n2. The evaluation metrics in table 1 are not well explained. Additionally, how should the visualization results in Figures 6 and 7 be interpreted? What signals indicate a better prediction in those visualizations?\n\n3. Since the proposed algorithm also uses the same background grid as in GraphCast that converts an unstructured mesh into a structured one, why does GraphCast suffer from reduced accuracy while the proposed algorithm does not? Additionally, could GraphCast serve as a competitive baseline as well?\n\n4. In the paper by Pfaff et al. (2020) introducing GraphMeshNets, I did not find any results regarding the dataset named SphereSimple. Did I miss something? Furthermore, why was GraphMeshNets not included as a baseline? I am also unclear about the evaluation metrics used in Pfaff et al. (2020) compared to those in the current paper. Did both papers use the same set of metrics? For the DeformingPlate dataset in Pfaff et al. (2020), it appears they reported only a single metric for rollout-all, with a value of 15.1 ± 4.0, whereas the current paper reports two metrics: one for displacement and one for stress. Could you clarify the differences between these two sets of metrics and how they can be made comparable?", "audience_yn": "Yes", "audience_explanation": "From the perspective of a non-expert, the proposed algorithm outperforms several recent methods on public datasets in terms of accuracy, efficiency, and scalability. In addition, the paper introduces a new dataset that may be of interest to researchers in the same field.", "requested_changes": "Please refer to the previous section for explanations regarding novelty, evaluation metrics, and other related points.", "broader_impact_concerns": "N\\A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OMJC8q3D3X", "reviewer": "Reviewer_UfKQ", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Authors propose an architecture for learning surrogate models of deformable bodies interaction. The main components of the approach are:\n1. GNN used as part of encoder\n2. Special procedure that transfer features from mesh to features on multilevel structure grid\n3. Sparse convolution on multilevel structured grid (for one dataset)\n4. Downsampling of the grid with Farthest Point Sampling algorithm\n5. Attention blocks of encoder and processor\n6. Decoder that transfer features back from structured grid to heterogeneous graph where physical observables are defined\n\nThe resulting architecture is tested on several benchmarks and is reported to have better accuracy and scalability when compared with baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claims are supported by empirical evaluation.", "audience_yn": "Yes", "audience_explanation": "The paper presents surrogate modelling for an applied problem in continuum mechanics. It may be of interest for AI4Science community that develop neural solvers for partial differential equations.", "requested_changes": "1. Authors write \"However, FEM typically incurs high computational costs and requires significant manual effort from engineers to ensure solver stability and convergence\" and reference https://arxiv.org/abs/2211.00713. Is it a reliable source for such claim? The article mentioned is on deep learning, so the authors of this article have incentive to motivate their research. Can the authors please consider backing up high costs of FEMs with some other source, or numerical test, or estimation?\n2. The idea to represent irregular data by projecting on regular grid appears in many publications. For examples in generative AI see https://arxiv.org/abs/2306.07473 and references therein. The oldest example from scientific computing known to me is https://ieeexplore.ieee.org/abstract/document/662670.\n\n   I suggest authors discuss prior art and include relevant references (not necessarily mentioned above) to improve the presentation of their ideas.\n3. For the description of Farthest Point Sampling authors reference https://arxiv.org/abs/1706.02413. Is it an appropriate reference? FPS was not introduced in this paper. I suggest authors to cite original sources of FPS, e.g., https://ieeexplore.ieee.org/document/623193.\n4. Farthest Point Sampling is an essential part of the architecture that ensures scalability. Anchor Attention described in https://arxiv.org/abs/2505.16463 seems to be related technique. Can the authors discuss difference between FPS and Anchor Attention?\n5. In Table 1 authors report RMSE for stress and displacement. Can the authors provide reference values of strain and displacement for the simulations from considered datasets (i.e., what is the order of magnitude for these quantities)? Alternatively, authors may want to provide relative error in place of absolute ones. Besides that, both stress and displacement are not dimensionless, so some form of normalisation is required.\n6. Figure 9 shows RMSE for different number of levels in multilevel structured grid. Can the authors explain why it is a U-shape curve? It looks slightly counterintuitive to me that accuracy deteriorates for large $L$.\n7. The other interesting hyperparameter is the number of latent tokens. Can the authors report results on how accuracy depends on the number of latent tokens? What is typical number of latent tokens for each dataset?\n8. In section 4.2.4 the latent tokens are selected based on FPS algorithm. Can the authors provide an illustration of points selected by FPS?\n9. Can the authors share details how FPS is implemented?", "broader_impact_concerns": "na", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Kt6pwLMZhI", "reviewer": "Reviewer_Zaye", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper puts forward a novel method named Adaptive Spatial Tokenization (AST) for effectively simulating the interactions among deformable bodies. The main contributions of the paper include:\n\n1. **Innovative Spatial Tokenization Method**: It presents the AST method. This method naturally clusters adjacent mesh nodes by partitioning the simulation space into grid cells and projecting the unstructured mesh onto a structured grid.\n2. **Attention-Based Model**: The authors develop an attention model that functions on spatial tokens within a latent space to simulate the interactions of deformable bodies.\n3. **Large-Scale Dataset**: The paper introduces the ABCD-XL dataset, which encompasses diverse scenarios of deformable body interactions.\n\n**Main Strengths:**\n\n- **Scalability Strength**: The method can handle large-scale meshes with more than 100,000 nodes. In contrast, existing methods encounter failures because of memory limitations.\n- **Computational Efficiency**: It reduces the complexity from O(n²) to O(n) by avoiding the costly construction of pairwise global edges.\n- **Adequate Experimental Validation**: Comprehensive experimental validations and ablation studies have been carried out on four datasets.\n- **Larger and More Generalized Dataset**: The ABCD-XL dataset appears to be challenging and has the potential to drive the development of this field.\n\n**Main Weaknesses:**\n\n- **Treatment of Physical Constraints**: There is an inadequate discussion regarding how to maintain physical conservation laws, such as the conservation of energy and the conservation of momentum.\n- **Lack of Introduction to the ABCD-XL Dataset**: Considering that the dataset is regarded as a contribution by the authors, the paper lacks an introduction to the specific statistical characteristics of this dataset and a comparison with other datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claims of the paper are supported by accurate, convincing, and clear evidence:\n\n- Comprehensive quantitative evaluation on four datasets (DeformingPlate, SphereSimple, ABCD, ABCD-XL).\n- Fair comparison with three strong baseline methods (MGN, BSMS, HCMT).\n- Provided detailed RMSE metrics and visualization results.", "audience_yn": "Yes", "audience_explanation": "The research findings of this paper have significant value and broad interest for the TMLR audience:\n\n- **Practical Application Value**: It has direct application potential in areas including materials science, mechanical design, and robotics.\n- **Computational Efficiency Improvement**: This paper offers a novel solution for industrial-grade simulations, which is of great practical importance.\n- **Large-Scale Simulation Capability**: It bridges the gap in existing methods when dealing with complex, large-scale deformable body interactions.", "requested_changes": "- Provide a more detailed introduction to the proposed ABCD-XL dataset, including its advantages over other datasets.\n- Lack of stability analysis of the method under extreme conditions (e.g., large deformation, contact separation).\n- Add an analysis of how physical constraints such as conservation of energy and conservation of momentum are maintained in the AST framework.\n- Conduct a sensitivity analysis of crucial hyperparameters, such as Lcell and dtoken. This analysis will help in understanding how variations in these hyperparameters impact the performance of the proposed method.\n- Add a comparison of accuracy and efficiency with commercial FEM software on the same problem.\n- Explore and discuss the scenarios and limiting conditions under which the method might encounter failures.\n- Commit to making the code and the ABCD-XL dataset publicly available upon acceptance. Open-sourcing the code and dataset will significantly enhance the reproducibility of the research.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)