MAVEN: A Mesh-Aware Volumetric Encoding Network for Simulating 3D Flexible Deformation
Authors: Zhe Feng
Shilong Tao
Haonan Sun
Shaohan Chen
Zhanxing Zhu
Yunhuai Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Deep Learning
Explicit modeling
Geometric Deep Learning
Simulation of Solid Deformation
Primary area: learning on graphs and other geometries & topologies

Abstract: Deep learning-based approaches, particularly graph neural networks (GNNs), have gained prominence in simulating flexible deformations and contacts of solids, due to their ability to handle unstructured physical fields and nonlinear regression on graph structures. However, existing GNNs commonly represent meshes with graphs built solely from vertices and edges. These approaches tend to overlook higher-dimensional spatial features, e.g., 2D facets and 3D cells, from the original geometry. As a result, it is challenging to accurately capture boundary representations and volumetric characteristics, though this information is critically important for modeling contact interactions and internal physical quantity propagation, particularly under sparse mesh discretization. In this paper, we introduce MAVEN, a \textbf{m}esh-\textbf{a}ware \textbf{v}olumetric \textbf{e}ncoding \textbf{n}etwork for simulating 3D flexible deformation, which explicitly models geometric mesh elements of higher dimension to achieve a more accurate and natural physical simulation. MAVEN establishes learnable mappings among 3D cells, 2D facets, and vertices, enabling flexible mutual transformations. Explicit geometric features are incorporated into the model to alleviate the burden of implicitly learning geometric patterns. Experimental results show that MAVEN consistently achieves state-of-the-art performance across established datasets and a novel metal stretch-bending task featuring large deformations and prolonged contacts.

Published as a conference paper at ICLR 2026
MAVEN: A MESH-AWARE VOLUMETRIC ENCODING NETWORK FOR SIMULATING 3D FLEXIBLE DEFOR- MATION
Zhe Feng":; :Shionng Tao", Haonan Sun', Shaohan Chen', Zhanxing Zhu Yunhuai Liu².
1 Academy for Advanced Interdisciplinaryy Studies, Peking University
2 School of Computer Science, Peking University 3School of Electrical and Computer Science, University of Southampton
ABSTRACT
Deep learning-basee approaches, particularly graph neural networks (GNNs), have gained prominence in simulating flexible deformations and contacts of solids, due to their ability handle unstructured physical fields and nonlinear regression on graph structures. However, existing GNNs commonly represent meshes with graphs built solely from vertices and edges. These approaches tend to overlook higher-dimensional spatial features, e.g., 2D facets and 3D cells, from the original geometry. As a result, it is challenging to accurately capture boundary representa- tions and volumetric characteristics, though this information is critically important for modeling contact interactions and internal physical quantity propagation, par- ticularly under sparse mesh discretization. In this paper, introduce MAVEN, a mesh-aware volumetric encoding network for simulating 3D flexible deforma- tion, which explicitly models geometric mesh elements of higher dimension achieve a more accurate and natural physical simulation. MAVEN establishes learnable mappings among 3D cells, 2D facets, and vertices, enabling flexible mu- tual transformations. Explicit geometric features are incorporated into the model to alleviate the burden of implicitly learning geometric patterns. Experimental re- sults show that MAVEN consistently achieves state-of-the-ar art performance across established datasets and novel metal arretch-bending task featuring large defor- mations and prolonged contacts. The code is available at https://github..omm cow./ee--eee--eeeeeee feng27/MAVEN.
INTRODUCTION
Flexible deformation and contact of solids are prevalent across a wide range of real-world scenarios, ranging from industrial manufacturing (Tao et al. 2025a blc), aeronautical engineering (Phanden al.] 202T), nuclear materials (Allen al. 2012). Accurate modeling of these behaviors and their evolution significantly enhances the understanding of these scenarios. Although many classical numerical solvers, such as Finite Element Methods (Dhatt et al., 2012) and Material Point Methods (Bardenhagen al., 22044), have been developed for solid systems, achieving the desired accuracy incurs high computational costs, as they rely on fine meshes and small time steps due to low-order approximations and the iterative solution of large linear systems. Recently, deep learning (DL) has rapidly emerged as a powerful tool for efficient physical simulation, showing great potential, particularly in molecular dynamics (Jumper al. 2021), fluid simulations (Li et al. 202T), and structural solid deformations (Pfaff et al. 2020).
Among these DL-baasdd solvers, graph neural networks (GNNs) have demonstrated superior per- formance in the domain of solid deformation, due to their ability to handle dynamic unstructured meshes and perform nonlinear regression on graphs (Sanchez-Gonzalez al. (2020) Gao al. [2022] Gladstone et al. 2024). To handle irregular solution domains, existing GNN-based methods input
"zhe_ fengeffn2222 (teng27@pku.edu.co 1 "oorespondenccrz//uuuu soton.ac..o. ac.uk, www.huiiliu@ppuu.cco

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Node-hased Graph Abstract
Mesh Geseritinn
Physies Evolution +
Mesh Defommation
GNN
(a) Mesh division from original physical states. The purple region represents fixed rigid body, while the blue region represents the deformable body subjected to clamping motion.
(b) Node-based graph construc-
Mesh Aware
Volumetric Encoding
Internal
Fails denect contacts
Interaction
Cell
Swap and Contact
Facet Vertex
Information Storage
detect cadius
(c) Mesh aware volumetric encoding based on 3D cells. 2D facets
(d) Node-based method ignores contact features even with large ra- dius
Figure The physical state on the continuous material domain discretized using structured meshes. Node-based methods construct point-edge graphs from the mesh and apply GNNs for com- putation. However, such abstraction may overlook contact interactions. A more effective approach should incorporate higher-dimensional geometric structures in the mesh, such as 3D cells and 2D facets, which retain accurate geometric information after discretization.
unstructured meshes, where the geometry is discretized into multiple connected simple cells using regular polyhedron (Figure [a)). GNNs abstract mesh vertices into graph nodes to capture internal physical interactions, using edges defined by mesh connectivity, shown in Figure ((b) Inter-object
contact typically detected via an interaction radius. Mesh edges serve as pathways for propagating
physical quantities, making the mesh structure and its associated graph a central representation of the physical system.
Although GNNs are effective, their accuracy deteriorates sparse meshes, which are commonly used for computation efficiency concerns practice (Allen al.) 2023). As illustrated Figure I(d). the distance between the points in GNN differs from the actual distance between surfaces, and this deviation worsens under coarser mesh configurations. As result, contact information may be missing without appropriate detection radius. Increasing the radius may help, but with the cost of computations, there are still no guarantees of complete and accurate contact information. In addition, GNNs model internal propagation in approximating integral kernels (Anandkumar al., 2020 Li et al. 2025), which is based on positional vectors and physical variables. However, with coarse meshes, nodes may not be sufficient to adequately sample neighborhood regions, hindering the accurate construction of characteristics of the localized physical field.
These limitations are mainly due node-based modeling by only using the vertices. These methods represent meshes as graphs with edge features encoding distances, but existing approaches rely- ing on topological representations often lose critical spatial features required for physical aceuracy. Crucially, in addition to the vertices, the mesh contains a much more comprehensive set of high- dimensional geometric elements, that is, 2D facets and 3D cells, as illustrated in Figure [(c] colored orange and yellow. Our idea is that such high-dimensional elements like 2D facets and 3D cells could be incorporated to enable the model to better characterize geometric structures within 3D continuous space. With this key incorporation, graph based contact modeling can explicitly cap- ture boundaries and contact as face-to-face geometries, making approaches more suitable for precise simulations. Additionally, though in the of coarse mesh discretization, the model might
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lead to inaccurate integral approximations based on discretely sampled vertices, volumetric cells could compensate and maintain stable computations by retaining geometric continuity.
To fully exploit high-dimensional geometric elements in mesh-based neural networks, we design a novel framework within an secodr-prrocesoor decoder architecture that explicitly embeds cell and facet elements into the model, thereby enhancing performance under sparse mesh conditions. Technically, we propose MAVEN, model based on Mesh-Aware Volumetric Encoding, in which we construct explicit nodes for each geometric element in the mesh, including vertices, facets, and cells. During each processing step, the vertex information is encoded into higher-dimensions elements using learnable position-aware aggregators. The internal interactions and external loads are then handled at the cell level through facet nodes, allowing information to propagate through the mesh via its higher-dimensional structures (cells, facets). This cell-facet co-design enables comprehensive geometric modeling beyond node-based approaches. MAVEN achieves state-of-the-arr performance extensive evaluations. The main contributions of this paper can be summarized follows.
We propose a paradigm that explicitly incorporates high-dimensional geometric elements into the simulation of 3D solid systems. This approach enables the model to capture precise geometric information and maintain stability under sparse discretization conditions.
We design MAVEN, model based on mesh-aware volumetric encoding that captures high- dimensional geometries by explicitly modeling both cell and facet elements. MAVEN fa- cilitates data transformation between elements through carefully designed transformation functions, and propagates information over a cell-facet graph using a modified two-stage message passing process.
We compare MAVEN with state-of-the-art methods on public elastic deformation datasets and metal bending problem, with elastic-plasti deformations and a coarse mesh. The experimental results demonstrate that MAVEN outperforms baselines solid deformation with an acceptable computational efficiency.
RELATED WORK
2.1 LEARNING PHYSICAL SYSTEMS WITH GNNS
Recently, the application of Graph Neural Networks (GNNs) for simulating flexible dynamics has emerged as a promising research direction (Sanchez-Gonzalee et al. 2020 Han el al. |22222022a Shlomi al. 2020 Gao al. 2022). As a baseline and essential method, MGN (Plaff al.] (2200 rep- resents meshes as graphs by treating vertices as nodes and using connectivity and proximity-base edges, within and across objects. adopts an relative positions as implicit geometric features and learning dynamics via message passing. Later studies primarily aim to improve message passing through more expressive architectures (Dwivedi hierarchical graphs to propagate infor- mation in various ranges (Cao ef al. 2023: Fortunato et al. 20022] Grigorev et al. 2023), and adopt a hybrid design that integrates both approaches ( YYu et al. 2024). Since these methods model physical features only on vertices, we refer to them as mude-based GNN approaches.
Processor-Deccder architecture, encoding
& Bressonn 202 Shao et al. 2022 Han et al. 2022b),
However, graph-based models often neglect essential high- digh-dimensional geometry, particularly in sparse meshes common in real-world scenarios. For inter-object contact, true interactions occur between surfaces, yet node-based GNNs approximate them via vertex distances, causing errors when contact regions extend beyond vertex discretization (Alen et al., 2023). For intra-object dynamics, some studies (Li et al. 2025 Anandkumar al. 2020) interpret message passing as an approxima- tion of local kernel function that performs discrete integration over information from neighboring graph nodes. In the sparse condition, meshes provide too few nodes to capture local geometry. Con- sequently, critical quantities such as volume and surface area are poorly estimated, and these errors propagate through message passing, leading to significant deviations in predictions.
2.2 GEOMETRIC ELEMENTS IN PHYSICAL SIMULATION
These limitations stem from node-based modeling that relics solely on mesh vertices, overlooking the rich set of high-dimensional geometric elements inherently present in the mesh. Mesh represen-
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tations also include 3D cell structures that accurately capture geometric partitioning in the continu- ous domain, and 2D facet structures that define boundaries between regions and encode inter-object contact information. These elements contribute to more stable computations, particularly under sparse mesh conditions. For example, classical numerical methods (Reddy 1993) [Bardenhagen et al., 2004) describe physical quantities within volumetric elements by defining family of shape functions (Zienkiewicz & Taylor] (205) that interpolate physical qualities throughout the cell, and contact penalty terms are imposed on the integrals over the boundary facets. Based on this modeling approach, numerical methods can maintain controlled errors even under sparse mesh conditions.
A limited number of studies have focused on incorporating nigh-dimensional geometric information into DL-based physical simulations to improve computational accuracy. PHYMPGN (Zeng et al.) 2025) follows discrete L.place-Beltte Beltrami operators (Reuter et al. 2009). using face areas and cosine values of neighboring triangles for single vertex as broadcast operator between node pairs. This operator is limited to 2D settings and presents significant challenges when extending to 3D mesh domains. FIGNet (Allen et al. 2023 Lopez-Guevara et al. 2024) constructs face-to-face edges to capture contact relationships. A Although effective for rigid bodies, these methods still face significant challenges in modeling internal propagation and dynamic deformations within 3D solids. To address this, we propose a novel bL-bdsed architecture for 3D dynamic deformation simulation that intrin- sically integrates high-dimensional geometric structures with hierarchical feature representations.
METHODOLOGY
3.1 PROBLEM DEFINITION
The evolution of Lagrangian system is initiated by an initial material domain Ro. together with field function U(0, z) that defines the initial physical quantities, such as displacement, velocity, and pressure, at each material point At each time t E [0, we focus on the current physical state U(t. of every point æ E Ωg- To enable discrete computation, the initial material domain Qn is partitioned, without overlap or omission, into set of regular tetrahedra (or hexahedra) cells C, which collectively form the mesh. The collection of vertices and surface facets from these regular regions defines the set of vertex V and the set of facets J of the mesh, respectively. Physical field information u at time is stored at each vertex U; E V to approximate the continuous domain, allowing the state of any point of material to be estimated by interpolation from the values at the vertices of the corresponding cell. The exact shape of each cell and facet at any given time is determined by the current state of deformation of its vertices. Excessive distortion or even fracture may occur as a result. In this work, we primarily consider scenarios in which the mesh does not undergo severe distortion, which aligns with the assumptions commonly made in industrial simulation settings.
The simulation trajectory of a physical system originates from an initial physical field u gov- erned by discretization mesh M = V.). The input of variable-time external forces (Fo, far ..... pkar) acting on the vertices drives the progression of the dynamic state, gen- crating the sequence of physical evolution {u", uºr _ u ukArj in discretization evolution time 0, At,... K&t- = T. The objective of the simulator is to predict next physical state (k+1)) from a history of previous states. In this paper. we consider {u", u (k=-1)) 1)Ar, t phAr] as input states. The rollout trajectory can be obtained by applying the simulator to the last prediction iteratively.
3.2 MODEL OVERVIEW
The overall architecture of MAVEN is illustrated in Figure 2 with the widely adopted "encoder- processor-decoder framework PPffff et al. 2020). Unlike node-based models, MAVEN additionally models each element in the cell set [C} and facet set {F} as individual nodes participating in mes- sage passing, thus enhancing the ability to capture high-dimensional geometric information. First, MAVEN performs feature extraction for all node types. The cell and facet nodes are initialized with their geometric characteristics, such as 3D volume, surface area, as well as 2D area and perimeter, while the vertex nodes retain their physical quantities as input features. Subsequently, we employ stack of processors to model physical interactions within and across solid objects. In each processor, the cell and facet nodes update their geometric representations from the nearby vertex nodes via geometric aggregator. A modified two-stage message passing is then applied to propagate information during cell-facet graph constructed by the relative geometric relationships
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Col Fast Gragh
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Topdig
Comnec
Two-
Geometric Aguregator
Passing
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Processor
Encoder
Geometric Aggregator
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The overall structure of MAVEN. MAVEN follows mncoder-processoof pipeline: it extracts geometric and physical features for vertices, cells, and facets, updates them through position-aware geometric aggregation and refined cell-facet message passing, and finally disaggregates the processed features back to vertices to produce smooth predictions.
Figure 2:
between elements. Finally, a disaggregation operation distributes the aggregated features back to the vertex nodes, generating smooth intermediate results. The final processor output is subsequently mapped back to the original domain to produce the predicted results.
In MAVEN, the cell and facet focus primarily on diverse features. The facet is a pivotal hub for information exchange, where external forces, object contacts, and the propagation of internal phys- ical quantities converge. The diverse information is integrated and subsequently transmitted back to their respective cells. (Allen et al. 2023) shows that faces can capture contact information more ef- fectively, which node-based models might otherwise neglect under sparse conditions. The cells fully characterize the geometric and volumetric properties of the 3D continuum domains. The adoption of volumetric features of adjacent cells as propagation coefficients for vertices significantly enhances the performance of 2D tasks (Zeng al. (2025), motivating our design the cell- fell-facet propagation framework, ensuring comprehensive geometric information within the model.
Next, elaborate on each key component of MAVEN. For convenience, in the following descrip- tion we denote A as the feature fusion operator that integrates multiple features into single repre- sentation, implemented via multilayer perceptrons (MLP) practice. Various A distinguished using the subscript and superscript notation.
3.3 ENCODER: SHOMETRY-INFORMEE FEATURE ENCODING
The MAVEN encoder constructs feature representations for the cell, facet, and vertex nodes while also processing external force conditions and inter-facet contact relationships.
Vertex node encoder For given vertex node V and its associated physical field quantities uh,, apply standard GNN practices encode quantities into a high-dimensional latent space to derive the vertex node feature h
h = (u((t)))
Cell and facet node encoder Since cells and facets not possess direct input features, we consider computing their representations from high-dimensions geometric properties. Inspired by the clas- sical finite-volume method (Eymard et al., (2000), we posit that both the internal volume and surface area of region are critical geometric descriptors. Accordingly, we use volume and total surface area as initial geometric features for each cell, while area and perimeter are used to characterize facet nodes. In addition, we incorporate the initial geometric attributes of each element to enhance the model's awareness of high dingh-iimnsional geometric variations over time. Let 2(c,), E(c,) be the volume and surface area of cell c, and aff:), X(fe) be the and perimeter of facet Se, MAVEN
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generates cell and facet features he; and h follows:
ho. - (S(C)) E(d), ((C), I(e)),, by - (o()) o(1)))
(2)
Here, all h he., and y. projected to the same latent dimension, which is set to 128 in practice.
Facet-to-facet features Instead of constructing edges between vertices, MAVEN establishes contact
connections directly between the interacting facets. We improve (Allen et al. 2023) by applying a simplified Bounding Volume Hierarchy algorithm (Clark) 1976) to detect all pairs faces within a collision radius For two contacting facets fs and fr. the translation equivariant vectors of each face edge include: (1) the relative displacement between the center points di P, two faces, (2) the vertices spanning vectors from one face relative the center point on another face d Xs Ps, d X,, Pr. and (3) the normal unit vectors of the face of the sender and receiver faces nts Rl. MAVEN generates face-to-face features hf.-tf. as:
h h A *+* ([d degleye (d. Iryee ng m,
External force features External forces are defined as scripted motions for specific surface vertices, which means that their next-step positions x are explicitly provided in the current step. We define the external force feature for each node h as its relative displacement to the next time step, and introduce flag indicate whether the node is scripted.
[1. 0,
1 if e is scripted
h
(4)
if is not scripted
In practice, scripted motions are typically applied over entire surface regions rather than isolated vertices, making it essential to impose constraints at the surface level. Therefore, we define scripted features on each facet h by concatenating motion-related features of its associated vertex nodes.
h = A (concat,, E. (h:,))
(5)
To ensure translational and permutation invariance, we sort the vertices of each facet by their dis- tances the facet centroid, thereby enforcing a unique representation.
3.4 PROCESSOR
All features extracted by the encoders are fed into processor module composed of stacked layers. the 1-th layer, the processor takes the vertex features of the previous layer h1/-1 and applies a geometric aggregator to incorporate the vertex information into the facet and cell nodes to generate geometric features hn and hf. Two-stage message passing is used to propagate physical information across high-dimensional elements, where messages are first exchanged subsequently to cells. Finally, an inverse disaggregation decoder maps the updated cell-level features back to the vertex nodes for residual updates, producing spatially smooth fcatures h over domain.
facets and
Geometric Aggregator Since vertex features updated through the processor, it implies that the features of the cells and facets must also be updated. We update the features each element by aggregating the features of all vertices that it contains. A straightforward approach is to concate- nate the initial features with those of all associated vertices, followed by an aggregation operation. However, since each element contains a relatively large number of vertices (e.g., eight vertices in a hexahedron), this approach results in significant computational overhead. Another approach is average the features of all vertices, following the conventional GNN. However, this leads vere homogenization of features between vertices and overlooks the relative geometric relationships between the nodes in their corresponding cells.
Inspired by the shape function (Reddy) 1993) in numerical solvers, which describes physical fields in cell using local coordinates, MAVEN constructs aggregation coefficients from each vertex the element based on the local coordinate system of the element. These coefficients are shared across all processor layers. Let de,,,, denote the position vector from the center of the cell c to vertices 0j E c; similar to d f.... Based on the local coordinate system, we employ an MLP to generate a centered normalized vertex aggregation weight dca.y E R for each cell.

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Oct, .... UC MLP
concat UK(
(de,.))
"U,...., UK-11 represents e; (6)
Similarly, coefficients vo 2 ( for each facet f; can be derived, where K and M denote the number of associated vertices for each cell and facet, respectively. The coefficients obtained are then utilized to perform weighted aggregation for element feature update. We also sort the vertices within each cell to ensure permutation invariance.
h = A heeo
Σ oE (KI1
Ocooh,)) hy; = A (h
Σ
(7)
PE
Message passing on cell-facet graph After extracting features, we construct a bipartite cell-facet graph G = (VG [C,F), Eg) to explicitly capture the topological and geometric relationships between volumetric elements and their boundary interfaces. The Eg contains pairs (q. f5) for fj q. MAVEN performs two distinct message passing steps, each dedicated to the facet and cell nodes, respectively. In the first stage, information is aggregated through facet nodes, which serve as "edges', bridging not only adjacent cells but also mediating interactions between external forces or contacts and the internal dynamics of the object. Inter-object contact interactions are represented on the facet level, where information from all face-to-face edges is aggregated. To incorporate context from adjacent cells, we similarly employ learnable aggregation scheme with af,,s.
5-511 h
Σ A (hf.-s... h/y)), h +F. A f.=h
(h h
F-J.
h
Σ
f b/y)
(eg.fi)ERo
(8)
In the second stage of message passing, each cell aggregates information from its associated facets. A similar strategy is adopted, employing symmetric aggregation coefficients ac,,f afiee to com- bine messages from multiple surfaces.
h -,
A
Σ
Oct,ty he Fd)
Ee
Geometric Disaggregator Finally, the nigh-dimensional geometric information encoded in each cell is retransmitted to its associated vertex nodes using the same symmetric aggregation coefficients an.ey = dej,u.- residual connection is applied update the vertex features. This disaggregation at the vertex level facilitates boundary-aware averaging of cell-level information, contributing to globally smooth predictions across the domain.
h h-v. A, V
Σ ErEC
he hey CC,)),
[+1 h
h² h + FFN(h + b)t)
(10)
where F-N represents the feed-forward network used in the Transformer (Vaswani al. 2017 The next layer uses h the input vertex features.
3.5 DECODER AND UPDATER
Our model decodes the features of vertices hú using MLP decoder to predict the velocity $1+1 and the physical quantities at+1 for the next state. The next position f++1 estimated by first-order integration *++ 3e+1 + x²
Training Loss We the one-step MSE loss training objective. Since other physical quantities may be included, MSE in flexible dynamics is calculated as follows:
1 |V
L =
1122 + |
| ||ott1 - 22+1
(11)
3.6 DISCUSSION
Here, briefly discuss how MAVEN differs from existing approaches.

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Compared to classical FEM methods, MAVEN learns complex, nonlinear physical behaviors di- rectly from data, avoiding the hand-crafted constitutive models required in FEM. generalizes across varying geometries and boundary conditions, enabling faster inference and improved scala- bility for large-scale simulations.
Compared to basic DL methods such MGN, MAVEN propagates physical information through high-dimensional geometric elements, including cells and facets. This approach introduces a small amount of additional overhead. However, enables MAVEN to accurately capture geometric infor- mation, which enhances the model's awareness of local neighborhoods and significantly improves its stability under sparse meshing conditions.
Compared to hierarchical approaches, MAVEN focuses mainly on accurately capturing local geometric details. Although hierarchical methods are effective at modeling long-range interactions, they offer limited benefits for precise geometric representation. Moreover, MAVEN can be readily extended to graphs after pooling through an automatic mesh, which we leave for future work.
Compared to PhyMPGN, which depends on the cotangent Laplacian for local geometric, MAVEN avoids the inherent limitations of Laplacian-basee -aaeed operators in 3D, where no symmetric, local, and linearly accurate purely geometric Laplacian exists (Wardetzky et al. (007). By explicitly modeling high-dimensional geometric elements and constructing operators through message passing, MAVEN provides more flexible geometric framework, better suited for 3D Lagrangian formulations, and can naturally adapt to other physical settings such as 3D Eulerian formulations.
Compared to FIGNet, which is tailored for rigid-body contact and lacks mechanisms for modeling volumetric physical propagation, MAVEN explicitly represents cells and performs message passing over nigher-dimensional geometric elements. This allows MAVEN to capture intra-object dynamics with much higher fidelity, especially under sparse mesh. In our experiments, we treat FIGNet as ablated variant using only facet-level information, while MAVEN's joint use of facets and cells yields better accuracy, underscoring the importance of modeling internal geometric structure.
EXPERIMENTS
Datasets To evaluate the efficiency MAVEN, we our model on datasets with differrnnt com- plexity in 3D solid simulation. The deforming plate (DP) (Pfaff et al., 2020) and cavity grasping (CG) (Linkerhägner et al.) 2023) are typical public datasets for the autoregressive elasticity task. The DP dataset contains relatively dense tetrahedral meshes, while the CG data set has coarser meshes. To further explore plastic scenarios, we establish Metal Bending dataset (MBD) inspired by (Clausen et al. 2000) in real-world manufacturing, representing a class of solid deformation involving elastoplastic deformation, large displacements, and very coarse hexahedral meshes. In addition to target geometry, also predict attendant physical quantities in experiments, including inner stress (Stress) and equivalent plastic strain (PEEQ). The rollout steps for all datasets are re- stricted to between 75 and 125 for consistent comparison. We briefly present the motion process of the dataset in figure 3 See Appendix A for more details.
Bending
Grasping
Forming
(a) Cavity Grasping
(b) Deforming Plate
(c) Metal Bending
Figure
Visual description of the dataset.
Baselines We comprehensively compare MAVEN against baselines with node-based graph simula- tors. These include classical node-based models MGN (Pfaff et al.) 20200) and Graph Transformer (GT) (Yun et al. 2019), as well as hierarchical models HCMT YYu et al. (024)) and HOOD (Grig-)
orev et al. 2023 To further distinguish the functions between the cell and the facet elements,
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Table 1: Rollout results(- x 102) for MAVEN and other baselines, with 50-step rollouts and full-
sequence rollouts. Our results are derived by averaging the root mean square error (RMSE) values computed across all intermediate steps and test datasets.
CG
DP
MBD
Model
Rollout
Pos
Pos
Stress
Pos
Stress
PEEQ
50
6,38
14.01
01.495,3644r9..m
686.1927us
12508.0744 272392.56
0.24
MGN
ALL
16.8 46.89
23.65xa.19
10,623,89004455
2012.16:29966
9737.58.287.61
1.45.00000
50
6,1 65157
14.7240.30
24,384,076
678.87=39.9
10368.29 292173..58
000000000
GT
ALL
16.692
26.77=0.52
32,171.330+224711
1406.61
14255.72s1208.5)
2.07
50
6. 14
14.46e0.47
12,333,3588s6k..e
251.86=52.23
19940.56=2662.0
0.45+0000
HCMT
ALL
16.87=022
24.94s0.76
30,317,188
188
2003.30e77.71
11539.27
1.30
50
6.96 6.96-0.083
14.27+0.32
23,474,655
3-2597778
623.57=23.47
11739.02Hw2...
0.37
HOOD
ALL
18.8 8.84-085
20.01+0.30
1762.41
8352.52-442.6
1.56.0000
50
6 0,26200066
14.7420.18
11.926,010=91.6
515.17
5583.7 1
0.22a9.12
FIGNet
ALL
17.59+a.1
26.51
31,4491,188-255
1030.57=15.00
5402.3 +805.40
1.09+0.52
50
13.78 5.21:0.0066
21,667,,48::10.22
276.73=44.16
4901.56x46.06
0.20200.ss
MAVEN
ALL
15.41=a11
23.41a0.32
27,907,490:1s5.00
810.42:24.08
4776.72s71.20
1.01
Improv.
13.07%
1.33%
5.49%
33.82%
11.90%
8.67%
adapt FIGNet (Allen et al., 2023) to propagate internal physical quantities through the vertices. The specific settings of the models are provided in Appendix
4.1 EXPERIMENTAL RESULTS
Rollout Results Table D shows the rollout results of all models, demonstrating that MAVEN consistently outperforms state-of-the-art methods predict all physical quantities. From fine- grained to coarse meshes, MAVEN achieves incremental average improvements of 3.41%. 13.07%, and 18.13% across the three datasets, respectively. This indicates that explicitly capturing high- dimensional geometric features is beneficial for physical simulation and becomes even more critical under sparser mesh conditions. the MBD dataset, geometry-basee methods (FIGNet, MAVEN) significantly outperform node-based approaches. In addition, the cell-element-wwarr architecture enables MAVEN to better capture variations three-dimensional volumes compared to FIGNet.
Visualization Figure presents the visualization results. Compared with other methods, MAVEN adopts a bell-based propagation approach, allowing physical contact information to transmitted more effectively throughout the entire deformable body. As result, MAVEN achieves lower emrors even in regions that are far from the deformation part. At the same time, the use of facet-based contact detection allows MAVEN and FIGNet to maintain stable contact even under particularly coarse meshes. More visualization results and analyses can be found in Appendix
4.2 ABLATION STUDY
We validate two key components in our model, specifically, the aggregators that explicitly compute geometric features, and the feature aggregation method based on local geometric coordinates. We conducted systematic ablation studies on the MBD and CG to evaluate the contributions of different component models, testing three models: 1) Our full model; 2) Model A, which replaces geometric- based aggregation coefficients with degree-averagin approach, is used to validate the effectiveness of our geometric aggregation strategy based on local coordinate systems; 3) Model B that replaces geometric input features of 3D cells and facets with zero padding, is used to assess the importance of explicitly computing geometric features; and 4) Model C, which removes explicit modeling of cell and facet nodes by averaging their precomputed geometric features onto neighboring vertex nodes and propagating them through standard message passing, is used to evaluate the role of explicilly representing digher-dimensional geometric elements.

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Ground Truth Step 0,50,104
Geometry (MAVEN) 60,80,104
Geometry (GT) 60,80,104
1992: 01
Ground Truth Step 0,50,99
Geometry (MAVEN) 50,75,99
Geometry (FIG) 50,75,99
Figure 4: Visualization of error maps. The first and second rows respectively show sample visual- izations from cavity grasping and metal bending datasets.
CG
MBD
Model
Rollout
Pos
Pos
Stress
5.21
276.73
4901.56
Ours
ALL
15.41
810.42
4776.72
50
6.26
402.79
5302.67
Model A
ALL
17.45
926.71
6683.94
50
5.41
391.40
6839.26
Model
ALL
15.933
1652.31
6680.39
50
6.05
632.71
9802.55
Model
ALL
17.08
1680.20
10375.86
Table 2: Ablation results on CG and MBD.
shows the averaged error results in the test datasets. We observe that replacing geometry- aware aggregation with simple averaging (Model A) performs significantly worse on the generally sparse CG dataset. This indicates that such sparsity requires capturing detailed intra-clement geom- etry to surpass standard node-based methods. Similarly. omitting explicit geometric features (Model B) leads severe degradation on the highly sparse MBD dataset. This suggests that in extremely sparse settings, traditional GNNs struggle to infer local geometric structure implicitly and therefore fail accurately capture underlying geometric topology. For Model C, which does not explicitly model high-dimensional geometric elements, its performance on both datasets is close to traditional node-based methods. This suggests that adding geometric features alone, without modeling higher- dimensional topology, is insufficient for nodes to fully capture surrounding geometric structure.
Table 4.2
These ablation results collectively demonstrate the effectiveness and necessity of MAVEN's de- sign choices, including the explicit modeling of higher- dimenssoill geometric elements, the explicit computation of geometric features, and the use of geometry aware aggregation for message updates.
awwae
5 CONCLUSION AND LIMITATIONS
We propose MAVEN, an architecture that models mesh geometry with high-dimensional features and learnable aggregation to simulate 3D solid contact and deformation on coarse meshes, out- performing baselines in physical propagation and contact representation. Despite these strengths, MAVEN remains sensitive to mesh quality due to its geometric modeling. In addition, as local op- erator, it does not yet natively support efficient long-range interactions. Extending the framework to thin-shell, surface-based. or Eulerian systems also requires additional geometry-awan adaptations. Detailed discussion provided in Appendix G
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ACKNOWLEDGMENTS
This work is supported partly bythe National Natural Science Foundation of China (NSFC) 62576013, National Key Reasearch Plan under grant 40.22224FFC2004004 the Jiangsu Provin- cial Key Research and Development Program under Grant BE2022065-1, BE2022065-3, and the Ningxia Domain-Specifes Large Model Health Industry R&D No. 2024JBGS001.
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A
DATASET
Deformation
Figure 5: Description Deforming Plate from
Pfaff et al. 2020)
Deforming Plate Dataset (Pfaff et al. (020) This dataset contains 1,200 3D dynamic simulations of a hyperelastic deformable plate pressed by a rigid solid (F Each sample records the geometry of the plate and the internal stress, with average of 1,271 points per simulation. In our setup, unroll this dataset with a step size of 100 iterations to maintain consistent sequence lengths across all datasets. Following the protocol of the original article, we split the data into 1,000 training samples, 100 validation samples, and 100 test samples.
Cavity Grasping (Linkerhägner et al. 2023> The dataset comprises three-dimensional dynamic simulation of deformable cavities subjected to gripping by rigid gripper (Fig. 6). containing a total of 840 samples. The gripper is modeled as two rigid bodies, corresponding to its jaws, which undergo motion in opposing directions. The deformable objects are cone-shaped cavities generated with randomly assigned radii in the range [50, 87.5]. Their material properties are specified as elastic, with Poisson's 's ratios drawn cyclically from the set -0.9, 0.0, 0.49. This dataset, primarily designed for autoregressive modeling tasks, provides temporal trajectories over 105 simulation steps. Each sample contains 1,386 points. Following the strategy of the original article, 600 samples are used for training, 120 for validation and 120 for testing.
[=1
80
Figure 6: Description Cavity Grasping from (Linkerhägner et al. 2023
Metal Bending Dataset To rigorously validate the capability of MAVEN, designed an industri- ally inspired test scenario featuring large deformations, coarse mesh discretization, and elastoplastic material behavior. This configuration mimics challenging real-world engineering applications such as metal-forming processes (Clausen et al., (000),, where computational methods must simultane- ously handle geometric nonlinearity, material nonlinearity, and under-resolved meshes while main- taining physical accuracy and numerical stability. The results are calculated by ABAQUS software (Abaqus) 201T).
As illustrated in Fig. this scenario involves clamping straight aluminum profile using a special- ized device, where the profile is first stretched beyond elastic limit to induce plastic yielding, and then progressively pressed against a curved steel die through controlled displacement; the resulting compressive contact forces generate permanent plastic deformation achieve the prescribed target geometry. The straight metal component is modeled slender component of size 2 200mmm² to accurately replicate field conditions. For discretization, we employ a 1 x 3 x 5mm mesh grid throughout the component. The material properties are configured as measured aluminum profile characteristics, with a Poisson's ratio of 0.37, Young's modulus of 69,000, and the true stress-strain curve depicted in Fig. aa.. The 3D rigid die geometry is determined by its cross-sectional profile
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Strech
Bending
Figure 7: Description of Metal Bending.
30
130
000
3.00
6.4
300
600
Ga Stadc
(a)
(b)
Figure 8: (a) True Stress-Strai Curve of Aluminum Profile. (b) Exemplar Motion Trajectory for Clamp Mechanism.
and characteristic guiding curve. To ensure continuous contact between the aluminum workpiece and the die during bending operations, this guiding curve must maintain convexity and smoothness. We construct the curve by combining two circular ares lying in the XY and XZ. planes, respectively, each defining principal curvatures, and then synthesizing them into composite spatial curve. In
d the ellipse / + F = 1(a > > 0) with a focal point at (c, 0), we employ three distinct uni- form distributions control the morphology of the die, which and a ~ [1770, 190 (unit:mm). The clamping tool's motion trajectory is generated by applying classical evolute algorithm along the characteristic curve (see Fig for illustrative example). The dataset contains an average of 1163 nodes per sample, with rollout lengths varying between 75 and 125 timestep. We generated total of 1000 trajectories, divided into 800 for training. 100 for validation, and 100 for testing.
L1[0.1, 0.3), U(001, 0.3]
B METRICS
In our autoregressive framework, consistent with the baseline methods under comparison, we em- ploy the Root Mean Square Error (RMSE) the evaluation metric. Given the predicted physical quantity Js and ground truth value y of current vertices, the RMSE is calculated as:
RMSE = 1 11
(12)
For positions, take the distance between two points the sole comparative physical quantity. To evaluate the performance across multiple trajectories and frames, we report the final error metric as the average value computed over all frames for each individual trajectory:
ERROR ΣM EJ RMSE,,
(13)
EMT
M
The number trajectories is denoted by M, where T₁ represents the length (number of timesteps) of each individual trajectory.
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C
IMPLEMENTATION
Table 3: Key hyperparameters and parameter numbers of models.
Deforming Plate Metal Bending Cavity Grasping
HIDDENS
[128, 128)
[128, 128)
[128, 128]
MGN
LAYERS
15
15
15
PARAMETERS(M)
3.85M
3.85M
3.85M
HIDDENS
[144, 144)
[144, 144]
[144, 144)
HCMT
LAYERS
12+3
6+9
10 + 5
PARAMETERS(M)
3.24M
3.24M
3.11M
HIDDENS
[96, 96]
[96, 96]
[96, 96]
FIGNet
LAYERS
15
15
15
PARAMETERS(M)
3.08M
3.48M
3.05M
HIDDENS
[128, 128]
[128, 128)
[128, 128]
HOOD
LAYERS
12+3
6+9
10+5
PARAMETERS(M))
3.56M
3.16M
3.47M
HIDDENS
[128, 128]
[128. 128]
[128, 128]
GT
LAYERS
15
15
15
PARAMETERS(M)
3.64M
3.61M
3.57M
HIDDENS
[96, 96]
[96, 96]
[96, 96]
MAVEN
LAYERS
15
15
15
PARAMETERS(M)
3.11M
3.15M
3.08M
Model Hyperparameters To ensure fair comparison, all models were evaluated under similar parameter budgets and computational costs, with detailed configurations provided in Table 3 Due to the larger paramcter count per block in MAVEN and FIGNet, these models employed fewer propagation layers. For HCMT and HOOD, while the optimal hierarchy layer count originally reported was 5 layers on the deforming plate dataset, we reduced it to 3 layers in our implementation because the hierarchical partitioning algorithm failed to correctly subdivide new meshes beyond the fifth layer. For all datasets, adopted a uniform batch size of in all experiments.
Training Implementation All models were uniformly trained in 1M steps. We used Adam opti- mizer (Kingma 2015), with learning rate decreasing from 10-4 10-5 All experiments were conducted using a single RTX 3090 24GB GPU and repeated three times for the calculation of the standard deviation. All models were trained using Mean Squared Error (MSE) as the loss function, where each physical quantity was first normalized via Gaussian standardization and then summed directly to compute the final loss. This standardized approach ensured a fair and controlled compar- ison to objectively assess the relative performance of each method.
Dataset Input and Detect Parameter The inputs and outputs of each dataset, as well as the cor- responding node-based contact detection radius and face-based contact detection radius r',, are shown in Table To ensure fairness, the node-based model and our model are able to detect nearly the same number of contact edges.
Table 4: Model input, output and contact detection parameters for dataset. S denotes stress, and P denotes PEEQ.
Dataset
Input
Output
rw
FF
noise
Deforming Plate
typer, of x0 St.
x
1
2/.
Se41
St.
0.03
0.01
0.003
Cavity Grasping
types, _! x 1-1
[+] x²
0.1
0.05
0.01
Metal Bending
type,, x
22 P. x
x/ Sh+1 - St. pp+1 P
0.3
0.1
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Geometry (FIGNet) 60,80,104
Geometry (HCMT) 60,80,104
Ground Truth Step 0,50,104
Geometry (MAVEN) 60,80,104
Geometry (GT) 60,80,104
US2E-C1
Geometry (HOOD) 60,80,104
Geometry (MGN) 60,80,104
Figure 9: Visualization of distance error map Cavity Grasping.
D MODEL EFFICIENCY
We briefly discuss computational efficiency of MAVEN. Thble the runtime performance of various baselines two datasets. The computational overhead of MAVEN primarily stems from geometric feature computation and inter-node mapping operations, particularly on hexahedral meshes. However, MAVEN still achieves efficiency improvement of 2922.66% over the Abaqus simulators (712.44ms per step) in the metal bending dataset.
Table 5: The inference time per step(ms) for each model on three dataset
Dataset
MAVEN
MGN
FIGNet
HCMT
HOOD
CG
25.48
18.37
24.15
55.05
51.37
22.02
DP
48.27
39.73
43.08
64.94
57.28
54.83
MBD
23.57
17.42
17.08
43.68
36.45
22.34
E VISUALIZATION
Fig. g and 10 present the complete visualization results for each dataset. It is worth noting that in the Metal Bending dataset, wessagg-passing-base methods MGN and HOOD exhibit severe mesh distortions, especially at contact regions. In contrast, graph-attention-bass methods HCMT and GT better preserve mesh shapes, though their overall structures become distorted. However, all of these methods suffer from severe interpenetration issues, as further illustrated in the rollout animations provided in the Supplementary Material. We believe that this phenomenon arises because individual nodes cannot accurately identify their intended contact regions. Enlarging the detection radius intro- duces many irrelevant points into the contact set, which, to some extent, is also exacerbated by the large discrepancies in mesh lengths across the x. y. and axes. More specifically, near the fixed end, the mesh exhibits severe interpenetration. Closer to the moving end, the deformable body is heavily constrained by the lower rigid body, preventing it from generating the correct motion and causing to stagnate. This indicates that baset- contact detection of critical importance when dealing with sparse meshes.
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Geometry (GT) 50,75,99
Geometry (HCMT) 50,75,99
Ground Truth Step 0,50,99
Geometry (MAVEN) 50,75,99
Geometry (FIG) 50,75,99
Geometry (MGN) 50,75,99
Geometry (HOOD) 50,75,99
Figure 10:
Visualization of distance error map on Metal Bending dataset. Gray indicates that the at this location exceeds the given error upper bound.
F STATEMENT ON THE USE OF LARGE LANGUAGE MODELS
We declare that LLMs mainly used in this paper improve the clarity and fluency of the text, and to a limited extent for code generation in technically mature modules, to reduce repetitive work.
G LIMITATION AND FUTURE WORK
In this section, we briefly discuss the limitations of MAVEN and outline several promising directions for future work.
The dependence on mesh quality. Since MAVEN explicitly models cell-level information, the ge- ometric quality of the initial mesh can substantially affect performance. This behavior resembles that of traditional numerical methods (e.g., FEM, FVM) rather than node-based GNN models. In our experiments on the Cavity Grasping dataset, we compared meshes generated by a basic triangu- lation procedure with those produced by higher-quality meshing algorithm. The results show that MAVEN's performance degrades markedly when operating on low-quality meshes. However, most existing deep learning datasets do not provide their original mesh discretizations, and meshes recon- structed from point clouds typically exhibit very low geometric quality. Consequently, generating more datasets with high-quality meshes, well reducing sensitivity mesh quality, constitutes an important direction for our future work.
Support long-range interaction. While MAVEN fundamentally designed as local operator that explicitly leverages high-dimensional geometric structures (facets and cells). it does not yet na- tively capture long-range interactions. Existing graph pooling strategies (e.g., kNN clustering, BFS grouping, global slicing) can, in principle, be directly applied to MAVEN cell-facet graph to expand the receptive field. However, these methods often introduce significant computational overhead and offer limited benefit in modeling a global high-dimensional geometric structure. Developing a geometry-awan hierarchical extension that supports efficient long-range information propagation therefore represents important direction for future work.
Extend to boarder range systems. MAVEN is designed primarily for elastoplastic solid sim- ulation on arbitrary 3D volumetric meshes, where explicit modeling of facets and cells provides high geometric fidelity. Although MAVEN can be adapted to thin-shell or surface-based geometries through appropriate redefinition of geometric features, and extend to Eulerian formulations ing fixed meshes and standard coundary-condition encodings, these adaptations require additional geometry-aware considerations and suitable datasets. Extending MAVEN into unified framework
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capable of supporting surface- based systems, thin-shell structures, and Eulerian physical simulations on sparse and irregular 3D meshes remains an important direction for future work.
19


Reviews:
[{"review_id": "C1YZvSL3TF", "reviewer": "Reviewer_31zk", "summary": "This paper proposes a mesh-aware volumetric encoding network to predict physically meaningful 3D deformation. The paper shows better performance compared with alternative baseline models.", "questions": "As all the examples shown have limited variations for the geometry, does the method generalize to more flexible shapes?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The overall idea of encoding meshes at different levels is plausible.\n\nThe performance improvements compared with existing methods.", "weaknesses": "The deformation evaluated is somewhat limited. For more diverse materials and object shapes, it would be useful to show the generalizability. \n\nThe proposed method essentially relies on volumetric information, but it is unclear whether the compared methods are surface or volume based. This leads to questions regarding whether the evaluation is fair.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "jSzhvZPhPg", "reviewer": "Reviewer_2viX", "summary": "This paper proposes a model that integrates volumetric geometric information into a GNN architecture to handle dynamic deformation simulation of meshes. Instead of using only vertices, as in conventional methods, the model integrates geometric features obtained from facets and cells and proposes a method for processing these elements. By incorporating this geometric information, the model demonstrated the advantage of achieving good performance in sparse meshes compared to other baselines.", "questions": "See Weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I agree that the existing baselines failed to adequately capture the volumetric geometric information of the mesh. It is a strength that they proposed an architecture capable of containing all information that can be extracted from all possible constituent elements of the mesh: vertex, facet, and cell.\n\nThe architecture, designed to be applicable regardless of whether the mesh is tetrahedral or hexahedral, is a good foundation for generalization.", "weaknesses": "1. The biggest concern is that the computational cost for calculating the surface area, volume, and perimeter of cells and facet at every step seems to be vast for a fine mesh. Please provide the computational cost of the process of calculating geometric information based on the number of nodes and edges.\n\n2. It was stated that facets within the search radius r are found using BVH when searching for contact surfaces. If this is the case, there is a need to specifically explain how the problem illustrated in Figure 1(d) is solved even in a sparse mesh. This seems to be the biggest differentiator from node-based methods, but I do not clearly understand how it is resolved.\n\n3. If we assume that all facets in a sparse mesh within the radius r are in contact, a different problem from node-based methods might arise: since the model recognizes contact even when the facets overlap only slightly, wouldn't this lead to the issue of perceiving a wider contact area than the actual contact surface? A case study on how MAVEN resolves the issue shown in Figure 1(d) seems necessary.\n\n4. The ablation study is conducted at too coarse a level, making it difficult to figure out what is effective for the model's performance. That is, it is difficult to clearly confirm whether the proposed model performs better because it actually incorporates more geometric information. Is the complex aggregator utilized superior to a simpler method of merely averaging and including the surrounding cell and facet information in the node features?\n\n5. Long-range interaction is one of the important problems for GNN-based simulators. However, this methodology does not seem to include a discussion on that aspect.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "tsUfQrZzLQ", "reviewer": "Reviewer_ZTpm", "summary": "The proposed paper, MAVEN, presents a graph neural network (GNN) model for simulating physics, particularly flexible solid deformations, including contacts. The key contribution of the work is the incorporation of additional explicit geometric features (cells and faces) into mesh graph network line of simulators, which previously considered only vertices and edges. This design reduces the burden of implicitly learning geometric patterns, enabling the network to capture 3D spatial structure and behaviour better.  \n\nThe authors describe how to integrate these higher-order geometric entities using geometric aggregators and disaggregators, along with additional message passing on cell–facet graphs. MAVEN is compared against MeshGraphNets and several follow-up models in this category. It consistently achieves state-of-the-art performance across a few established datasets and on a new metal stretch–bending benchmark that features large deformations and sparse meshes.", "questions": "1.\tHow can MAVEN handle surface meshes, such as thin shells or 2D manifolds, without volume or Eulerian systems? Could the model be extended to such cases?  \n\n2.\tAround L285/L318, the paper states: “To ensure translational and permutation invariance, we sort the vertices of each facet by their distances to the facet centroid, thereby enforcing a unique representation.” It is not clear how this sorting leads to invariance. Please provide a detailed justification. \n\n3.\tWhat are the optimisable parameters in the method? In Section 3.5, a list of additional optimisable parameters or MLPs compared to the baseline method (MGN) could be added.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1.\tThe core idea that incorporating higher-dimensional elements, such as 2D facets and 3D cells, enables better geometric representation of volumetric solids is a strong conceptual insight. \n\n2.\tThe paper carefully explains how to integrate these geometric features, introducing geometric aggregators/disaggregators and modified message-passing schemes. The authors analyse design choices such as averaging versus learning coefficients in a local coordinate system and demonstrate where this new model is most beneficial (for coarse mesh discretisation and contact scenarios). Overall, it extends the expressive capacity of GNNs for physical simulation without much computational cost.\n\n3.\tThe draft is clearly written. It systematically and thoroughly investigates different aspects of the idea. Considering that GNN-based simulation has received qwuite a bit of traction, this work provides nice insights, valuable to this community.", "weaknesses": "1.\tIt is unclear whether MAVEN can handle beyond volumetric solids. MeshGraphNets can model a wider range of systems, such as cloth (thin shell) or fluid dynamics, whereas MAVEN seems focused only on deformable solids. It would be useful to clarify whether MAVEN can handle surface-based or thin-shell geometries (such as cloth) and whether a version using only faces, without volumetric cells, is feasible. This would also make the comparison with surface-based GNNs such as HOOD fairer. \n\n2.\tThe discussion focuses exclusively on Lagrangian (mesh-based) systems. The authors should comment on how MAVEN could extend to or differ from Eulerian formulations. Eulerian systems are explored in the original MeshGraphNet and its variants, but MAVEN's volumetric encoding will prevent GNN simulation in such scenarios. \n\n3.\tThe paper would benefit from positioning the approach relative to Physics-Informed Neural Networks (PINNs), see [Karniadakis et al.]\n\n[Karniadakis et al.] - Karniadakis, George Em, et al. \"Physics-informed machine learning.\" Nature Reviews Physics 3.6 (2021): 422-440.\n\nMinor comments and suggestions\n\n1.\tThe term *flexible deformations* (used in the title and abstract) is somewhat vague. Consider using elastic or soft-body deformations, which more accurately describe 3D deformable solids.  \n\n2.\tThe overview figure’s caption could be expanded to explain the method flow better.\n\n3.\tThere are a few typos:  \na.\tL292 likely refers to a geometric aggregator rather than an encoder.  b.\tIn Figure 4, the label GT in the top-right should probably read FIG.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "RPvufhb8lg", "reviewer": "Reviewer_3Wkt", "summary": "MAVEN explicitly models high-dimension geometric mesh elements for physical simulation.\n\n**I am not an expert in this field, so an accurate evaluation is difficult for me. I will defer to the assessment of other reviewers.**", "questions": "See Weakness section", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- It is important to utilize more diverse 3D information when performing mesh-based simulation.\n\n- The work demonstrates both accuracy and computational efficiency; despite using more information, the computational efficiency did not significantly degrade.", "weaknesses": "- As the authors themselves mentioned in the Related Works section, existing studies have also used high-dimensional information such as Cell.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)