Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
Authors: Sungwon Kim
Namkyeong Lee
Yunyoung Doh
Seungmin Shin
Guimok Cho
Seung-Won Jeon
Sangkook Kim
Chanyoung Park
Venue: ICML 2025 poster (2025)
Keywords: Mesh Neural Networks
3D analysis
Primary area: deep_learning->graph_neural_networks

Abstract: Mesh-based 3D static analysis methods have recently emerged as efficient alternatives to traditional computational numerical solvers, significantly reducing computational costs and runtime for various physics-based analyses. However, these methods primarily focus on surface topology and geometry, often overlooking the inherent thickness of real-world 3D objects, which exhibits high correlations and similar behavior between opposing surfaces. This limitation arises from the disconnected nature of these surfaces and the absence of internal edge connections within the mesh.  
In this work, we propose a novel framework, the Thickness-aware E(3)-Equivariant 3D Mesh Neural Network (T-EMNN), that effectively integrates the thickness of 3D objects while maintaining the computational efficiency of surface meshes. Additionally, we introduce data-driven coordinates that encode spatial information while preserving E(3)-equivariance or invariance properties, ensuring consistent and robust analysis. Evaluations on a real-world industrial dataset demonstrate the superior performance of T-EMNN in accurately predicting node-level 3D deformations, effectively capturing thickness effects while maintaining computational efficiency.

Thickness-wwre E(3)-Equivariant 3D Mesh Neural Networks
Sungwon Kim
Namkyeong Lee?
Yunyoung Doh Seungmin Shin Guimok Cho 3
Seung- Won Jeon Sangkook Kim
Chanyoung Park 12
Abstract
Mesh-based 3D static analysis methods have re- cently emerged as efficient alternatives to tradi- tional computational numerical solvers, signif- icantly reducing computational costs and run- time for various tbyyics-based analyses. How- ever, these methods primarily focus on surface topology and geometry, often overlooking the in- herent thickness of real- world 3D objects, which exhibits high correlations and similar behavior between opposing surfaces. This limitation arises from the disconnected nature of these surfaces and the absence of internal edge connections within the mesh. In this work, propose novel frame- work, the Thickness-aware E'3)-Eqqivariant 3D Mesh Neural Network (T-EMNN), that effectively integrates the thickness of 3D objects while main- taining the computational efficiency of surface meshes. Additionally, we introduce data-driven coordinates that encode spatial information while preserving E33)--quivariance invariance proper- ties, ensuring consistent and robust analysis. Eval- uations on a real-world industrial dataset demon- strate the superior performance of T-EMNN in accurately predicting node-level 3D deformations, effectively capturing thickness effects while main- taining computational efficiency.
Telkero Put
Neassn Corra ation (1)
L2 Norm (4)
Figure I. The left figures show mosh, with two different target nodes (*), their thickness paired nodes (*), thickness distance (-). and nearby nodes within radius (+). The right figures compare Pearson correlation and L.2 Norm between the target node's odees defor- mation and its thickness paired / nearby nodes within radius.
complex systems. Traditionally, computational numerical solvers like finite element methods (FEM) (Klocke et al., 2002; Felippa, 2004) have been the primary tools for such tasks. While accurate, these solvers often involve high com- putational costs and extended runtimes, limiting their scala- bility for real-time or large-scale applications.
In recent years, mesh-based 3D analysis methods (Plaff et al., 2020; Suk al., 2021; Trang al., 2024) have emerged as a promising alternative to traditional numerical solvers. By employing a graph-based representation of 3D object surfaces, these methods facilitate enhanced computa- tional efficiency while accurately modeling and preserving the intricate surface topology and geometric properties of the objects. However, existing mesh-based methods focus solely on modeling the surfaces of 3D objects, overlooking their thickness. Real-world objects such as plates, baskets, and layered materials inherently possess thickness, where in- teractions between opposing surfaces significantly influence one another. This is because many surface-related physical behaviors (e.g., deformation and bending) are directly in- fluenced by bending, shear, and buckling stiffness, all of which depend on thickness (Okafor & Oguaghamba, 2009; Wang, 2010; Renton, 1991; Gürdal al., 2008). Without internal interactions to model the empty space enclosed by these surfaces, these methods fail to capture the coupled dy- namies and correlations between opposing surfaces, leading to inaccuracies when applied to objects with thickness.
1. Introduction
Advances in static analysis have become essential across var- ious fields, including structural engineering (Whalen et al., 2021), materials science (Panthi et al., 2007; Wei et al., 2019), and geophysics (Ren & Tang, 2010; Schwarzbach et al., 201 1). These analyses enable detailed physics-based predictions, such as deformation, stress distribution, and load testing, which are critical for designing and optimizing
'Graduate School of Data Science, KAIST, Dacjeon, Republic of Korea FIndustrial & Systems Engineering, KAIST, Daejcon, Re- public of Korea 'LG Electronics, Pyeong-tack, Republic of Korea. Correspondence to: Chanyoung Park <cy. @ kaistac.kr's kr">.
To quantitatively illustrate the significance of these inter- actions, we present an analysis in Fig. 1, highlighting the strong relationship between opposing surfaces that collec- tively define the thickness of an object. On these surfaces, we identify two nodes that are positionally aligned on each opposing surface (i.e., with coinciding normal vectors) as a thickness node pair. To verify the highly correlated and simi-
Proceedingy of the 42nd International Conference Machine Learning, Vancouver, Canada, PMLR 267, 2025. Copyright 2025 by the author(s).

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Thickness-aware E()--Equivariant 3D Mesh Neural Networks
lar behavior of thickness node pair, we compare the Pearson correlation and L2 Norm between the deformation of the tar- get node (o in Fig. 1) and its thickness paired node (+). We also include comparisons in terms of the mean deformation of nodes within a defined radius' (+). as existing mesh-based methods rely on adduu--asedd aggregation (Pfaff et al., 2020; Anandkumar et al., 2020). As shown in Fig. 1, thickness node pairs exhibit significantly higher correlation and simi- larity compared to the mean of nearby nodes within radius, demonstrating that the modeling relationship among thick- ness node pairs would be beneficial in accurately modeling the behavior of 3D objects with thickness. However, mesh- based objects, which represent the geometry and topology of surfaces, face challenges in accurately modeling these in- teractions due to the lack of connections between opposing surfaces within the mesh.
coordinate-based information altogether. However, these simplifications limit their ability to capture global spatial contexts and interactions, which are critical for accurate and comprehensive analysis.
To address these challenges, we employ data-driven coor- dinates, allowing the model to directly use 3D coordinate features as neural network inputs. This approach allows for robust and expressive representations of object geometry while effectively capturing global spatial contexts across the entire shape. Specifically, our proposed data-driven co- ordinates ensure invariance to B(3)-transformaiion. This approach guarantees consistent spatial information, achieves computational efficiency, and preserves the richness of spa- tial representation.
The key contributions of this study are as follows:
Motivated by these findings, this work presents a novel framework that effectively incorporates the inherent thick- ness of 3D objects, enhancing mesh-based methods by en- abling precise interaction modeling between opposing sur- faces while maintaining computational efficiency.
Thickness-Aware Framework: We propose Thickness- aware E(3) EEuu--quiat 3D Mesh Neural Networks (T- EMNN) that accurately models interactions between op- posing surfaces while retaining computational efficiency.
Data-Driven Coordinates:
Our framework incorporates data-driven coordinates to ensure consistent and robust spatial representation of 3D objects.
In addition, while considering thickness helps capture geo- metric properties that influence structural behavior, learning spatial information (i.e., coordinates) is also important, as it enables the accurate representation of the spatial continuity inherent in field variables such as stress distribution and deformation. Therefore, incorporating spatial information in learning-based surrogate models would further enhance their ability to generalize physical behavior patterns by cap- turing spatial relationships. However, when utilizing spatial information, it is crucial to consider that preserving E(3)- equivariance is essential in static analysis, as transforma- tions such as rotation, translation, and reflection (i.e., the E(3)-group) do not alter material properties.
Eqiivaquiva or Invariance: The proposed model preserves E(3)-equivariance or invariance, ensuring ro- bustness to transformations such as translations, rotations, and reflections.
Validation on Real-World 3D Objects: We validate our approach on real-world 3D objects in industry, demon- strating accurate node-level 3D deformation predictions in practical scenarios.
2. Related Work
To ensure the -quivvarre of spatial information in a mesh neural network, we consider leveraging E(3)- equivariant graph neural network approaches. These meth- ods often transform the original coordinate system into higher-dimensinn dimenssoall representations or utilize specialized equivariant functions, such as spherical harmonics (Worrall & Brostow, 2018; 8; Gasteiger et al., 2020; Brandstetter et al., 2021; Batatia et al., 2023; Batzner et al., 2022). Although effective, these techniques typically introduce substantial computational overhead, making them less suitable for real- world industrial applications, particularly for large-scale meshes with a high number of nodes and edges. In con- trast, methods that avoid such computationally intensive techniques often adopt constrained representations, rely- ing solely on local geometric properties (e.g., relative dis- placements, cross products) (Satorras et al., 2021; Trang et al., 2024; Du et al., 2022) or excluding directional and
2.1. Mesh-Based 3D Representation
Meshes (Kato et al., 2018; Pfaff et al.. 2020: Feng et al., 2019; Trang et al., 2024; Rubanova et al., 2021; Li et al., 2018b) encode both surface and volumetric characteristies through vertices, edges, and faces, facilitating geometric and topological analysis. Recent neural architectures (Smirnov & Solomon, 2021; Li et al., 2023; Singh et al., 2021; La- hav & Tal, 2020) leverage mesh structures for enhanced 3D analysis by incorporating geometric and topological features. For instance, MGN (Pfaff et al., 2020) models physical inter- actions via graph- message passing with distances, dis- placements, and world distances between connected nodes. MeshNet (Feng et al., 2019) captures structural information from mesh faces through kernel-based operations, while Milano et al. (Milano et al., 2020) incorporate both edge and face geometries using attention mechanisms.
'The radius was set to the average thickness of the shape for consistency.
Beyond learning mesh geometry and topology, several meth- ods address the limitations of wressage-passing where lo- cal communication constrained by mesh density (Gao
2

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Thickness-aware Eq)--Equivariant 3D Mesh Neural Networks
& Ji, 2019; Han al., 2022; Fortunato et al., 2022). To enhance global information exchange, hierarchical pooling techniques have been introduced to extend message propa- gation beyond immediate neighbors (Cao et al., 2022; Han al., 2022; Janny et al., 2023; Yu al., 2023).
equivariant if the following holds for all g E E(3):
(TT (T()))))
where Tg X - X and TV > - y are transformations applied to x E X' and y E 3 respectively.
Furthermore, the geometric and topological richness of mesh graphs has driven research into equivariant representa- tions. Many approaches (Shakibajahromi et al., 2024; Trang al., 2024) build on H[n)-equivariant graph neural networks (EGNN) (Satorras et al., 2021), leveraging their efficiency for large-scale mesh data. EGNN ensures Eq3)-equivariance via message passing while avoiding direct nonlinear spa- tial encoding. EMNN (Trang et al., 2024) extends EGNN by generating E(3)-invariant messages that incorporate ge- ometric features like face areas and normal magnitudes, enhancing equivariant modeling while preserving efficiency. However, both approaches struggle to capture global spatial relationships, as they rely on relative positions and direc- tional updates rather than fully embedding spatial features.
Invariance, on the other hand, is a special case of equivari- ance where the output remains unchanged under transfor- mations, satisfying: (TT (x)) = o(x).
3.3. Thickness in the Mesh
Since traditional meshes lack explicit thickness information, we first define thickness node pair as pair of nodes where one resides on one side of the surface and the other on the opposing side, aligned along the normal direction of the origin node. The thickness paired node of e; is denoted as (w:)) E V. where T(us) A vú and is defined as:
T(u;) = arg min ||x; - (x, - d n 2, oV,
(I)
3. Preliminaries
subject (x, xx) noode < O, where nooee is the unit outward normal vector at node 0i d >> 0 is a scalar for the ray projection distance, and || I] is the Euclidean distance. This definition ensures that T(x)) lies on the opposing side
3.1. Notations
A shape is represented as mesh M = E). where the nodes V correspond to unique coordinates in 3D space, and the edges E' define the connectivity between nodes. Let the coordinate of a node vs E V be denoted as x; E R3, representing its position in 3D space. In this work, we de- note the mesh as the surface mesh, where its nodes and edges are only located on the surface of the shape, ensur- ing that the mesh is water-tight. Each shape is associated with experimental conditions C, which influence the results of simulation. The goal of this study is to predict the de- formation of each node along the x, y. and axes, given the shape and the experimental condition e E C as inputs. Formally, the deformation at a node v; is represented as Ax;- = Ar, Ay, Az,), which is the output of the model.
of the surface by requiring (x, - xz) - -noode < 0, avoiding the selection of nodes on the same surface.
The distance between thickness node pair is then defined as the thickness of the mesh at node vs. Formally, the thickness t(v.) at node o; is given by:
t(u:) = ||x XT(x))l-
(2)
This definition enables the quantification of mesh thickness at any node, providing a geometric measure of the spatial separation between opposing surfaces.
4. Methodology
Each face in the mesh is defined a triangle consisting of three connected nodes, and the outward-facine normal vector of a face fA: is denoted as ne A node v; E V is sur- rounded by a set of faces forming local disk-like structure, ensuring geometric continuity. The normal vector of node noode is defined as the average of the normal vectors of its surrounding faces: mode = [..) E, E eT(e,) n ne where F(u,) is the set of faces adjacent to node v, and [F(r;)) is the number of such faces.
Our method, T-EMNN, extends the mncode-process-decco framework of MGN (Pfaff et al., 2020), introducing key innovations for handling 3D shapes with thickness while in- corporating spatial information in an B(3)-eqqivariant man- ner. We employ data-driven coordinate transformation (Sec. 4. to integrate spatial information. T-EMNN consists of an encoder (Sec. 4.2.1). 1), a surface processor (Sec. 4.2.2), a thickness processor (Sec. 4.2.3), and a decoder (Sec. 4.2.4), effectively processing surfaces with thickness. The overall framework is shown in Fig. 2.
3.2. Equivarine and Invariance
E(3)-equivariance ensures that a function's output trans- forms consistently under transformations from the Eu- clidean group B(3), which includes translations, rotations, and reflections. For mapping á X + V, Φ is E(3)-
4.1. Coordinate Transformation: E(3)-Invrriant Data-driven Coordinate System
Each node E V has an original coordinate x rep- resenting its position in the input coordinate system. To ensure E(3)-invariance, we transform the coordinates into a
3

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Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
Geometric Encoder
Thickness Processor
Decoder
Processor
Geanenn oné.
Invorse Transformation
L-Jeyar
Transtormation
Spatial Encoder
Z
be Conditon rrb.
:Oiginal System
J. Sytte
Spans erbb.
Figure Overview T-EMNNN
principal axis orientation, regardless of the shape's s initial alignment or transformations.
b
Step 4. Coordinate Transformation. The adjusted coordi-
X
Xen Xieox
nates X; are transformed into the in3)-invariant coordinate system using the rotation matrix R:
b2
Figure 3. Our proposed data-driven coordinate system.
inv R
system defined by the shape itself, independent of its orien- tation or alignment in the original coordinate system. This transformation is achieved through the following steps:
Preserving Original Coordinate Mapping. For each shape, the center of mass Xem and the rotation matrix R are stored to enable the output predictions to be transformed back into the original coordinate system. During the inverse transfor- mation, the predicted invariant coordinates x!" are mapped back to the original system as:
Step 1: Adjust Coordinates to Center of Mass. The center of mass X.m of the shape is calculated as:
1
Σ
nig
Xem
x
orig
Rx inv
Xem-
(8)
This ensures that the predictions remain consistent with the original coordinate system while benefiting from the E(3)- invariant representation during processing. The transformed coordinates x."", along with the stored x; and R. allow seamless mapping between the input and output spaces.
where [V] is the total number of nodes in the mesh. Each node's coordinate is adjusted relative to the center of mass:
orig
Xam
(4)
Step Principal Axis Generation. As shown Fig. 3, we
We provide formal proof of the invariance in Appendix H.
obtain three orthogonal basis vectors b1, b2, b3, each in R3, by applying Principal Component Analysis (PCA) to X E NK3, where X composed of X defined in Eq. 4, and the number of nodes within the mesh. These basis vectors form the principal axes of the shape and are combined into rotation matrix R = [b] b2 b3] 3333
4.2. Thickness-aware Mesh Neural Network (T-EMNN)
4.2.1. ENCODER
For every node v; E V and edge eij E E within the sur- face mesh M = (V, F),, we encode their features using respective MLP encoders.
Step 3. Direction of Principal Axes. To ensure consistent aligmment of the principal axes, the direction of each basis vector bie (1,2.3) is determined using reference vector v, which connects the center of the bounding box Xbbox to the center mass Xem The direction is adjusted as follows:
Geometric Encoder. To preserve E(3)-invariance during processing, T-EMNN utilizes only invariant features for ge- ometric encoding (e.g., distance from the center mass or neighboring node) that remain robust under any transforma- tion of the E(3) group. Formally, the encoded features for node vú and an edge Gij are given as:
b, b, if bf v2 b, bf v 0,
where b. determines whether the basis vector b, aligns with the direction of v, and Xem Xthox is the reference vector. The center of the bounding box Xabox is computed Xam Xeas where
z (0) Pnodee (f:),
(0) ij
Dedge
where f; E Rd and fūj E Ree represent the initial feature embeddings of node v; and edge eij. Pmote and dedge are MLP-based encoders designed for nodes and edges. The specific initial features utilized are detailed in Tab. 1.
as Xbhox
Tningev v Ze
maxxEV Iep] maX maxxev Zu.
Xmin
min minggy 2o]
Xmax
(6)
The outputs of the geometric encoders, z E Hd and e Rdd are later used as the input embeddings for the first layer (1 = 0) of the processor modules.
are the minimum and maximum coordinates of the point cloud along each axis. This adjustment maintains consistent

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Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
Spatial Encoder. To incorporate spatial information (i.e., co- ordinates) into the analysis, T-EMNN utilizes an MLP-based spatial encoder to derive the spatial embedding z E IR:
2: goord = @ccord (x:")) )
(10)
[a] Thickness
[b] Witth
where x E 2 represents the transformed coordinates of node V/.
Figure 4. The concept of thickness (left) and width (right).
Sec 3.3. In brief, thickness is characterized by the spatial separation between opposing surfaces, with thickness paired node ((x)) of node v, defined as the closest node on the op- posing surface along the inward normal direction of a given node v; The thickness t(u)) defined in Eq. 2 represents the distance between opposing surfaces.
Condition Encoder. The condition embedding h. E IR is encoded as:
h Ooand (c):
(11)
where doand processes the experimental condition vector E IRs,, where is the number of experimental conditions
such as temperature and pressure.
In addition, to account for biickness-related interactions, we introduce a thickness edge Ci,bhcck connecting 2 to T(ee), with its feature fhickck E IR? defined as:
Note that while the initial geometric embedding of node 2;
(U) (0) and edge ejj are passed through the processor, the spatial embedding zą and the condition embedding he are integrated with the processed geometric embeddings by the decoder after the processor. This allows the processor to focus on capturing the geometric features of the shape.
fr. = [t(())) n. Darl,
(14)
where n. Diy is the dot product between the normal vectors at 1/ and T(;)). Including n; D47 ensures that the model accounts for the alignment of normal vectors between thick- ness pair nodes. This additional attribute complements the
4.2.2. SURFACE PROCESSOR
thickness ((xz), providing a more comprehensive represen- tation of pairwise interactions for improved processing of geometric relationships.
The surface processor in T-EMNN focuses exclusively the edges eij E E of the surface mesh M, without con- sidering interactions between thickness node pairs. This processor updates the embeddings of the surface edges and nodes, capturing the geometric and topological relationships of the mesh surface. The update rule for the edge embed-
Thickness Threshold. However, an important observation we had is that not all thickness node pair inherently rep- resent the real-world meaning of thickness. For example, in the case of a wide flat plate (Fig. 4), the thickness node pairs on the side surfaces (Fig. 4(b)) are more representative of the width rather than the actual thickness of the object (Fig. 4(a)). The distinction between these two concepts is not discrete but rather ambiguous.
dings (!) E Rd is defined as:
(1+1) ei
( .2; )2
(1)
(12)
where fill is an MLP with residual connections, which updates the edge embeddings Ø based on the embeddings of the connected nodes vú and uy. Then, the update rule for the node embeddings 2; E Rdd is defined as:
To address this, we define a Thickness threshold r that dy- namically regulates the interactions between nodes. Specifi- cally, thickness edges are incorporated only for nodes whose distance their thickness paired node is within the thresh- old (1(1;) S 7), ensuring meaningful interactions between opposing surfaces with strong dynamic relationships. It is important to note that the thickness threshold T is not pre- defined but instead learned in a data-driven manner. By training real-world data, the model dynamically adapts to identify the optimal threshold T that captures interac- tions between opposing surfaces without relying on manual tuning or prior assumptions. This data-driven approach en- hances the framework's 's robustness and adaptability across various applications.
surt. Z
<- ( (1)
Σ
(+1)) bj
(13)
JEN()
where f another MI.P with residual connections and (i) denotes the set of neighboring nodes of u,. It updates the node embeddings (1) by aggregating the updated edge embeddings eij
((+1) from its neighboring edges.
The updated node embeddings z: 2,u(l(0) Rdd serve the input for the corresponding 1-th layer of the thickness pro- cessor. This ensures that the geometric relationships refined by the surface processor are directly utilized by the thickness processor to model interactions between opposing surfaces.
Thickness Activation Function. The thickness activation value I; is calculated for each node u; determine the contribution of its thickness edge in the propagation process. The activation is defined as:
4.2.3. THICKNESS PROCESSOR
The concept of thickness in meshes, and the methodology for identifying thickness node pair, has been detailed in
1 = 1+ (((L))TT)
(15)
5

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Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
where ((v)) represents the thickness of node v/. is learn- able threshold, and is a scaling factor set to 3. controlling transition sharpness. By doing so, edges with t((;) < are
producing the node-level prediction p. E R in the trans- formed E(3)-invariant data-driven coordinate system.
Inverse Transformation. To map the prediction back to the original coordinate system, the stored center of mass Xem and rotation matrix R for the shape are used. The final deformation in the original system is calculated as: arig P
assigned weights close to I, while edges with ((vz) > T are excluded from the propagation process.
Weighted Message Passing. In the thickness processor, each node v2 is connected to its thickness paired node ((;)) through a single thickness edge tiici The embedding for this thickness edge djichick Rdd is initialized in the first layer using dedicated encoder, Dutick, which maps the thick- ness edge feature f; liick to its embedding:
R P + Xem-
This ensures that the predicted deformations are consistent with the input shape's 's original orientation and alignment, while benefiting from the E(3)-invariant processing during model computation.
(0) e rthick f- Ohtick (hick))
(16)
where tick is defined in Eq. 14. For subsequent layers, thick E Rd is updated using the output from the previous (1-1) layer thick This ensures effective propagation of thickness and normal alignment information through the layers.
5. Experiment
5.1. Dataset Description
We evaluate T-EMNN using a dataset from real-world in- jection molding applications. Injection molding, a common manufacturing process, involves injecting molten plastic or metal into a mold, letting it solidify, and then taking the fin- ished product out of the mold. Product quality depends on factors like temperature, geometry, and gate design, with de- formation being critical. This dataset is well-suited for eval- uating T-EMNN as its geometries exhibit thickness across all surfaces, enabling thickness-relates interaction modeling. Additionally, node spatial positions significantly influence deformation, underscoring the importance of geometric and spatial information. The predominantly "basket-like struc- tures capture both surface-level and mhickness-based inter- actions well. More details, including data split and initial features, are provided in the Appendix A.
(1+1) The updated embedding for edge @ichick at each layer is computed as:
(1+1) iithick + I;
[ (c thick 2,
e.M (t) thick
s1rf.())
((.))
(17)
where fM is an MLP that processes the features of the thickness edge thi as well as the node embeddings strf.()) and (T(,.) The activation value 1; ensures that only relevant thickness interactions contribute during mes- sage propagation. Then, the updated embedding of node Us, (1+1) RRd is then computed as:
z;
z: 11+1) < (z, (1) .hickik)
(18)
where Miick is another MLP that combines the current node (1+1) embedding z. (t) with the updated edge embedding dhhcikk
5.2. Baselines
By focusing on a single thickness edge per node, this message-passing scheme maintains computational effi- ciency while retaining the ability to model relationships between opposing surfaces. The architecture of the thickness processor ensures seamless integration of both surface-level and thickness-baas based geometric information.
As baselines, we include multiple traph-based neural net- work methods to evaluate T-EMNN against existing tech- niques. MGN (Pfaff et al., 2020) models physical interac- tions using graph-based message passing but lacks E(3)- equivariance. EGNN (Satorras et al., 2021) ensures E(3)- equivariance through message passing while maintaining computational efficiency by avoiding direct nonlinear en- coding of spatial features. Building upon EGNN, EMNN (Trang et al., 2024) optimizes this framework for mesh data by generating -(3)-invariant messages that incorporate geo- metric information from mesh faces.
4.2.4. DECODER
In the decoder, T-EMNN produces the final node-level pre- dictions by combining the geometric embedding 2; spatial embedding 2: goaie, and experimental conditions ha-
First, the geometric and spatial embeddings are concate- nated and processed as follows:
5.3. Evaluation Settings
We assess the models under two distinct conditions: 1) in- distribution, where the test data retains the same aligned coordinate system as the training data, and 2) out-of- distribution, where the original coordinate system is ran- domly rotated, resulting in a misalignment with the train- ing data. Note that the out-of-distribbtioo scenario is de- signed to assess how well the methods adapt to objects
z
final
@cambine (z,, z (oord)) IR..
(19)
where Doomhine integrates geometric and spatial features. Then, the combined embedding 2f(mall is concatenated with he and passed through the decoder:
juv p
didecode ( (Z. final hh]):
(20)
6

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Thickness-aware Eq)--iquivariant 3D Mesh Neural Networks
Table 1. Model Performance in In-Distribution and Out-of-Distriuttinn Settings, averaged over 3 seeds with standard deviation (in
parentheses). Bold indicates the best performance among the methods. Descriptions of each feature are provided in Appendix A.
Medel
Equifactance
Spatid
Thicknon
Inpuit of
Edge Fisture
Nodo Fisture
(rigginal)
(ostated)
information
edgs
fwod
RMSE (1)
MAE
R' (1)
RMSE
MAE
R² (T)
(4) MLP (6) MLP
0.2518 (01961)
0.1164 ot (0155
0.8064 (000224)
0.4780
0.2393 00312455
X-"
0.2544 (0/0015)
0.1045 00 0008)
119154 (0/0016)
0,2545 (0.0015)
0.1071 (0 00077
[19555
(0,0009)
MGN
0.2854
0.1178 (0.001))
(Q.199)
0.1938
900
MGN
0.2241
0.0538 (0,0029)
0.056
0.9446 (0,0033)
O0 RONN
155005 ea 2097)
54.363 ( 1000)
-14341
545 6425)
19 7900)
-122011
(1039
( PONN
x
(12844 (0.0011)
0.1220 02 (171)
01880 (0/0056)
0.4576 01544
1958 (2 (09-))
018044 (047706
POENN
0.2270 (0
0.0863 05 (993))
0.9129 (0/0036)
0.2271 (0.0019))
0.0967 (2 (2.0099
0.9443
(0/0317)
EMNN
Mi-F1
166.077 (1.5220)
58,467 (2,0000)
16034,0 (723.)
201.450 (1.922))
92.237 (1.3000)
34302.7 (64462)
EMNN
Mi-Fi
0,3006 (211266)
0.1284 (8.0131)
0.8626 (00005))
(0.0193)
0.2034 (0.002)
0,7972 (8.007))
Wi-11
0.2210 (0.005))
0.0937 (0.0034)
(50034))
0,2210 (0.005))
0.0963 (0.0052)
(0.0012)
T-ENNN
0,2132
(60053)
0.2131
0
0.1513
Tw 6o
0.32
0330
Count
3.04
339 225 Epoch
350
375
357
0.30
Thickness EN2
Thicsness thressoks
Thicaness threshold
Thilnnes miresools
Figure 5. Learning curve of the thickness threshold during train- ing across three seeds (left), and the distribution of thickness values f(v.) with the cutoff threshold (red dotted line, t(xx) = r) used for message passing in the thickness processor (right).
Figure 6. Performance comparison of T-EMNN with fixed thick- ness threshold. The value 5.68 corresponds to the learned thickness threshold T-EMNN when using a learnable threshold.
with transfreaoo coordinates, where maintaining E(3)- equivariance becomes essential.
patterns in the data. However, when the coordinate system lacks E(3)-qqivvariant properties, performance significantly deteriorates when testing data exhibits different coordinate distribution (i.e., out-of-distribution results of (a) in Tab. 1). This underscores the critical role of -3)-evuuvrrii in ensuring the robustness of the coordinate system.
We assess the model performance using three metrics: 1) RMSE, which evaluates the effectiveness of handling out- liers, 2) MAE, which measures the consistency and accuracy of the model's predictions, and 3) R², which quantifies the model's s ability to explain variance, ensuring reliability for real-world industrial applications.
Moreover, for baselines that do not explicitly leverage la- tent spatial embeddings (c.g.. (c) MGN, (f) EGNN, and (i) EMNN in Tab. 1), they solely rely on geometric relation- ships between neighboring edges or faces within the mesh graph. However, when observing R² values, relying solely on local geometric relations (c.g., coordinate differences, distances) proves insufficient for accurately predicting re- gional behavior across the entire shape.
5.4. Experiment Results
To evaluate data-driven coordinates, we modify the base- lines to use spatial embeddings (i.e., 2,00n) derived from either the original (x) or proposed (xx")) coordinates (Sec. 5.4.1). Additionally, we assess whether the learned thickness threshold 7 optimally facilitates message passing between opposing surfaces (Sec. 5.4.2).
Impact of Data-driven Coordinate. To address the miss- ing spatial information in bascline methods, we incorpo- rate an additional spatial encoder, as defined in Eq. 10, and combine it with the geometric embedding via concate- nation, as described in Eq. 19. When integrating our pro- posed data-driven coordinate system into the baselines, an inverse transformation of the outputs is applied to ensure E(3) -euuviaavve
5.4.1. MAIN RESULTS
In Tab. 1, we observe T-EMNN, which incorporates spa- tial information into the model while enabling propagation between opposing surfaces with strong relationships, out- performs other methods. The detailed analysis is as follows:
In Tab. 1, methods utilizing our data-driven coordinates ((e) MGN, (h) EGNN, and (k) EMNN) consistently outperform their counterparts using the original coordinate system ((d) MGN, (g) EGNN, and (i) EMNN). The enhanced alignment provided by our proposed data-driven coordinate system significantly improves the representation of spatial relation- ships, leading superior performance in downstream tasks. Furthermore, the data-driven coordinate system achieves this not only by preserving E(3)-qquivariance ((h) EGNN and (k) EMNN) but also by enabling E(3))-quivari in
Impact of Spatial Information. Spatial information plays critical role in capturing localized patterns, which are es- sential for accurate interpretation in downstream tasks. To evaluate the isolated impact of spatial information, we first evaluate the performance of MLPs that use only vertex co- ordinates as input ((a) and (b) in Tab. 1). The results demon- strate that spatial information alone is sufficient to achieve strong performance in terms of R² score, highlighting its importance in representing meaningful relationships and
7

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Thickness-aware E())-Equivariant 3D Mesh Neural Networks
(a) Ground Truth
(b) MLP + xiov
(c) MGN lav
(d) EGNN +
(e) EMNN 2
(0 T-EMNN
(g) Thickness Edge
Figure 7. Visualization of error magnitude (RMSE). The ground truth shows deformation magnitude (a), while (b f) illustrate prediction errors. Additional examples are in Fig. 15 (Appendix). In (g). (+) and (+) represent a thickness node pair, (-) its thickness edge, and ( the shortest path within the mesh of length 6.
RMSE
MAE
Table 2. Ablation study on the thickness edge feature, frnnick.
0224-
922-
Method
RMSE
MAE
R²
022
w/o thickness w/o dot product
0.2156 (00644)
0.0908 (.007))
0.9148 (0.0089)
0.1
0.2191 (0.1433
0.0912 (0.0067)
0.9134 (0.0163)
M4M
M4GN
MUN
SGMN
T-EMNN
0.2132 (0.0046)
0.0892 (0.0025)
0.9228 (0.0063)
Figure 8. Comparison of baselines with x and their extension with our thickness edges.
models that originally lacked it, such as (e) MGN. This ca- pability transforms previously limited models into robust systems capable of handling transformations effectively, thereby greatly enhancing their ability to capture and repre- sent spatial relationships.
(a) Below the threshold r
(b) Above the threstidd
Figure 9. Visualization of thickness edges with E(v:) below the Iearned threshold 7 (left) and those above т, filtered out by the thickness processor (right). Detailed in Fig. 16 (Appendix).
ness t(;;) and the dot product of normal vectors n;y The thickness feature encodes the weight of the edges based on the distance between thickness pairs, while the dot product of normal vectors imposes geometric alignment by quan- tifying the directional consistency between the normals of the paired nodes. By jointly leveraging these features, the model effectively learns edge weights that promote high- quality message passing between opposing surfaces, thereby enhancing overall performance.
5.4.2. IMPACT OF THICKNESS-AWARL FRAMEWORK
To assess the effectiveness of the thickness-awarc frame- work, we examine whether the learned thickness threshold r optimizes message passing between opposing surfaces and distinguishes 'thickness" from 'width.' As shown in Fig. 5, r converges to 5.68 across three seeds with low variance, fil- tering out 3.83% of thickness edges exceeding this threshold (Sec. 3.3). In this section, threshold learned by our model impacts performance and accurately captures the actual thickness.
verify whether the thickness
Qualitative Analysis. Fig. 9 illustrates that the learned thickness threshold effectively filters out noisy thickness edges while retaining meaningful ones, thereby enhancing the model's 's ability to process relevant interactions.
Performance Impact of Thickness Edges. To confirm that = 5.68 is optimal, we compare the performance of T-- EMNN using fixed thresholds ranging from 0 to 20 (Fig. 6). The results show that performance peaks when the thresh- old is near 5.68. Morcover, when the threshold set to zero removing thickness edges- peee--reeffreeeeeeeee degrades sigmificantly. Similarly, thresholds above 10 introduce noisy information from irrelevant nodes representing 'width,' lead- ing to performance deterioration.
In Fig. 7 (b-e), observe baselines struggle to predict around edges, where stress concentration leads to distinct physical behavior. This is because the inherent locality of surface-man mesh, where requires GNN-based methods to take at least six propagation steps along the shortest path (- in Fig. 7(g)). In contrast, T-EMNN facilitates single-hop interactions between opposing surface neighbors through thickness edges (- in Fig. 7g)), allowing for broader mes- sage passing and improving performance in these crucial regions in Fig. 7(f)
To further examine the importance of thickness edges, we incorporate them, along with our thickness processor, into the baseline models. As shown in Fig. 8, all baseline models exhibit improved performance when incorporating thickness edges compared to their counterparts without them. This result validates the importance of the thickness in 3D ob- jects, and its effective integration can improve the models' capability in static analysis.
5.4.3. EVALUATION UNDER DYNAMIC SETTING
To evaluate the dynamic capabilities of our frame- work part-ppattlularll the thickness processor- conduct next-timestep deformation prediction using the Deforming Plate dataset (Pfaff et al., 2020). This experiment demon- strates how the thickness processor enhances the model's ability to handle dynamic scenarios.
Ablation Study of Thickness Edge Features. In Tab. 2, we analyze the performance impact of the proposed thickness edge features, fi.bick comprising two components: the thick-
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Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
A Acknowledgements
Table Performance evaluation on the Deforming Plate dataset for predicting deformation at the next timestep.
This work was supported by LG Electronics; by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00335009))
Method
Input of dlownd
RMSE (x10")
MAE (x10")
and by the National Research Foundation of Korea (NRF) funded
T-EMNN
17.420
0.7007
wio thickness
(3.122)
(1,039)
(0.0224)
14.983
5.851
0.7579
by Ministry of Science and ICT NRS-2022-NR0687533
T-EMNN
(0.671)
(0.195)
(000200)
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are no immediate ethical con- cerns associated with this work. However, with any ad- vancement in machine learning, its application in real-world scenarios should be carefully evaluated to prevent potential biases or unintended consequences.
Figure 10. Visualization of thickness edges in the Deforming Plate dataset: edges with t(v:) below learned threshold (left), and edges with ((vs) above which are filtered out by the thickness processor (right).
References
Alon, U. and Yahav, E. On the bottleneck of graph neural networks and its practical implications. arXiv preprint arXiv:2006.05205, 2020.
We first construct thickness edges based on the mesh at the initial timestep and keep them fixed throughout all subse- quent timesteps. The prediction task is to estimate the defor- mation of each node at the next timestep. As our data-driven coordinate system is designed for static analysis and to miti- gate misalignment within the dataset, we instead adopt the original coordinate system for dynamic analysis. Further- more, because method assumes single material, we incorporate an additional node feature: the shortest distance from each node to the actuator the global coordinate sys- tem. This allows the model capture external interactions applied by the actuator on the deforming plate.
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6. Conclusion
We introduced T-EMNN, thickness-aware E(3)- equivariant 3D mesh neural network that captures complex geometric interactions, including thickness, by integrating a learnable thickness threshold for effective message passing while filtering irrelevant connections. By leveraging data-driven coordinate system and a transformation mecha- nism that preserves s(u)aiquuvariaanc, T-EMNN achieves state-of-thee thr performance in predicting node-level 3D deflection with high accuracy and computational efficiency. Comprehensive evaluations on real-world industrial datasets validate its robustness and practicality, making it an effective solution for applications like injection molding.
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Thickness-aware E())-Equivariant 3D Mesh Neural Networks
A. Data Details
The dataset consists of 504 valid samples derived from 28 unique geometries, each with experimental conditions (i.e., 28 18 504). The training and validation sets include 23 geometries, with 80% of the samples used for training and the rest for validation. One geometry from the training set was entirely reserved for validation. The test set comprises all 18 experimental conditions for 5 geometries and the 18th experimental condition for the remaining 23 geometries, resulting in 113 test samples. The average mesh consists of approximately 54, 127 nodes and 324,771 edges.
Figure 11. Examples of Dataset Shapes.
The experimental conditions consist of eight types for each shape: pack pressure, pack time, projected area, gate size, injection value, volume, melt temperature, and mold temperature. By varying these combinations, total of 18 experimental conditions are generated.
MGN utilizes the coordinate difference between nodes i and j (i.e., Xij) and their distance (i.e., Xxy [I) as edge features. For node features, MGN incorporates the normal vector of node (i.e., n;), the geodesic distance from the gate (the injection position of molten plastic or metal) to node i (i.e., 9:), and the radius from the center of mass node (i.e., r,). EGNN, EMNN and T-EMNN, which ensure -quiivaiiee, use only ||x;y]] || as edge features and and r, as node features.
B. Implementation and Experimental Setup
Our model is implemented using Python 3.10.13, PyTorch 2.0.1, Torch- Corch-Geomettrc 2.4.0, and trimesh 3.23.5. All experiments were conducted on an NVIDIA GeForce RTX 4090 with CUDA 12.2.
Each experiment was run for 200 epochs per seed with learning rate of 0.001 and a weight decay of 5e-4. ensure stable optimization of the learnable thickness threshold т, we employ an adaptive learning rate scheduling strategy. Specifically, we utilize the ReduceeLRnnllteen algorithm, which dynamically adjusts the learning rate when monitored metric plateaus. We set the patience to 5, the initial threshold to 1, and the reduction factor to 0.5, applying it exclusively to the learnable thickness threshold.
C. Baseline Details
In this work, we compare three prominent graph-based methods. MGN (Pfaff et al., 2020) utilizes iterative message-passing steps, while EGNN (Satorras al., 2021) and EMNN (Trang et al., 2024) are designed for F(3)-equivariance. To ensure consistency, we rely on the official codes provided by the original authors for all baselines.
MGN: hithr//wwwhhhhoooooooo/////////e htt.:///ee...aotoooooo.eoeeeenee tre//mastermmeshgrppneet
EGNN: http:///h//ttthbb.ooo/////s/////
EMNN: http://ggtthhcccm//////mmlaaaaaa
To ensure fair comparison, we configure all baselines and our proposed method, T-EMNN, with three message-passing
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Thicknes--aware E(3)-Equivariant 3D Mesh Neural Networks
layers and 32 hidden dimensions. All results are reported based the performance of the best validation model selected within 200 epochs across 3 different seeds. The spatial and condition encoders consist of two linear layers with ReLU activations. Since our task involves static analysis without time-step predictions, all baselines directly predict the target field variables, specifically the 3D deformation.
D. Hyperparameter Details
Thickness Threshold
RMSE
0339
MAE
A2
10
0.50
8.09
0:10
Apra
Alpha
Figure 12. Analysis of Hyperparameter a.
Hyperparameter a is scaling factor that influences the transition sharpness. Specifically, as increases, the thickness activation value 7; in Eq. 15 discretely masks the thickness edges whose thickness values are near the threshold. In contrast, when a is small, 4 smoothly attenuates the weight of thickness edges, gradually approaching zero beyond the threshold.
A smaller ensures more stable learning of the thickness threshold by providing smoother transition. However, it may introduce noise from softly masked thickness edges, potentially degrading the overall model performance. As shown in Fig. 12, when a is between 3 and 10, the thickness threshold converges within the range of 4 to Conversely, when 1, longer edges contribute message passing, leading less precise control over edge masking. When large (e.g..
10), edges around the thickness threshold strictly masked, but the convergence becomes unstable due to discrete masking effects, resulting in fluctuations.
Based on these observations, we set a to 3, which provides stable convergence for learning the thickness threshold while maintaining acceptable performance.
E. Computational Efficiency
Model
Input of Ocoord
Speed (it/s)
GPU Memory (MB)
MLP
32.51
693 693
MLP
32.72
MGN
21.18
1,656
MGN
orig inv
22.58
3,954
MGN
22.29
3,952
EGNN
23.85
5,700
EGNN
orig
23.59
5,740
EGNN inv
23.66
5,740
EMNN
18. 18.99
7,250
EMNN
oig
19.75
7,488
EMNN
19.99
7.322
T-EMNN
X inv
20.21
3,714
Table 4. Comparison of training speed (iteration/sec) and GPU memory usage (MB) across diffferent models.
Our model is based on MGN, with an additional thickness edges module that introduces minimal computational overhead.
This ensures that the computational cost remains similar to MGN while improving structural representation.
Training Speed (it/s). T-EMNN achieves 20.21 it/s, comparable to MGN and slightly higher than EMNN (18.99-19.99 it/s)
This indicates that the added module does not significantly impact training speed.
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Thickness-aware B()--qquivrrintt 3D Mesh Neural Networks
GPU Memory Usage. T-EMNN consumes 3,714 MB of GPU memory, much lower than EMNN (7,250-7,488 MB) and similar to MGN. This shows that approach maintains efficiency while reducing memory overhead.
F. Result Details
0.4
0-
-0.4
0.
-50000
ID3
Figure 13. It scores for all test data. In the shape IDs, 's' indicates seen shapes included in the training data, while 'us' refers to shapes. The number following 's' 'us' represents the shape type, and "c' denotes the experimental condition, followed its corresponding type (c.g., "18")
In Figure 13, we present the detailed results for each test dataset under Out-of (OOD) settings. While GNN- based methods incorporating our proposed data-driven coordinate system (e.g.. MGN w/ inv. coord., EGNN w. inv. coord., EMNN w. inv. coord., and T-EMNN) generally exhibit superior performance across all shapes, methods that utilize the original coordinate system (e.g., MGN w. coord., EGNN w. coord., and EMNN w. coord.) perform significantly worse. Furthermore, MGN, EGNN, and EMNN, which rely solely on deep representations of geometric features, fail to accurately predict the target deformation.
G. Surface Mesh vs. Volume Mesh
GPU Memory Usage (4)
Performance (t)
0.11x
Volume mesh
1900 Memory
Score
Inference Time
0.19k
Volume Mesh Surface Mesh
:0
15
23
Volume mesh
Surface
[seclepoch)
mesh
Figure 14, Comparisons between volume mesh and surface mesh. The methods used for comparison are based on the MGN framework with coordinate embeddings from our proposed coordinate system. GPU memory usage represents the average GPU consumption across the test dataset, and inference time reflects the total time required to process the test dataset.
Mesh representations can be categorized into surface and volume meshes, each with unique strengths and limitations. Surface meshes, representing only the outer boundary of 3D objects, are computationally efficient and excel at capturing geometric and topological properties. However, they cannot model internal structures, which are critical for analyzing physical interactions within the object.
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Thickness-aware E()--Equivariant 3D Mesh Neural Networks
Volume meshes, in contrast, extend to the interior of objects, enabling high-fidelity analyses such as FEM for properties like density, thermal gradients, and stress. However, shown in Figure 14, they computationally expensive and require significant effort to preprocess CAD models into high-quality meshes. Additionally, their dense internal connectivity poses challenges for tasks focused on understanding geometry and topology. In GNN-based methods, nodes in volume meshes receive messages from all surrounding nodes, which processed based relative distances and directions. This excessive connectivity complicates the network's 's ability to clearly capture the overall shape and structure, ultimately hindering performance in geometry-driven tasks.
Surface meshes, with their simplified representation of outer boundaries, are better suited tasks prioritizing geometry and topology. However, they lack edges that connect opposing surfaces represent thickness and capture critical correlations. This limitation challenges GNNs to effectively model interactions between these surfaces, often necessitating extended message- -aaaggggaannngggggg steps. Such methods can lead to issues like over-smoothing (Li et al., 20) or over-squashing (Alon Yahav, 2020), limiting their overall effectiveness.
To overcome these limitations, propose the Thickness-AMw Mesh Neural Network (T-EMNN), which combines the computational efficiency of surface meshes with enhanced modeling of internal interactions. By incorporating thickness- aware message- meessage-passing mechanism, our method captures correlations between opposing surfaces while preserving the geometric and topologiaal strengths of surface meshes. This enables robust and efficient analysis of 3D objects with intrinsic thickness, effectively bridging the gap between surface and volume meshes.
Ground Truth
MLP + lnv
MGN l
EGNN + lav
EMNN llv
T-EMNN
Figure 15. Visualization of error magnitude (RMSE). The ground truth represents the magnitude of deformation, while each method's ss figure illustrates prediction emor (RMSE) relative to ground truth.
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Thickness-aware Ei)--iquivarian 3D Mesh Neural Networks
Figure 16. Detailed visualization of thickness edges with t(1)) below the learned threshold 7 (left) and those above т, filtered out by the thickness processor (right).
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Thickness-aware B())-Equivariant 3D Mesh Neural Networks
H. Invariance/ Proof of the Proposed Data-driven Coordinate System
Note: Depending on whether the 'Inverse transformation" in Sec. 4.2.4 applied or not, we can choose between E(3)- equivariance and invariance.
First, we will prove that our transformation algorithm T is invariant to translations E R and to rotations and reflections for any orthogonal matrix Q E R3x3 Specifically, we aim to show that the function T satisfies the following property, demonstrating its invariance. We will then discuss how applying the inverse transformation leads to equivariance.
T(Qx + g))
(21)
We seek to demonstrate that x the same for any Q and 9. proving that the transformation yields a consistent result regardless of the specific values of Q and The transformation algorithm T consists of four main steps:
Step 1: Center of Mass Adjustment
The center of mass of the original coordinates is given by (Eq. 3):
orig
Xem
x;
WEV
The adjusted coordinates of the original points are then (Eq. 4):
x;
orig
Xem
After applying the translation vector g and the orthogonal matrix Q, the center of mass of the transformed coordinates becomes:
Xem,T
Σ
(Qx, orig + g)
(22)
V
VEEV
Thus, the adjusted coordinates in the transformed space are:
1 [v] 23VV
Qx orig + X Qx orig +g
(xx, orig + +9)
(23)
Note that the translation vector g cancels out final expression. This shows that the transformation is invariant to the translation g. Therefore, after Step 1, the adjusted coordinates &; unaffected by the translation y, and subsequent steps only consider the effects of the orthogonal matrix Q.
Step 2: Principal Axis Generation
To generate the principal axes, we compute the covariance matrix Cx of the coordinate matrix X using the formula:
Cx
x
(24)
V
After applying the orthogonal matrix Q to coordinate matrix, the covariance matrix of the transformed coordinates Cox becomes:
1
Cox
(Q
x Cx
(25)
V
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Thickness-aware B())-Equivariant 3D Mesh Neural Networks
Thus, the orthogonal matrix does affect the principal axis directions within the data; it only alters the basis. Let orig be the rotation matrix corresponding to the three principal basis vectors of the original coordinate matrix X such that:
R orig [bi, b2, ba] R3x3
(26)
Then, the rotation matrix Rr for the adjusted coordinates becomes:
RT = QReig
(27)
Step Determining the Direction of Principal Axes
It is important to note that the basis vectors generated by the PCA algorithm do not have an explicit sign convention, meaning basis vector could be represented as [1, -1, 1] or 1-1, -1.. To resolve this ambiguity, introduce data-driven criterion to determine the consistent sign of the basis vectors, such using a reference vector that connects the center of the bounding box to the center of mass, as described in Section 4.1, Step 3.
While the proposed criterion in Step generally yields consistent signs for the basis vectors across most real-world meshes, it has limitation in cases where the shape exhibits perfect symmetry with respect to the three principal axes. Specifically, if the dot product between a principal axis b, and the reference vector is zero (i.e., b, = 0) in Eq. 5, the sign of that axis cannot be deterministically aligned. However, such perfectly symmetric meshes are rare in practice, especially for constructed real-world geometries.
For the proof, we assume that the signs of the basis vectors are already aligned.
Step 4: Coordinate Transformation
For the final transformation, we apply the rotation matrix R onig to the adjusted coordinates x; to obtain the data-driven invariant coordinate inn:
x RT T T (QR ( (ORTTTTTOX
R RoigT X,
(28)
Thus, iny is invariant to the translation vector 9 and the orthogonal matrix Q. This shows that the transformation results in a data-driven invariant coordinate system that possesses in3a-innar..
Final Prediction and Equivariance
When we further process the 'inverse transformation' after decoding in Section 4.2.4, the final prediction inM R in the invariant coordinate system is inverse-transformmed follows:
p: Rp! + X (QR ori ( (Roig p.) T Q(p xcm) + V
Σ
(Qx²re + +g)
(29)
This simplifies to:
p: = Q(p arig
x;
orig
+
Σ
(Qx ong + g)
(30)
V
V
VEV
V.E
Thus, have:
p: = Qp orig +
(31)
This shows that the final prediction preserves the equivvaareee Therefore, depending on whether the 'inverse transformation is applied or not, choose between eqiivvaiieee and invariance.
18


Reviews:
[{"review_id": "Wt5HHVAwKz", "reviewer": "Reviewer_otWZ", "summary": "The paper introduces **Thickness-aware E(3)-Equivariant Mesh Neural Networks (T-EMNN)**, a framework designed to address the limitations of existing mesh-based 3D analysis methods, which often overlook the inherent thickness of real-world 3D objects. The authors argue that thickness plays a critical role in physical behaviors such as deformation, bending, and stress distribution, particularly in objects like plates, baskets, and layered materials. T-EMNN integrates thickness information into the mesh representation while maintaining computational efficiency and preserving E(3)-equivariance (invariance to translations, rotations, and reflections). \nThe paper also introduces a **thickness threshold** to dynamically regulate interactions between opposing surfaces, ensuring that only relevant thickness-related interactions are considered. Experimental results show that T-EMNN outperforms existing methods, including MGN, EGNN, and EMNN, in both in-distribution and out-of-distribution settings.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for taking the time to review our work. For a comprehensive response, please refer to the attached [link](https://shorturl.at/gOsz6). All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).\n\n---  \n\n**A1. Additional Datasets for Challenging Meshes (Response to Methods And Evaluation Criteria and W3)** \n\nWe evaluate our method on four additional datasets (using 5 seeds for reliability):  \n- **Deforming Plate** (Node-level prediction - Position of Next timestep)  \n- **Basket** (Graph-level prediction - Max pressure)  \n- **Circular Plate** (Node-level prediction - Deflection of each node)  \n- **SimJEB** [1] (Node-level prediction - Magnitude of the displacement)\n\n**To visualize the thickness edges on other challenging datasets**, we include the learned valid thickness edges in Figs. L4, L5, and L6. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right). \n\n**Regarding performance**, as shown In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the \"In Dist.\" setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the \"Out-of-Dist.\" setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance. \n\nMoreover, to evaluate the **dynamic capabilities of our framework**—specifically the thickness processor—we perform next-timestep deformation prediction using the Deforming Plate dataset (as described in the MGN paper) to demonstrate how our thickness processor helps the model handle dynamic scenarios. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1, where our model successfully accounts for the deforming plate’s thickness, leading to improved performance.\n\n---\n\n**A2. Discussion on Limitations in Scenarios Where Thickness is Not Dominant for Prediction (Response to Claims And Evidence)**\n\nSince not all objects have thickness, our method may not be very helpful for such shapes. However, in these cases, we adjust the threshold where thickness influences performance in a learnable manner. In such situations, **the thickness threshold will adapt so that no thickness edge contributes to the prediction.** If thickness is not important, the thickness edge will be ignored, and the model will ultimately be equivalent to the MGN with our proposed data-driven coordinate architecture.\n\n---\n\n**A3. Equivariance/Invariance Proof (Response to Theoretical Claims, W2 and Q2)**\n\nPlease refer to Sec. L1 for the detailed proof. Our model can be either invariant or equivariant, depending on whether the 'inverse transformation' is applied.  \n\n---\n\n**A4. Surface Mesh vs. Volume Mesh (Response to W1 and Q1)**\n\nPlease refer to Appendix G for a comparison of GPU memory usage, performance, and inference time between volume-based and surface-based methods. We use the same model architecture (MGN) for a fair comparison, with the distinction that the input data is either volume mesh or surface mesh. The surface mesh utilizes only 11% of the GPU memory and takes 19% of the inference time compared to the volume mesh, while also delivering better performance. We argue that the performance degradation of the volume mesh arises from the over-smoothing or over-squashing problem that occurs when the mesh density is too high in GNN-based methods. For more details, please refer to Appendix G.\n\nRegarding the DetailRecon, which was published in January 2025, we did not recognize the paper, but we will address it in the related work section of the revised version. Moreover, it is worth noting that FEM is not a neural network-based method, but rather a traditional numerical technique used for solving partial differential equations. While FEM remains a cornerstone in engineering simulations, it differs fundamentally from neural network-based approaches like ours, which aim to learn patterns from data rather than relying solely on predefined physical models.", "claims_and_evidence": "The claims made in the paper are well-supported by both theoretical and empirical evidence. However, the paper could benefit from a more detailed discussion of the limitations of the proposed method, particularly in scenarios where thickness is not the dominant factor in deformation.", "methods_and_evaluation_criteria": "The proposed methods are well-suited for the problem at hand. The experimental design is sound, with a clear comparison to state-of-the-art baselines (MGN, EGNN, EMNN) and ablation studies to validate the contributions of individual components (e.g., thickness edges, dot product of normal vectors). \nCould the proposed show more visual examples on the thickness reconstruction for challenging mesh, such as the leaves, cloth, etc.", "theoretical_claims": "The paper does not present formal theoretical proofs but provides a clear explanation of the E(3)-equivariance and invariance properties of the proposed data-driven coordinate system. The authors justify their approach using geometric principles and demonstrate its effectiveness empirically.", "experimental_designs_or_analyses": "The experimental design is robust, with the following key strengths: Dataset, Baselines, Ablation studies, Out-of-distribution testing. One potential limitation is the lack of comparison with volume-based methods (DetailRecon: Focusing on Detailed Regions for Online Monocular 3D Reconstruction), which could provide additional context for the performance of surface-based approaches like T-EMNN.", "relation_to_broader_scientific_literature": "The paper builds on several key areas of research, Mesh-based 3D representation, E(3)-equivariant networks, and Thickness modeling.\nThe paper situates itself well within the broader context of 3D mesh analysis and equivariant neural networks, but it could benefit from a more detailed discussion of how T-EMNN compares to volume-based methods, which are commonly used in structural analysis.", "essential_references_not_discussed": "It is ok for me.", "other_strengths_and_weaknesses": "**Strengths**:\n1. The introduction of thickness-aware modeling and data-driven coordinates is a significant contribution to the field of 3D mesh analysis.\n2. The validation on a real-world industrial dataset demonstrates the method's applicability to practical problems like injection molding.\n3. The model's performance under out-of-distribution settings highlights its robustness to transformations.\n\n**Weaknesses**:\n1. The paper focuses on surface meshes but does not compare T-EMNN with volume-based methods, which are commonly used in structural analysis.\n2. While the paper provides a clear explanation of the proposed methods, formal theoretical proofs could strengthen the theoretical contributions.", "other_comments_or_suggestions": "NA", "questions_for_authors": "1. How does T-EMNN compare to volume-based methods (like FEM, DetailRecon: Focusing on Detailed Regions for Online Monocular 3D Reconstruction) in terms of accuracy and computational efficiency?\n2. Can the authors provide formal proofs for the E(3)-equivariance and invariance properties of the proposed data-driven coordinate system?\n3. How well does T-EMNN generalize to other types of 3D objects beyond the industrial dataset used in the experiments?", "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": "OFY8MxCJ6o", "reviewer": "Reviewer_xjH9", "summary": "This paper presents a novel graph neural network that incorporates thickness edges—connections between opposite sides of a surface mesh—to enable thickness-aware processing. To maintain E(3)-equivariance, it introduces a data-driven coordinate transformation. The model is evaluated on an injection molding dataset, where it outperforms various baselines.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. For a comprehensive response, please refer to the attached [link](https://shorturl.at/gOsz6). All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).\n\n---  \n\n**A1. Problem Definition and Dynamic Analysis**\n\nThe task is to predict node-wise 3D deformation for 28 unique shapes with thickness, each under 18 condition combinations. In Figure 7, we visualize the deformation by taking the L2-norm of each node's deformation, resulting in a scalar value per node. These shapes are tested under diverse conditions, which are defined as \"static analysis,\" distinct from the dynamic analysis in MGN, which models material interactions. Static analysis focuses on how deformation changes under varying conditions (e.g., pressure), while dynamic analysis, such as the deforming plate in MGN, examines interactions over time (e.g., velocity). \n\nAlthough they serve different purposes, we also perform next timestep deformation prediction with the Deforming Plate dataset to demonstrate how our thickness processor helps the model handle dynamic situations. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1 and Fig. L6, where our model successfully handles the deforming plate's thickness, improving performance.\n\n---  \n\n**A2. Additional Datasets** \n\nPlease refer to our _rebuttal A2 to reviewer \"pfVr\"_ due to character limits.\n\n---\n\n**A3. Equivariance/Invariance Proof**\n\nPlease refer to Sec. L1 for the detailed proof. Our model can be either invariant or equivariant, depending on whether the 'inverse transformation' is applied. We will make sure to be more rigorous in distinguishing between the terms \"equivariance\" and \"invariance\" in the revised version.\n\n---\n\n**A4. Thickness Node Pair Definition.** Please refer to Fig. L7 and L8 in Sec. L3 for a comprehensive description. The dot product between the normal vectors of $v_i$ and $T(v_i)$ is included as an attribute of the thickness edge between $(v_i, T(v_i))$ to address the issue you raised. To better account for these non-perpendicular edges, we use the dot product to measure how well the normal vectors are aligned in the edge attribute.\n\n**A5. Ray Projection Distance ($d$).** We use the ray.intersects_location function from the trimesh library to find intersections between the mesh and the ray. Since the ray has no distance limit, it may extend to the far opposing side, leading to what is more accurately described as width rather than thickness, as shown in Fig. 4 of the paper. To address this, we introduce the thickness edge feature as a distance measure and apply a learnable thickness threshold to filter out incorrect thickness edges, without requiring a distance limit $d$.\n\n**A6. Selection of $\\tau$.** The learnable threshold $\\tau$ is essentially a thickness threshold that helps with the prediction. It defines the range of thickness edges where message passing is beneficial. As distance increases, the interaction or correlation between opposite nodes tends to decrease. This aligns with the design we intended, and the reason it is made learnable is to allow the model to dynamically adjust this threshold.\n\n**A7. Clarification of Node Features $g_i$  and $r_i$**  Node features are described in Apx, lines 606-607. We will reference them in the main text for clarity.\n\n**A8. Thickness Activation Function.** Please refer to Fig. L9 for details. The activation function in Eq. 15 includes $\\alpha$, which controls transition sharpness. Increasing $\\alpha$ sharpens the influence based on thickness, but the influence is gradually reduced, not abruptly cut off, to ensure stable training. For a detailed analysis of $\\alpha$, see Apx. D. The term \"exclude\" was meant to indicate a gradual reduction, not a hard cutoff.\n\n**A9. Regarding MGN.** While our task does not require dynamic modeling, we chose MGN as a baseline for its strong architecture, on which we build by adding a thickness processor and a data-driven coordinate system. As stated in line 142, our method shares the core MGN structure, making a fair comparison and emphasizing the impact of our additions.\n\n---\n\n**A10. Essential References Not Discussed**\n\nWe would like to clarify that MGN is mentioned in line 141, where we note that our architecture is derived from the core MGN structure. However, for clarity, we will specifically reference it in Sec 4.2.2.\n\n---\n\n**A11. Regarding the gravity.** Thank you for your insightful comment. We are indeed considering a coordinate system where equivariance operates beyond the direction of the gravitational field as part of our future work.", "claims_and_evidence": "There are two key issues with the claims and supporting evidence in the paper:\n\n1. **Limited dataset evaluation** – The model is evaluated on a single novel dataset. While the results appear promising, testing on an additional dataset (or ideally two) would provide stronger support for the claimed superior performance. Given that the authors used MGN as a baseline, I wonder if they considered the *deforming plate* dataset, where a plate undergoes deformation over time. Applying their method to this dataset, particularly with a surface mesh representation, could further demonstrate its advantages.\n2. **Unsubstantiated Equivariance Claim** – The paper states that the model “preserves E(3)-equivariance and invariance,” yet provides no theoretical proof to substantiate this claim. In my view, the lack of formal justification weakens the argument. Additionally, the phrasing raises a conceptual issue: how can the model be both equivariant and invariant? From my understanding, invariance is not merely a special case of equivariance, as it completely removes the output transformation rather than modifying it in a structured way. Instead, I would consider it a related but distinct concept. This imprecision in terminology further underscores the need for a more rigorous theoretical analysis to clarify and support the claim.", "methods_and_evaluation_criteria": "# Method\n**Thickness Node Pair Definition and Mapping Consistency**\n\nIn Section 3.3, the paper introduces the concept of thickness node pairs, defined by a transformation $T$ that maps each node $v_i$ to another node on the opposite surface. However, the definition provided in Equation (1) does not ensure that applying the transformation twice returns the original node; that is, $T(T(v_i)) \\neq v_i$. This issue becomes evident in materials where one surface is skewed relative to the other, resulting in a gradual thinning. The paper does not address how this scenario is handled, which raises concerns about the robustness of the thickness node pairing method in such cases. See for reference at the first draft I created for illustration: https://imgur.com/a/nmmI9oI\n\n**Ray Projection Distance ($d$) as a Hyperparameter**\n\nFrom my understanding, the ray projection distance $d$ is a crucial hyperparameter in defining thickness node pairs. In materials with multiple layers, determining the appropriate value of $d$ is essential to accurately pair nodes between surfaces. If the thickness varies between different sheets, selecting a single $d$ becomes challenging, potentially leading to incorrect thickness pairings. The paper does not discuss strategies for choosing ddd in such complex scenarios, leaving a gap in the methodology for materials with non-uniform thickness. In the second image provided in https://imgur.com/a/nmmI9oI , I don't see how the correct thickness pairs can be computed using a global value $d$. \n\n**Selection of $\\tau$**\n\nI like the idea of selecting a data-driven $\\tau$, but I am unsure if a global $\\tau$ can suffice for non-uniform thickness objects. Going back to the second image in  https://imgur.com/a/nmmI9oI , a $\\tau$ of 2 would exclude all thickness edges on the left side, while a $\\tau$ of 6 would also include the \"width\" edges on the right side. Is my intuition here correct?\n\n**Clarification of Node Features $g_i$ and $r_i$**\n\nTable 1 references node features $g_i$ and $r_i$, but there is no explanation of these features in the main text or Appendix A. Clarifying the model inputs is important for understanding the model's functionality. Including descriptions of these features in the main body or referencing a specific section in the appendix would enhance the paper's clarity and comprehensibility.\n\n\n**Thickness Activation Function**\n\nRegarding the activation function for thickness pairs, the authors write \"“while edges with $t(v_i) > \\tau$ are excluded from the propagation process.” This is not precise, the activation function for $t(v_i) = \\tau$ is 0.5, see the graph here: https://imgur.com/a/6Pyfzfr\nAs far as I understand it, the activation function has to be shifted to the left.\n\n**Overall Assessment**\n\nDespite these concerns, the proposed architecture for handling thickness appears sensible. Addressing the issues related to the transformation mapping, the selection of the ray projection distance $d$, and the clarification of node features would strengthen the paper's contributions and provide a more robust framework for thickness-aware processing in graph neural networks.\n\n# Evaluation:\n\nI did not fully understand the exact problem being addressed in the dataset. Unless I missed it, this information does not appear to be provided in the main text or in Appendix A. Figure 7a) presents a “Ground Truth” with an output value/vector(?) per node, but it is unclear what the referenced “deformation magnitude” represents.\nWhat is the exact challenge of the dataset? What is given and what needs to be predicted?\n\nExisting baselines, such as MGN, typically handle deformation tasks over time, addressing challenges like rollout stability by introducing noise. However, I suspect that this is not the case in the current task. If so, does the comparison to MGN only refer to its architecture and encoder-processor-decoder structure rather than its dynamic modeling capabilities?\n\nIn summary, the goal of the task is not clearly defined, and incorporating a dynamic deformation over time would provide a more appropriate basis for comparison with the baselines.", "theoretical_claims": "The paper does not provide theoretical proofs to support its claims. As mentioned earlier, the claim that the model “preserves E(3)-equivariance and invariance” lacks formal justification, making it difficult to assess its validity. A rigorous proof or at least a more detailed explanation would be necessary to substantiate this claim.", "experimental_designs_or_analyses": "I encountered difficulties in fully assessing the experimental design due to ambiguities in the dataset's objectives and the lack of precise definitions for the evaluation metrics used. The metrics—Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²)—are mentioned as abbreviations without explicit descriptions of their computation methods. While they are in general well-known, a precise defintion of them would strengthen the reproducibility of the paper's results.\n\nAdditionally, using 3 seeds for repetition is on the lower side, especially if only one dataset is used. Using confidence intervals instead of the standard deviation can also strenghten the statistical significance of the results.", "relation_to_broader_scientific_literature": "The paper introduces a novel \"Coordinate Transformation: E(3)-Invariant Data-driven Coordinate System\" in section 4.1, which is not addressed in the related work section. This contribution could benefit from contextualization within the broader literature, especially related to coordinate transformations in geometric deep learning and invariant representations. Including references to works that explore coordinate transformations preserving E(3) invariance could strengthen the discussion.\n\nAdditionally, the paper mentions thickness handling, but only indirectly through mesh density and hierarchical pooling. It would be valuable to relate these ideas to prior work that explicitly addresses thickness handling in similar contexts. There may be relevant research on mesh processing or pooling methods that directly deals with the manipulation of thickness or geometric features in 3D data. If present, incorporating such references could provide a clearer connection to existing research and help position the paper's contributions more effectively in the broader field.", "essential_references_not_discussed": "In Section 4.2.2 the authors propose their surface update. These are the exact update rules given in the Message Passing Neural network used for example by MeshGraphNet and originates from the paper \"Graph Networks as Learnable Physics Engines for Inference and Control\" by  Sanchez-Gonzalez et. al., 2018. I suggest to cite one of the papers and clearly distinguish here between own contribution and existing architectures.", "other_strengths_and_weaknesses": "The general idea of improving the surface mesh by incorporating thickness edges is crucial for learnable physic simulators. I really like the idea and I can clearly see the scientific gap. The data-driven coordiante frame looks promising, I am wondering how this can be extended to multiple objects and their interaction, as well as how this is handling deformations over time. Also, what if gravity is a concern, does this not break the symmetry? Although this is a general problem with equivariant networks, it would strengthen the paper, if this is discussed.", "other_comments_or_suggestions": "- 094, right: Mention MGN=”Mesh Graph Net”\n- Inconsistent notation in Eq. 14: Normal $n_i$ uses only the node index $i$, while $n_{T(v_i)}$ uses the complete node $T(v_i)$ as an index.", "questions_for_authors": "Main questions are discussed in \"Methods And Evaluation Criteria\".", "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": "tGOmZJtDL4", "reviewer": "Reviewer_8wBp", "summary": "This paper presents the Thickness-aware E(3)-Equivariant Mesh Neural Network (T-EMNN), a novel graph neural network designed to efficiently integrate the thickness of 3D objects into mesh-based static analysis. The authors introduce an innovative thickness-aware framework that explicitly considers interactions between opposing surfaces through dedicated \"thickness edges\". The method employs a learned thickness threshold to distinguish genuine thickness relationships from irrelevant spatial connections (e.g., width) and uses data-driven coordinates to ensure E(3)-equivariance. The authors validate their method using a real-world industrial dataset focused on injection molding applications. T-EMNN significantly outperforms state-of-the-art graph-based methods in terms of prediction accuracy (RMSE, MAE, R²) while maintaining computational efficiency.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your thorough review. For a comprehensive response, please refer to the attached [link](https://shorturl.at/gOsz6). All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).\n\n---\n\n**A1. Practical Utility of the Method (Response to Weakness and Q1).**\n\nThe pipeline is illustrated in Fig. L10 to aid comprehension of the proposed approach. In the actual industrial process of injection molding product development, the process follows the stages of **design -> analysis -> mass production**. According to this process, designers must verify the product's defect status through analysis before mass production. However, **there is a bottleneck in the design -> analysis phase**, and we aim to address this using T-EMNN.\n\nThe reason the design -> analysis process is a bottleneck is that the work is divided between the designer, who specializes in design, and the analyst, who specializes in numerical analysis. **The designer does not have numerical analysis expertise and must rely on the expert analyst.** This process requires iteration. When the designer modifies the design, they send it to the analyst for analysis. The analyst then reports the results to the designer, who evaluates whether there are any defects based on the analysis. If defects are suspected, the designer modifies the design and sends it again, creating a repeated cycle (design -> analysis -> design -> analysis). **This iterative process typically takes about a week for a single product** (based on the Basket dataset), leading to significant delays in mass production.\n\nWe expect that using the **T-EMNN we developed will dramatically reduce this iterative process.** First, **T-EMNN allows designers to self-validate their designs** without requiring any special analysis skills. Furthermore, we confirmed that with **T-EMNN, it takes less than 3 minutes** (based on the Basket dataset) to go from preprocessing to checking the analysis results for a single injection molding product. Thus, with T-EMNN, designers can quickly self-validate designs that are unlikely to have defects. This allows designers to filter out designs with low defect potential and send only those to the analyst, significantly reducing the iteration process. This approach will drastically shorten the time to mass production.\n\n---\n\n**A2. Generalizability of the Thickness Threshold (Response to Q2)**\n\nWe evaluate our method on four additional settings (using 5 seeds for reliability):  \n- **Deforming Plate** (Node-level prediction - Position of Next timestep)  \n- **Basket** (Graph-level prediction - Max pressure)  \n- **Circular Plate** (Node-level prediction - Deflection of each node)  \n- **SimJEB** [1] (Node-level prediction - Magnitude of the displacement)\n\nRegarding performance, as shown in Table L1, the thickness processor successfully handles even next-timestep prediction. In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the \"In Dist.\" setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the \"Out-of-Dist.\" setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance. \n\nTo qualitatively demonstrate our model’s ability to process thickness, we include the learned valid thickness edges in Figs. L4, L5, and L6. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right).\n\n---\n\n**A3. Discussion for Handling Non-watertight Surfaces (Response to Q3)**\n\nWe believe that **T-EMNN can be applied even in the case of non-watertight surfaces.** However, when creating the thickness edge, we utilize a method where a ray is cast from the starting node in the opposite direction of the normal (i.e., into the interior of the object) until it reaches the opposing surface. If the object is not watertight, there may be cases where the ray does not touch the opposite surface. In such cases, if the ray does not touch a specific node, the thickness edge can simply be excluded. This should not pose any issues for training and prediction.\n\nHowever, for our T-EMNN, the watertightness of the 3D shape data is not a critical factor. **What matters is whether the physical properties between thickness pairs are similar.** If they are similar, we expect that using T-EMNN will still yield good prediction results.", "claims_and_evidence": "Good to me.", "methods_and_evaluation_criteria": "The evaluation criteria (RMSE, MAE, R²) and comparison methods (MGN, EGNN, EMNN) used by the authors are appropriate and clearly demonstrate the effectiveness of the thickness-aware design. Visualizations in Figure 7 illustrate clearly the error distributions and improvements achieved by T-EMNN. The visualization in Figure 9 show the effectiveness of the threshold. All of these are good. However, the practical utility of the method could be strengthened by demonstrating more clearly the application pipeline in industrial scenarios like injection molding.", "theoretical_claims": "Good to me", "experimental_designs_or_analyses": "The experimental designs, particularly the inclusion of in-distribution and out-of-distribution conditions, are sound. The ablation study (Fig. 6, Table 2) effectively demonstrates the contribution and optimization of the thickness threshold and thickness edge features, validating the robustness of the proposed methods.", "relation_to_broader_scientific_literature": "The proposed work effectively builds upon and extends existing mesh-based neural network methods (MGN, EGNN, EMNN), clearly positioning itself within the broader scientific literature by addressing critical limitations related to thickness modeling and E(3)-Equivariance. It offers notable improvements in practical application contexts such as structural engineering and manufacturing.", "essential_references_not_discussed": "The reference is good to me", "other_strengths_and_weaknesses": "The proposed method effectively addresses a significant limitation of existing surface-based mesh methods by integrating thickness-awareness into mesh neural networks. Additionally, the proposed data-driven coordinate transformation notably enhances robustness against spatial transformations, as demonstrated by the clear performance improvements observed when integrating this component into existing baseline methods. The paper further provides comprehensive and thorough evaluations, including meaningful ablation studies that help clarify the contribution of each component.\n\nHowever, the paper currently lacks demonstrations of practical applications beyond synthetic validation scenarios. Including concrete examples demonstrating direct applicability in real-world industrial tasks would significantly strengthen the paper. Furthermore, the application pipeline description could be more detailed to better bridge theoretical contributions and practical utility. Lastly, the paper would benefit from a discussion regarding potential extensions to handle non-watertight surfaces (i.e., surfaces with holes), as the current methodology does not appear to strictly require watertight constraints.", "other_comments_or_suggestions": "Good to me.", "questions_for_authors": "1. Could you further elaborate on the practical integration of your model into an existing industrial workflow, particularly for injection molding? Would this integration introduce significant overhead?\n\n2. How sensitive is the learned thickness threshold τ across significantly different geometries or scales, and how generalizable is this threshold to other applications beyond injection molding?\n\n3. How to handle non-watertight inputs? For example, a \"solid\" surface with \"small\" holes no that. Does the proposed method robust to that kind of inputs?", "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": "oDR2PzzjEx", "reviewer": "Reviewer_pfVr", "summary": "This paper considers the thickness of the mesh when predicting the deformation. T-EMNN is proposed to preserve E(3)-equivariance and invariance. Experiments show that the proposed method outperforms baselines and the introduction of thickness benefits the baselines as well.\n\n## Update after rebuttal\nI appreciate the authors' efforts to address my concerns and I have updated my score to 3.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. For a comprehensive response, please refer to the attached [link](https://shorturl.at/gOsz6). All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).\n\n---\n\n**A1. Novelty of Our Work.**\n\nWe would like to organize our contributions into two key aspects: **1)** Thickness-Aware Framework and **2)** Data-driven Coordinates, which effectively achieve E(3)-equivariance. Given your concern about the novelty of the GNN architecture, ___we will focus on emphasizing the Thickness-Aware Framework in this rebuttal___, rather than the data-driven coordinates.\n\nPerhaps our architecture may appear simple, but we would like to emphasize that the ___input meshes do not inherently contain any information regarding the \"thickness.\"___ As a result, a basic GNN without any thickness processing faces challenges in interacting between opposing surfaces, despite their high correlation, as shown in Fig. 1 in the introduction.\n\nOur novelty lies in how the GNN model ___adaptively addresses the \"thickness,\"___ even though this information is not provided in the input data. To achieve this, __we define the \"thickness edge\" from scratch and address potential issues__, such as the negative correlation between opposing surfaces when the thickness is too large to share the correlative reaction. To overcome this challenge, we propose a __learnable thickness threshold__ that adaptively identifies the optimal thickness edges. As a result, our method demonstrates strong performance, particularly on the $R^2$ metric, showcasing its ability to predict more realistic behavior.\n\nWe would like to emphasize that our method is not distinguished from others simply by utilizing several conditional embeddings through concatenation. In fact, __the architecture of the baseline methods was modified to allow for the use of exactly the same conditional embeddings, including both condition and coordinate embeddings, as in our method, ensuring a fair comparison.__ Our distinct contribution lies in whether the model can handle opposing surfaces with high correlation, and our model successfully achieves this by defining and processing the thickness in a learnable way.\n\n---\n\n**A2. Evaluation on Four Additional Datasets.**\n\nWe evaluate our method on four additional datasets (using 5 seeds for reliability):  \n- **Basket** (Graph-level prediction - Max pressure)  \n- **Circular Plate** (Node-level prediction - Deflection of each node)  \n- **SimJEB** [1] (Node-level prediction - Magnitude of the displacement)  \n\nThe dataset and results are presented in Table L2 and Figures L1-L5. In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the \"In Dist.\" setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the \"Out-of-Dist.\" setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance. \n\nTo qualitatively demonstrate our model’s ability to process thickness, we include the learned valid thickness edges in Fig. L4 and L5. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right).\n\n- **Deforming Plate** (Node-level prediction - Position of Next timestep)  \n\nMoreover, to evaluate the dynamic capabilities of our framework—specifically the thickness processor—we perform next-timestep deformation prediction using the **Deforming Plate** dataset (as described in the MGN paper) to demonstrate how our thickness processor helps the model handle dynamic scenarios. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1 and Fig. L6, where our model successfully accounts for the deforming plate’s thickness, leading to improved performance.\n\n---\n\n**A3. Reorganization of related work.** Thank you for pointing this out. We will ensure that the related work regarding methods related to E(n)-equivariance/invariance is clearly addressed and reorganized separately.\n\n> [1] Whalen et al. \"SimJEB: simulated jet engine bracket dataset.\" Computer Graphics Forum. Vol. 40. No. 5. 2021.", "claims_and_evidence": "The claims are generally clear.", "methods_and_evaluation_criteria": "My main concern is about the limited dataset. Is there any other dataset? More samples or different kinds of objects would strengthen this paper.", "theoretical_claims": "Please refer to \"Claims And Evidence\".", "experimental_designs_or_analyses": "The evaluations seem correct.", "relation_to_broader_scientific_literature": "Please refer to \"Essential References Not Discussed\".", "essential_references_not_discussed": "Since “E(3)-equivariance and invariance” is one of the core idea in this paper, the author may discuss neural network with the property of “E(3)-equivariance and invariance” in a separate paragraph. For example, the author may adjust L85-106 as part of the related work.", "other_strengths_and_weaknesses": "The E(3)-equivariance is achieved by adjusting the inputs, while the GNN part does not contribute to this property. The proposed method mainly adopts the basic GNN module and introduces several conditional embeddings through concatenation. The main contributions mentioned above seem limited.", "other_comments_or_suggestions": "Please refer to the \"Questions For Authors\".", "questions_for_authors": "I summarize my main concerns as follows:\n1. Limited dataset. (Methods And Evaluation Criteria)\n2. Please provide more details to emphasize the novelty of this paper. (Other Strengths And Weaknesses)\n3. Reorganization of related work. (Essential References Not Discussed)", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors introduce a thickness-aware variant of E(3)-equivariant mesh neural networks, integrating surface thickness into mesh processing frameworks. The paper was well-received by the reviewers, resulting in three positive (2x WA, 1x A) and one WR recommendation. The reviewers welcomed the introduction of thickness as a contribution of high practical significance and highlighted its effectiveness and robustness. Concerns remained regarding the number of evaluated real datasets as well as questions regarding clarity in equi-and invariances. The authors provided answers to the individual concerns, convincing all but one reviewer.

I will follow with an accept recommendation. While one reviewer remained with a negatively leaning score, I believe his strongest concerns (regarding equi-and invariance) have been addressed sufficiently by the authors.
Decision: Accept (poster)