Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
Authors: Shilong Tao
Zhe Feng
Haonan Sun
Zhanxing Zhu
Yunhuai Liu
Venue: ICML 2025 poster (2025)
Keywords: Deep Learning
Multi-solid Systems
Explicit modeling
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We propose Unisoma to solve multi-solid systems in an explicit modeing paradigm.

Abstract: Multi-solid systems are foundational to a wide range of real-world applications, yet modeling their complex interactions remains challenging. Existing deep learning methods predominantly rely on implicit modeling, where the factors influencing solid deformation are not explicitly represented but are instead indirectly learned. However, as the number of solids increases, these methods struggle to accurately capture intricate physical interactions. In this paper, we introduce a novel explicit modeling paradigm that incorporates factors influencing solid deformation through structured modules. Specifically, we present Unisoma, a unified and flexible Transformer-based model capable of handling variable numbers of solids. Unisoma directly captures physical interactions using contact modules and adaptive interaction allocation mechanism, and learns the deformation through a triplet relationship. Compared to implicit modeling techniques, explicit modeling is more well-suited for multi-solid systems with diverse coupling patterns, as it enables detailed treatment of each solid while preventing  information blending and confusion. Experimentally, Unisoma achieves consistent state-of-the-art performance across seven well-established datasets and two complex multi-solid tasks. Code is avaiable at [https://github.com/therontau0054/Unisoma](https://github.com/therontau0054/Unisoma).

Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
Shilong Tao Zhe Feng'
Haonan Sun
Zhanxing Zhu 2
Yunhuai Liu
Abstract
Implicit Modeling
Domain-wise Learning Graph-wise
Multi-solid systems are foundational wide range of real-world applications, yet modeling their complex interactions remains challenging. Existing deep learning methods predominantly rely on implicit modeling, where the factors influ- encing solid deformation are not explicitly repre- sented but are instead indirectly learned. However, as the number of solids increases, these methods struggle accurately capture intricate physical interactions. In this paper, we introduce novel explicit modeling paradigm that incorporates fac- tors influencing solid deformation through struc- tured modules. Specifically, we present Unisoma, a unified and flexible Transformer- capable of handling variable numbers of solids. Unisoma directly captures physical interactions using contact modules and adaptive interaction allocation mechanism, and learns the deforma- tion through triplet relationship. Compared to implicit modeling techniques, explicit modeling is more well-suited for multi-solid systems with diverse coupling patterns, as enables detailed treatment of each solid while preventing informa- tion blending and confusion. Experimentally. Uni- soma achieves consistent state-of-the- the-tart perfor- mance across seven well-established datasets and two complex multi-solid tasks. Code is avaiable at http://gghub..com//teromoo0/0///
Learning
Contact
Deformation
Load Contact
Deformation
Explicit Modeling
Load
Figure I. Implicit Modeling and Explicit Modeling. Yellow planes are rigid solids. Orange and balls are deformable solids.
in industry, robotic gripping (Zhang et al., 2020), and com- posite materials research (Clyne & Hull, 2019) are typical application scenarios. From a scientific standpoint, multi- solid systems can often be formulated as high-dimensional, nonlinear Partial Differential Equations (PDEs). However, the large solution space, strict contact conditions, and com- plex dynamics pose significant computational challenges for traditional numerical methods (e.g.. Finite Element Methods (Sifakis & Barbic, 2012) and Material Point Methods (Sul- sky et al., 1995)), making them difficult to meet real-time or high-fidelity requirements (Umetani & Bickel, 2018). Re- cently, Deep Learning techniques have been increasingly adopted in scientific computing, showing promise for vari- ous physical simulations (Cai al., 2021; Lu al., 2021; Li et al., 2020a). Despite these advances, most research to date has focused on few-solid systems or other physical scenarios such as fluids. There remains a notable gap in applying deep learning to multi-solid systems, and a system- atic approach capable of characterizing complex physical interactions among multiple solids is still underexplored.
model
1. Introduction
Multi-solid systems encompass multiple rigid and de- formable solids with distinct mechanical properties (e.g.. elastic, plastic, hyperelastic). The motion of these solids and their contact interactions induce forces (also referred to as loads), thereby driving deformations and changes in the physical quantities of deformable solids, which are our primary focus. In these systems, diverse material behaviors, contact phenomena, and exerted loads converge into plex physical interactions, which in turn lead to intricate deformation behaviors (Braess, 2001). Consequently, how to effectively distinguish and capture these interactions is the key challenge in multi-solid analysis.
Multi-solid (rigid and deformable) systems play an essential role in broad range of real-world scenarios, such indus- trial manufacturing (Khan & Turowski, 2016), mechanical manipulation (Mason, 2001), and aerospace (Mouritz, 2012). For example, multi-point metal stamping (Li et al., 2002)
'School of Computer Science, Peking University, Beijing, China "School of Electrical and Computer Science, University of Southampton, UK. Correspondence Yunhuai Liu <-un- huai. pku.edu.en>> Zhanxing Zhu <z.zhu @ ao..o....o.....
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Presently, research specifically dedicated to multi-solid sys-

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
tems remains relatively scarce. In addressing solid prob- lems, data-driven approaches predominantly rely on im- plicit modeling, where the factors that influence the defor- mation of deformable solids are not explicitly structured but instead implicitly captured through the overall learning process. These methods can be broadly categorized into two types. As shown the upper branch of Figure 1, the first category (Raissi et al., 2019; Li et al., 2020a) is domain- wise, treating the entire computational domain as input and merging all solids into one large PDE problem. Then it is solved through Physics-Informed Neural Networks (PINNs) (Cai et al., 2021) or Neural Operators (Boullé & Townsend, 2023). Although this approach simplifies setup and ensures global consistency, it can become computationally expen- sive and lacks fine-grained insight into individual solids. As the number and diversity of solids increase, the effective cap- ture of intricate physical interactions becomes increasingly challenging, restricting its scalability. The second category Saanchez-Gozzale et al., 2018; 2020a) is graph-wise, using graph topologies to represent each solid as a subdomain and dynamically add or remove edges between solids based on a physical distance threshold. The GNN-based message passing (Gilmer et al., 2017) is then utilized to transmit and aggregate information between nodes. In this process, the edges merely act as conduits for information flow, and the interactions influencing the deformation of solids are still implicitly learned through the node-level propagation. How- ever, only relying on independent edges to learn physical interactions between different solids may introduce blend- ing and ambiguity (Yifan et al., 2020). In addition, the local nature of graph kernels limits its efficacy for long-range predictions involving long time span (Li et al., 2024a).
contact constraint to describe the deformation of single de- formable solid in the system. We first utilize distinct contact modules to capture each pair of contact constraints in the system. Then we propose an adaptive interaction allocation mechanism to compute the individual equivalent load and contact constraint for each deformable solid. Afterwards, for each deformable solid, we adopt a deformable module to model the influence of equivalent load and contact constraint its deformation under deformation triplet relationship. Unisoma achieves state-of-the-at performance extensive evaluations. Our main contributions are as follows:
We propose solve multi-solid systems in an explicit modeling paradigm. This kind of paradigm brings high degree of flexibility, enabling effective control and capture of essential physical interactions.
We design Unisoma, a unified Transformer-basee model with an explicit modeling approach capable of flexibly handling varying numbers of solids. Unisoma captures and integrates physical interactions with ex- plicitly structured modules, and learn the deformation under the deformation triplet relationship.
We evaluate Unisoma on two rulte tasks across seven datasets spanning different complexity, comparing it against over ten advanced deep learning models. Unisoma consistently achieves superior per- formance improvements under all test scenarios.
2. Related Work
Currently, there is a relative scareity of research focused multi-solid systems. For solid problems, data-driven frameworks are generally divided into two paradigms.
As shown in the lower branch of Figure I, unlike previous approaches, we propose to solve multi-solid systems within an explicit modeling paradigm, which directly models the factors (such as contact constraints and loads) that influ- ence the deformation of solids through the model structure. This design is especially well-suited to multi-solid scenarios where coupling patterns are diverse, allowing for tailored treatment of different pairs with more fine-grat ane-ggriied model- ing. In this way, we can distinguish and capture essential physical interactions more controllably and accurately. It avoids the issues of information confusion and the difficulty in fully learning complex physical interactions in multi-solid scenarios that are common in implicit modeling methods.
Implicit Modeling In this paradigm, the factors that influ- ence the deformation of solids are not explicitly structured but instead implicitly learned from data. Methods belonging to this paradigm can be classified into two categories. The first domain wise category treats the entire computational domain as input, merging all objects into one PDE prob- lem. In the solid field, PINNs have been widely applied (Haghighat et al., 2021; Rodriguez- Torrado et al., 2021; Okazaki et al., 2022). often with elaborately designed loss functions (Bai et al., 2023; Chiu et al., 2022) and domain- wise techniques (Diao et al., 2023; Li et al., 2023a), Neural Operators, such as FNO (Li al., 2020a) and its variants (Gupta et al., 2021: Deng et al., 2024), have demonstrated success in simple solid problems with regular geometries. Many variants (Li et al., 2020b; 2024a) are proposed to tackle irregular geometries. In particular, the geometric Fourier transform proposed in Geo-FNO (Li et al., 2023b) is vastly used, which projects the irregular geometry into uni- form latent structure. Transformer-ban based models (Vaswani,
Technically, we propose Unisoma, unified and flexible Transformer-based (Vaswani, 2017) framework with explicit modeling capable of handling variable numbers of solids and two important tasks (long-time prediction and autore- gressive simulation) in the multi-solid scenarios, as the first attempt of the explicit modeling to our best knowledge. Specifically, define a deformation triplet consisting of deformable solid, equivalent load, and an equivalent
2

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Unisoma: A Unified Transformer-bascr Solver for Multi-Solid Systems
A11 Loads
Loads Allocation
Deformation Triplet
Decoder
Deformation Module
Contact Module
Contacts Allocation
Processor
Contact Module
Contacts Allocation
Processor
Contact Module
Deformation
Module
Encoder
Loads Allocation
A11 Loads
Deformation Triplet
Figure 2. The overview of Unisoma. The contact module and deformation module are attention-base The loads allocation module and contacts allocation module deploy adaptive interaction allocation mechanism.
2017) have gained traction in continuum mechanics, uti- lizing advanced attention mechanisms (Liu et al., 2024; Li et al., 2024b; Wu et al., 2024; Xiao et al., 2024; Li et al., 2024c) and physics-informed inductive biases (Li et al., 2023d; Hao et al., 2023; Shao et al., 2022). However, all these models are primarily evaluated on simple few-solid scenarios. When tackling multi-solid systems, they strug- gle to effectively capture distinct physical interactions due to the lack of fine-gre tne-grained insight into individual solids, particularly as the number and diversity of solids increase.
ing for multi-solid systems to our best knowledge. It explic- itly models physical interactions with structured modules and learns deformation through deformation triplet relation- ship. Compared to implicit modeling, explicit modeling is better suited for multi-solid systems with diverse coupling patterns, it allows fine-grained treatment of each solid and avoids information blending and confusion.
3. Method
Problem Setup Consider a system with multiple ob- jects composed of Nd deformable solids u {uf N/ xxxxCa Fi=1 with different material properties, N rigid R xCx N
The second graph-wist wise category encodes the physical ob- jects as graphs, and leverage GNNs (Scarselli et al., 2008) to capture the interactions in local neighbor with radius threshold. In the solid field, GNN-based simulators have been vastly used in particle-based simulations (Li et al., 2019; Sanchez-Gonzalez et al., 2020b) and mesh-based sim- ulations (Weng et al., 2021; Han et al., 2022; Cao et al., 2023). MGN (Pfaff et al., 2021) utilizes message passing blocks (Gilmer et al., 2017) to learn the dynamics of physi- cal systems. Many subsequent works have enhanced perfor- mance through hierarchical structures (Grigorev al., 2023; Yu et al., 2024; Fortunato al., 2022), mhysics-informen guidance (Würth et al., 2024; Perera & Agrawal, 2024) and diverse geometrical information (Linkerhägner et al., 2023; Allen et al., 2022). However, when independent edges are used as the learning medium, it may cause ambiguity and blending of the physical interactions between the solids. Additionally, graph-based simulators still fall short in learn- ing global interactions and struggle to address long-time predictions involving long time span (Li et al., 2024a).
solids u" = {uf E
RMN/x Ri=l and N acting forces fisl yielded by the move-
(loads) u' {uf ment of solids in the system. They all mesh points described in the Lagrangian view. A point of de- formable solid d, or rigid solid r, is recorded as vec- tor of length C2 or C,,, composed of 3D space coordi- nates and its properties (like Young's Modulus, Poisson's Rate, Stress and Friction). A point of load f, is vector of length Cý recording the 3D space coordinates of the solid from which it originated, and its next movement or posi- tion. For example, let u,, and u,,+1 denote the current and next state of a moving solid, the load is represented as Concat (o,rddu,,.). Coord (oord(u;e44)) (u,++1) Coord(u, (u)))) or Concat CCorrd (,,)). Coord(u,,+1)) (1))+))), where Coord(.) indi- cates the 3D space coordinates the solid.
We consider two types of tasks: 1) Long-Time Prediction (Raissi al., 2019; Li et al., 2020a; Wu al., 2024): Directly estimate target physical quantities (like geometries and inner stress) of deformable solids with long time duration and without intermediate process. It only infer- ences one time as: P(ū,,4T u;,,), where T spans many time steps. 2) Autoregressive Simulation (Pfaff et al., 2021; Yu al., 2024; Li al., 2019): Autoregressively
In contrast, we propose an explicit mod- eling paradigm, explicitly incorporating factors influencing solid deformation through specialized modules. Technically, we propose Unisoma, a unified and flexible Transformer- based framework, the first attempt of the explicit model-
Explicit Modeling
3

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
simulate the deforming process of deformable solids with small time increments. It inferences many times as: P( (u,,), 20),,)1)))-- P(0,,44/0,,,T-i)
(i.e., x Linear(u)) and bpgg E RIxC be the deep fea-
ture of edge connecting mesh points Xp and Xg. The edge augmented physics-aware tokens are formalized as:
Wi = {Softmax
3.1. Model Overview
w c (p-g)
= {Softmax [Softmax(Linear(ww
(Linear(w; (wo + Wg])}
The overview of Unisoma is illustrated in Figure 2, fol- lowing the "encoder-prooe processor-dccoder" fashion (Battaglia et al., 2018). First, instead of processing the multiple solids in the original input domain, we embed each object into edge augmented physics-aware tokens (Section 3.2). We then adopt stacked processors to explicitly model physical inter- actions among solids, including equivalent loads, equivalent contact constraints, and solid deformation (Section 3.3). Throughout the process, tokens representing rigid solids remain unchanged, while those representing deformable solids are updated. This reflects the physical principle that rigid solids are non-deformable. Finally, the outputs of the final processor are mapped back to the original domain, re- sulting in the predicted outcomes. Now we elaborate on each key component of Unisoma.
EN 8/3 + + E EennEE Ep 8 ((pa))
2 E Was +7 "yEep EE (p-2)s En Wigg, + + "'oon EF (n.)),
(I)
E Wis + +7 EennEE W Waas
Here, s {s, E ERNXC) 3==1 are termed slices with number of M, whose weights w E RNXM are yielded from deep features x by Linear(-) projection and Softmax(-) operation. The weight Wa, indicates the degree that the i-th mesh point belongs to the j-th slice with 2j=1 Wi,j = 1. The mesh points with close geometry and physical features are more likely to be assigned to the same slice. As shown in Figure 3, we incorporate the mesh edges into the slice components s" = {sf E IR are derived from the weights of the connected points. The physics-aware tokens z {z, E R R M then are en- coded by spatially weighted aggregation and normalization. Notably. although we utilize graph-like representation of the objects, it differs from GNN-based methods that capture interactions in the node level with edges. We transfer the graph into the slice domain and explicitly model physical interactions at the object level with attention mechanism. Remark 3.1. Why incorporating edge features by spatially weighted aggregation? ? Firstly, the aggregation follows the transformation of the original slice operated on mesh points (Wu et al., 2024) and the (1) is unified projection form including mesh points and edges. The (1. degenerates to the original slice if edge feature is not taken into account. Secondly, the weights of edge features are derived from the weights of mesh point features. This aggregation em- phasizes point pairs that are connected by edges, thereby further enhancing the local information during the aggrega- tion process. Normally, the number of edges is more than the number of points. To avoid the point features being inundated by edge features, we utilize the ratio of points and edges = N/|E| for quality control.
M
F x M J==1. whose weights w [R| [R|
3.2. Encoder: Edge Augmented Physics-Aware Tokens
Originally, the objects in the physical domain are described as large number of mesh points. We first need to embed them into tokens. A naive approach treats each point as token, unfortunately leading to high computational cost and redundant representation (Yu et al., 2022). Transolver (Wu et al., 2024) proposes the concept of slice, which as- signs mesh points with similar physical states to the same learnable slice token, enabling attention to capture intrinsic physical correlations more effectively. However, they only consider point-level features and spatially aggregate points in the whole domain. This leads to the loss of local relation- ships of mesh points, which are important to model local interactions like contact and local deformation. Therefore, we incorporate the mesh edges into embedding and propose Edge Augmented Physics-Aware Tokens.
mesh points
mesh edges
The embedding is performed for each object instead of the holistic domain for flexibly explicit modeling. We denote the embedding tokens for deformable solids u rigid solids u², and loads u' as d = {d, E RM JizI. r {r. R Jim and f = {f, E R RMXxO) Mi respectively.
mesh
slice
Figure 3. Emhedding mesh points and mesh edges into slices.
For each object, we first utilize the k-nearest neighbor (kNN) method to construct the edge set E based on input mesh points. The edge attribution is the relative distance of the connected neighbors. Let x E N xC be the deep features of an object with N mesh points projected by linear layers
3.3. Processor
After embedding these objects into edge augmented physics- aware tokens, we propose an extensible processor to learn

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
the physical interactions in an explicit manner. As shown in the right part of Figure 2, the design of the processor revolves around the deformation triplet. We separately con- sider the loads and contact constraints and then capture their effects on the deformable solids. First, we design a contact module to capture each pair of contacts that may occur in the system. Next, we propose an adaptive interaction allo- cation mechanism to learn the equivalent load and contact constraint for each individual deformable solid. Finally, we introduce a deformable module to characterize the influ- ence of the equivalent load and the contact constraint on the deformation of the corresponding solid.
representing different objects in the original input space is generally not effective. Unlike the residual connections (He et al., 2016), which are mainly designed to combine features of the same object, summing features from distinct objects can lead to significant information loss. Specifi- cally, this operation causes an indiscriminate mixture of two feature spaces, obscuring the unique physical attributes of each object. In particular, tokens at corresponding po- sitions may encode erucial physical properties, but their summation with unrelated tokens may distort these details, However, we argue that the addition operation is effec- tive in slice domain. As mentioned before, we separately embed each object in the system into their own slice do- mains with size M. In fact, this separate embedding is a special case of the holistic embedding. Technically, for deep features x" and x of any two objects, we embed them into slice nnd I MxC with slice weight way [wT, Wā.
Deformation Triplet In a multi-solid system, the main factors influencing the deformation of a deformable solid are loads and contact constraints (Shabana, 2020; Wriggers & Laursen, 2006; Bathe, 2006). Loads are the forces that act on the system, causing stress and strain on the deformable solids, thereby inducing deformation. Contact constraints refer to the geometric conditions that a solid experiences when in contact with other solids. These constraints alter the force conditions experienced by the solid, thereby affecting its deformation. Therefore, the key insight of explicitly mod- eling the two factors in deformation guides us propose triplet relationship (d,, f, E) to represent the deformation of a single deformable solid dl. Here, f E MxC is the equiv- alent load, which represents the comprehensive effect of all loads in the system on the solid d, and ē E RMCC X is the equivalent contact constraint, comprehensively modeling all contacts experienced by the solid d, in the system.
w N' o w Wi.j wh
w No
Ei=l
NN* ae
Concat(x" " .x))
a 2/
No+
W aa
Li=
(3)
No
Wi
Σ 1=
Following this formulation, embedding each object individ- ually can be formulated as:
Na
W. x. Na W + E/=1 Who
W. + .. im1
23
(w = =0)
No
Contact Module A pair of contacts can occur between two solids. In multi-solid system, multiple pairs of contacts may exist. It is hard to distinguish between these contact constraints if we merge all solids as one input (Haghighat et al., 2021; Li et al., 2020a). Therefore, we model each pair of possible contacts individually. As shown the right part of Figure 2, for any two solids g and gj that are likely to contact, where gi, gj E d Ur and 1 I j S N' + N",, we propose contact module to capture the contact constraint ca between them, formulated as follows:
E
(4)
Wf,x
EM ig
For clarity, we omit the mesh edges here, and the com- plete form can be found in Appendix B, where it still holds. Through (4), we can observe that the separate embedding is special case that the slice weights for other objects are O. This indicates that we build pure slice domain on the holistic input domain but only projected by a single object. Furthermore, the slice 2,.
as can be reformulated as follows:
Q, K, V = Linear(g; (g: +gz)
QKT
(2)
03 EN' w
No + 1i=1
C2 = Softmax
V
X
2,
Ei=l W + i=
ΣN W
No
We add these two tokens as the input and employ the atten- tion mechanism to capture the intricate contact constraint. Totally N contact modules are arranged to learn No pairs of contact constraints c = {c& E IR MxC)NN tem with different numbers of contact pairs, can easily extend contact modules in the width direction. It brings vast flexibility and applicability for various multi-solid systems. Remark 3.2. Why is addition operation effective in slice domain? Directly adding two nigh-dimensional features
EN wi + (E w
2
(5)
- wh, + Di=1 Wiy
Jk=1 For sys-
2s Oz, + (1 - 0)2²
Here, a0 can be seen as the composition of zd and z through coefficient f, referred to as slice composition. Ac- cordingly, the operation in (4. is termed slice decompo- sition. We first construct multiple pure slice domains during
5

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
embedding. Through slice composition, we merge two slice domains that are contact-related. In practice, we adopt direct element-wise addition in Eq.(2) as a simple, parameter-free realization of this linear combination. This design achieves comparable performance to the learnable form, while re- ducing complexity. Although this simplified form does not perform explicit averaging (e.g. (e.g. + 2f)),. the resulting features are subsequently processed by normalization and attention layers (e.g., in the contact module), which miti- gates effects from scale differences. We then apply attention mechanism to capture the physical interaction within the composed slice domain. This avoids information loss and minimizes interference from unrelated objects.
constraint, rather than implicitly learning the triplet relation- ship from data as in implicit modeling methods. This avoids the information confusion and ambiguity caused by the ex- cessive number of solids. Besides, using an independent deformation module for each deformable solid allows for precise, individualized modeling of each solid, enhancing scalability and flexibility.
Decoder Afterwards, the transited tokens of deformable solids à = fd, E IR M points. Because these tokens are normalized during embed- ding, the decoding of mesh points and edges is naturally decoupled. We directly decode tokens with formulation proposed in (Wu et al., 2024):
fim
Fim1
are projected back to mesh
Adaptive Interaction Allocation In a multi- system, the deformation of a deformable solid will be influenced by all loads and contact constraints in the system to vary- ing extents. For example, among all loads, those that act directly on the deforming solid should have a more signif- icant impact on its deformation than those far away from it. Therefore, we propose an adaptive interaction allocation mechanism comprehensively take account all loads and contact constraints. Technically, the equivalent contact con- straint ē E R MCC and the equivalent load f E IR XC are formulated as follows:
M
6
W. W.ydj
1 1 No
(8)
Each token d is projected back by weighted broadeast, where the weights are the same as those in the forward embedding in (I) The û {ā e x - the prediction of the original deformable solids u
[i= are
4. Experiments
Nc
cf
We evaluate Unisoma on extensive experiments, including two essential tasks and seven well-established datasets, cov- ering multi-solid systems with varying complexity.
c Linear(c;), č
Σ
No
i=
(6)
N
C = Lfnear(f.),
F
4.1. Experiment Settings
Lim f
Datasets Our experiments span different complexity from few solids to multiple solids. All of them are in 3D space. Deforming Plate (Pfaff et al., 2021), Cavity Grasping (Link- erhägner et al., 2023), Tissue Manipulation (Linkerhägner et al., 2023) and Rice Grip (Li et al., 2019) are public datasets, widely followed on autoregressive task. To ex- plore more complex scenarios, we construct three datasets: Bilateral Stamping, Unilateral Stamping and Cavity Ex- truding, which are inspired by metal stamping (Wang & Budiansky, 1978) in industrial manufacturing and robotic gripping (Zhang al., 2020). Except for the target geom- etry, we also predict attendant physical quantities in some experiments, including inner stress (Stress) and equivalent plastic strain (PEEQ). The summary of datasets is recorded in Table 1. See Appendix D for more details about datasets.
The weights are learnable and deprived from correspond- ing physical interactions. Through the mechanism, those interactions have significant influence on the correspond- ing deformable solid having higher weights. At the same time, this allocation process can also be seen as the slice composition with prior inductive biases. In particular, the allocation weights for each deformable solid are different and are learned individually, extending the flexibility.
We adopt distinct deformation mod- ules for each deformable solid. The input for i-th module is the deformation triplet (d,, f, Z). Similarly, we employ attention mechanism and slice composition to model the deformation interaction as follows:
Deformation Module
Q,K, V Linear(d; + +f+++ C)
Baselines We comprehensively compare Unisoma against more than ten baselines within the implicit modeling paradigm. These include typical neural operators: GNO (Li et al., 2020b), Geo-FNO (Li et al., 2023b), GINO (Li et al., 2024a), LSM (Wu et al., 2023); Transformer solvers: Galerkin Transformer (Cao, 2021), Factformer (Li et al., 2024b). OFormer (Li et al., 2023c), ONO (Xiao et al., 2024), Transolver (Wu et al., 2024); and GNN-based mod-
QKT
(7)
d = Softmax
Here, à = (d. E MMC J. are updated tokens of de- formable solids. Within the i-th deformation module, we explicitly establish the relationship between the i-th de- formable solid and its associated equivalent load and contact
6

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
Table Summary of experiment datasets, which include various complexity. #Content records the number of deformahle solids (DS), rigid solids (RS), loads (L) and contact pairs (C). #Mesh represents the mean number of discretized mesh points, including deformable solids and rigid solids.
DATASET
#DIM
#MATERIAL
#CONTENT 1 D 1 RS ILIC
#MESH
#TARGET
DEFORMING PLATE
3D
HYPERELASTICITY
1271
GEOMETRY STRESS
CAVITY GRASPING
3D
ELASTICITY
I DS 2 RS 2L2C
1386
GEOMETRY
TISSUE MANIPULATION
3D
ELASTICITY
I DS 2 RS 2 2
362
GEOMETRY
RICE GRIP
3D
ELASTO-PLASTICITY
DS 2 RS C
1106
GEOMETRY
BILATERAL STAMPING
3D
ELASTO-PLASTICITY
2 DS 2 RS 2 5
13714
GEOMETRY STRESS PEEQ
UNILATERAL STAMPING
3D
ELASTO-PLASTICITY
2 DS 17 RS 16 L 18C
49386
GEOMETRY STRESS PEEQ
ELASTO-PLASTICITY
CAVITY EXTRUDING
3D
3
4800
GEOMETRY STRESS PEEQ
ELASTICITY
els: GraphSAGE (Hamilton et al., 2017), GraphUNet (Gao & Ji, 2019), MGN (Pfaff et al., 2021), HOOD (Grigorev et al., 2023). HCMT (Yu al., 2024). These advanced deep models are widely used for continuum mechanics, and most have demonstrated success in few-solid scenarios.
commonly used in autoregressive settings (Pfaff et al., 2021; Linkerhägner et al., 2023). We directly predict the physical quantities for the step with the maximum applied load, using the first step. Unisoma achieves consistent state-of-the-an performance across these benchmarks. The deforming plate only involves two solids and one pair of contact. There is just one contact module and the adaptive interaction al- location module is omitted. Under these conditions, the advantage of explicit modeling is less pronounced, so the accuracy of advanced models remains similar. For tissue ma- nipulation, one jaw of the gripper holds the tissue in fixed position, while the other jaw moves. Because the physical interactions of these two states differ significantly, accurate modeling is more challenging. By explicitly modeling these contact constraints and loads, Unisoma achieves noticeable improvements in this benchmark.
Implementation For Unisoma, we set the number of pro- cessors, the hidden channels and the slice number to 2, 128 and 32, respectively, across all experiments. All experiments are conducted on a single RTX 3090 GPU and repeated three times. We utilize Relative L2 and Root Mean Square Error (RMSE) as evaluation metries for long-time prediction and autoregressive simulation, respectively. See Appendix E and C for detailed implementations and metrics.
Table 2. Performance comparison on long-tine time prediction task. Relative L2 is recorded. A smaller value indicates better perfor- mance. The hest result is in bold and the second best is underlined.
We further evaluate more complex scenarios involving mul- tiple solids. Metal stamping (Wang & Budiansky, 1978), one of the most crucial industrial processes, is examined in two settings (see Figure 4 and Appendix D): one featuring bilateral stamping by two rigid dies, and another involving multi-point unilateral stamping by sixteen rigid dies. The processed metal is hollow, and rubber matcrial is inserted to support its cross-section. Note that both scenarios are challenging, as they require the model to account for numer- ous solids and their interactions. As presented in Table 3, Unisoma also excels in complex scenarios when compared to implicit modeling. Beyond geometry, Unisoma delivers the best performance on stress and PEEQ, which are es- sential for subsequent design analysis. is worth noting that Transolver outperforms other baselines most targets, illustrating how slices effectively capture complex interac- tions. The performance margins with Unisoma highlight the strengths of explicit modeling in complicated setups. More- over, in the bilateral stamping, random cuts in the metal can lead to occasional contact between the stamping dies and the rubber. Even if contact does not occur, we still
TISSUE MANIPU-
DEFORMING PLATE
CAVITY
GRASPING
LATION
GEOMETRY STRESS GEOMETRY GEOMETRY
GEO-FNO
0.0931
0.5048
0.1361
0.0764
GINO
0.1071
0.6015
0.1523
0.0781
GNO
0.1223
0.6362
0.1273
0.0769
LSM
0.1319
0.7002
0.1073
0.0399
GALERKIN
0.1368
0.7046
0.15
0.0832
FACTFORMER
0.0945
0.5135
0.1085
0.0886
OFORMER
0.1091
0.6136
0.1112
0.0316
ONO
0.1269
0.6592
0.1191
0.0513
TRANSOLVER
0.0933
0.4968
0.1077
0.0363
GRAPHSAGE
0.1194
0.6160
0.1 194
0.0794
MGN
0.1183
0.6232
0.
0.0825
UNISOMA
0.0892
0.4713
0.0984
0.0253
4.2. Experiment Results
Long-Time Prediction As presented in Table 2, we first
conduct long-term prediction task on three public datasets,

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
Table Performance comparison on long-time prediction task. Relative 1.2 is recorded. smaller value indicates better performance.
The best result is in bold and the second best is underlined. Under the same target, the left column represents the results for the metal, while the right column represents the results for the rubber.
BILATERAL STAMPING
UNILATERAL STAMPING
GEOMETRY
STRESS
PEEQ
GEOMETRY
STRESS
PEEQ
GEO-FNO
0.0125
0.0117
0.0390
0.1612
0.4269
0.4731
0.0109
0.0154
0.1113
0.1124
0.3771
0.3574
GINO
0.0299
0.0243
0.0543
0.2621
0.5523
0.6409
0.0188
0.0219
0.1262
0.1438
0.4651
0.4885
GNO
0.0285
0.0281
0.0463
0.2524
0.5830
0.5909
0.0426
0.0424
0.1351
0.1814
0.6413
0.6859
LSM
0.0598
0.0615
0.0556
0.3616
0.7001
0.7347
0.0367
0.0357
0.1327
0.1335
0.5889
0.6026
GALERKIN
0.0599
0.0594
0.0563
0.3442
0.6962
0.7335
0.1007
0.1010
0.1725
0.2794
0.7986
0,8555
FACTFORMER
0.0297
0.0105
0.0459
0.1378
0.6266
0.4218
0.0178
0.0166
0.1095
0.1035
0.3886
0.3022
OFORMER
0.0334
0.0268
0.0495
0.3178
0.5551
0,5157
0.0119
0.0098
0.1103
0.0981
0.3269
0.2675
ONO
0.0301
0.0275
0.0464
0.3291
0.5176
0.5231
0.0433
0.0437
0.1377
0.1311
0.5140
0.4748
TRANSOLVER
0.0083
0.0081
0.0346
0.1425
0.3592
0.3276
0.0079
0.0075
0.0839
0.0862
0.2938
0.2693
GRAPHSAGE
0.0227
0.0229
0.0440
0.2196
0.4936
0.4495
0.0275
0.0260
0.1059
0.
0.5441
0,5106
MGN
0.0195
0.0197
0.0421
0.2059
0.4646
0.4141
0.0391
0.0383
0.1252
0. 0.1417
0.5805
0.5694
UNISOMA
0.0057
0.0052
0.0278
0.1039
0.2817
0.2265
0.0071
0.0063
0.0807
0.0773
0.2591
0.2114
Table Performance comparison on autoregressive simulation task. We record RMSE-all, the average RMSE of the whole rollout traeectory and all samples. A smaller value indicates better performance.
CAVITY GRASPING (x10-3)
TISSUE MANIPULATION
RICE GRIP (x10-3)
CAVITY EXTRUDING (x10-3)
(x10-3)
GRAPHS
13.41 + 0.36
13.19 0.31
26.68 + 1.86
65.81 + 1.58
128.48 + 6.05
104.44
4.79
GRAPHUNET
14.27 0.11
21.31 1.17
22.48 + 1.23
58.83
+ 0.88
96.26
1.44
82.55
+ 1.69
HOOD
12.43 + 0.34
9.87 0.17
19.25 + 0.43
53.98
+ 1.84
89.93
2.69
81.02
+ 2.68
HCMT
19.32 + 1.82
17.59 + 1.81
18.36 + 0.47
64.91
#
110.18
5.98
97.09
1 4.42
MGN
12.89 0.46
9.56 + 0.29
19.61 1 0.78
57.37
1.62
95.87
1.17
86.28
1 4.93
UNISOMA
9.50 + 0.54
7.51 + 0.28
17.68 + 0.36
11.98 + 0.27
19.37 + 0.72
19.46 + 1.82
employ contact modules for potential interactions, and the adaptive interaction allocation mechanism controls the rele- vant constraints. This demonstrates both the scalability and flexibility of explicit modeling.
sage passing is constrained within a limited radius, which weakens the long-distant propagation of the influence of contacts. This limitation becomes more pronounced as the number of contacts increases. In contrast, Unisoma lever- ages explicit modeling to construct relationships between contacting solids and captures their influence on the defor- mation. The attention mechanism and slice composition enable global modeling, while the incorporation of mesh edges in the embedding preserves local features. As re- sult, the performance improvement in multi-solid scenarios is more significant. This modeling approach, which bal- ances global and local considerations, also highlights the flexibility of explicit modeling.
Autoregressive Simulation We further evaluate the perfor- mance of Unisoma on the autoregressive simulation task. In this task, the process starts from the first step, and each subsequent step is predicted based on the previously pre- dicted step. This is more challenging because it requires accurate modeling to prevent physical shifts and error ac- cumulation during the simulation. We compare Unisoma with advanced GNN-based simulators. As shown in Table 4, Unisoma achieves the best results across all datasets. In scenarios involving only one deformable solid, the physical interactions between the deformable solid and rigid solids are relatively straightforward to handle, resulting in simi- lar performance between Unisoma and other GNN-based simulators. However, in scenarios with multiple solids, the advantage of explicit modeling becomes evident. For the cavity extrusion, the materials of the three deformable solids vary, and the number of contacts and loads is higher. Relying solely on message passing between nodes is insufficient to implicitly capture all physical interactions. Moreover, mes-
-uttrt-dis (OOD) generalization To further ex- amine the generalizability of Unisoma, we experiment with OOD long-time prediction task on the cavity extruding dataset. In this dataset, the Young's modulus and Pois- son's ratio of the inner elastic material in the cavity are uniformly sampled from the ranges [30000, 70000] MPa and [0. 1, 0.45). The data is divided into two parts: the first part serves the training and validation set, with parameter ranges of [30000, 65000] MPa and [0.1, 0.42]. The samples out of these scopes are used as the test set. The numbers
8

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Unisoma; A Unified Transformer- Solver for Multi-Solid Systems
Ground truth
Geometry (Unisoma)
Geametry (aaettyyyTTrnllvverr
PEEQ(Unisona)
PEEQ(TransoLvver)
0.4
02
0.0
Ground truth
Geometry (Unisoma)
Geometry(Transolver)
Stress (Unisoma)))
Stress (TransSSra)))
Ground truth step 60, Geometry (Unisona) step 90, 180, 110, 120 t
Geometry (MGN) step 90, 100, 110, 120
Figure 4. Visualization of error maps. The first, second, and third rows respectively show sample visualizations from the hilateral stamping, normalized among same physical quantities the same samples.
unilateral stamping, and cavity extruding datasets. The error
Table generalization experiment on cavity extruding dataset. Relative L, is recorded. "Outer", "Middle", and "Inner" refer to the layers of cavity from outside to the inside. See Appendix G for full results, where Unisoma still achieves best.
Table Efficiency comparison. running time is measured by the time to complete epoch, which contains 10 iterations. The for Unisoma is set as 4 efficiency lest. Other edge-related methods set k 3 or 4 based the memory usage. We record the max GPU memory in one epoch due to the dynamic mesh number.
OUTER
MIDDLE
INNER
GEOMETRY GEOMETRY GEOMETRY STRESS
BILATERAL
UNILATERAL STAMPING
STAMPING
GEO-FNO
0.0117
0.0177
0.0198
0.2526
GNO
0.0238
0.0444
0.0449
0.3727
PARAM
TIME (S)
MEM
PARAM
TIME (S)
LSM
0.0249
0.0436
0.0339
0.3111
(M)
(GIB)
(M)
(GIB)
FACTFORMER
0.0202
0.0269
0.0308
0.2739
GINO
2.70
75.64
17.63
5.61
181.26
22.82
GNO
0.56
98.64
21.79
0.43
261.73
21.31
OFORMER
0,0134
0.0204
0.0229
0.2596
ONO
0.0213
0.0357
0.0372
0.2511
OFORMER
2.63
127.59
13.94
2.06
479.85
23.01
ONO
3.27
66.55
5.46
5.13
311.60
21.85
TRANSOLVER
0,0162
0.0334
0.0336
0.2643
TRANSOLVER
3.81
83.97
6.35
5.07
402.35
16.33
GRAPHSAGE
0.0211
0.0339
0.0416
0.3571
MGN
0.0178
0.0280
0.0385
0.3526
UNISOMA
2.85
70.96
0.93
5.21
152.55
1.10
UNISOMA
0.0077
0.0157
0.0179
0.2348
cations. This approach reduces the memory required for intermediate states during computation, allowing for more flexible memory allocation and release. As result, the model can handle larger problem scales, making it far more applicable and meaningful for real-world applications.
of samples in the training, validation and test set are 900, 83 and 217. As presented in Table 5, Unisoma handle OOD samples well, where consistently performs best unseen materials. These results indicate that Unisoma also captures some generalizable physical interactions. further highlighting the advantage of explicit modeling.
5. Conclusions
Efficiency provide the model efficiency comparison Table We can observe that the GPU memory usage of Unisoma significantly lower than other models, in- cluding Transolver, which also uses slice. Even when the number of mesh points around 50,000 (unilateral stamping), the memory consumption still remains around GiB. Uni- soma employs explicit modeling, processing each object separately. It breaks down the large matrix multiplications in implicit modeling into smaller, multiple matrix multipli-
This paper introduces new paradigm of explicit model- ing to solve complex multi-solid systems and, based it, designing unified and flexible Transformer architecture named Unisoma. Compared with implicit modeling mod- els, Unisoma explicitly represents the key factors affecting solid deformation and employs deformation triplct to more accurately capture the diverse interactions among multiple solids. Extensive experiments are provided verify the performance, OOD generalizability and efficiency.

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Unisoma: A Unified Transformer-baser Solver for Multi-Solid Systems
Acknowledgements
Cao, Y., Chai, M., Li, M., and Jiang, C. Efficient learning of mesh-based physical simulation with bi-stride multi-scale graph neural network. In International Conference on Machine Learning, pp. 3541-3558, PMLR, 2023.
This work is supported partly by National Key Reasearch Plan under grant No. No.2024YFC2607422 ral Science Foundation of China (NSFC) 61925202, and the Jiangsu Provincial Key Research and Development Program under Grant BE2022065-1 1, BE2022065-3.
the National Natu-
Chiu, P.-.., Wong, J. C.,, Ooi, C., Dao, M. H., and Ong, Y.-S. Can-pinn: A fast physics neural network based on coupled-automatii-nuuan method. Computer Methods in Applied Mechanics and Engineering. 395:1 14909, 2022.
differentiation
Impact Statement
This paper presents work whose goal to advance the field of Machine Learning for Multi-solid Systems. Note that this work mainly focuses the scientific computation. We are entirely committed to ensuring ethical considerations are taken into account when doing our research. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Clyne, T. W. and Hull, D. An Introduction to Composite Materials, An Introduction to Composite Materials, 2019,
Deng, J., Li, X., Xiong, H., Hu, X., and Ma, J. Geometry- guided conditional adaptation for surrogate models of large-scale 3d pdes on arbitrary geometries. In Larson, K. (ed.), Proceedings of the Thirty-Third International Joint Conference Artificial Intelligence, IJCAI-24, pp. 5790-5798. International Joint Conferences Artificial Intelligence Organization, 8 2024.
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f' fr Unisoma: A Unified Transformer-basel Solver for Multi-Solid Systems
A. Overall Structure
The overall structure of Unisoma is as follows:
Encoder Given system with multiple objects composed of N deformable solids u {uf M J with different
material properties, N" rigid solids u = {uf E I NT x ] and N loads u' = {u/ E R x
x 1
we first embed
each of them into edge augmented physics-aware tokens by Eq.()::
x Linear(u d, Encoder(x;)
isis N'
X Linear(u;) r; Encoder (x{),
15is N" {f E RM
= Linear(u:) (u,),
f, Encoder(x:)) );
1
Here, {d; E R.MCC tokens of corresponding objects u", u", and w/, respectively.
RiE r fr. E RMXCJ Ji and f -
Ji=1 are edge augmented physics-aware
Processor We first characterize the contact constraints in the system. For two solids that may contact with each other with high probabilities, we utilize contact module to capture the contact constraint. For any two solids gi and gj that are likely to contact, where g., gy d UUr and 1 < j & Na + N the contact constraint between them is formalized as:
Q K, V Linear( g: + +gj)
C₁ Softmax QK"
V
VC
Here, totally No contact modules are arranged to learn N pairs of contact constraints {cs E RRAXCIN
fk=1
Then we deploy adaptive interaction allocation mechanism to integrate loads and contact constraints, as follows:
N
c Linear(c;),
Ci F= FFN(() + è
No
df
f = Linear(f.),
The с and are the equivalent contact constraint and equivalent load. There is little difference their operation. As described above, the contact module attention-basse based. When the number contacts increases, the parameters grow and make the model unstable, Therefore, we move the FFN(-) to the back of contact constraints allocation, instead of arranging distinct FFN(-) for each contact module, to reduce parameters. With regard loads, they are from encoders. The FFN(-) is unnecessary for them.
Subsequently, we adopt deformation modules to capture the deformation of deformable solids. Notably, each deformation module processes one deformable solid, and the equivalent load and contact constraint for each deformable solid are different. For i-th deformable solid, its deformation is modeled as:
Q K. V Linear(d, (d: + +
QKT
d' = Softmax
V
VC
d= d' + FFN(
Here, d= (å, E M are updated tokens of deformable solids.
Decoder We decode updated tokens back to mesh points with formulation proposed (Wu et al., 2024):
a FFN(Decoder((..) +x+) FFN Σ +, ISiSN
Each token d is projected back to mesh points by weighted broadcast, where the weights are the same those in the forward embedding in (1). The üd = (ūf e xCa)N' are the predicted states corresponding to the original input deformable solids nd The residual connection with x is introduced to prevent the vanishing gradient problem.
13

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
B. Slice Composition and Decomposition
For deep features x E R
and
x
RN RN
XC of any two objects, embed them into slice z E IR MMC
follows:
N=+
ad Contact(x* x²); + EEeg EE wens 08 iJ
e0g
03 ]
1i=1
(p.g).s
(9)
No+NN
+ + EZze ERMEER wes (D-9)
Here, Gpgg E and p.g E E corresponding edge sets, Since apply k-NN separately to different solids, using the same value of k for each, it follows that Ea n E = Ø and 7 = k. Therefore, similar to wa the weight we can also be separated according to and B, allowing the (9) to be rewritten as:
Na + +yEgg EB (ee)s S. + E i= W Wi + +YEeg BEE (p.2), P.g
EB we
DE w
08 {j
ii=
(10)
Σ wio + +7 Ees EB
w (pw)s i== Wij + Eej well (p.a)s
As mentioned in Section 3.2, the edge weights w is deprived from point weights w When w = 0, we have we = 0. Following this formulation, embedding each object individually can be formulated as:
No Σ
w ; +, EE w 1=
aa (palls PM + +EM w Wiy + +7E.0 ERD we e (w = w = 0)
(pre), p.g
zj
EE Wis + EB Wet
(p.a).s + EE Wis +7E EE woo
(pa))s
(11)
No
w va + Σ pe. EBa we Mi
(pi4).s P..
En W Eog
EF (p.4).s
Therefore, we can also conduct slice decomposition with mesh edges, which builds pure slice domain on the holistic input domain but only projected by a single object.
Furthermore, the slice z can be reformulated as follows:
we + E Eeg EB (ni)s Gg
No No 1i=1
w
x + Eoi we
(mo))
o
EN wh, +7 E EB (poo)s
Σ ii
No
+ +.Engnen wes
(pa)l
Ei= why +7 Ees -e, EB (mm))) )2
No
we
N2
w/y
+ Eegg we (mn)s)as
(12)
ΣM w + Ees EE wee
EN whs + "e we
(pe))s
(p.))
2 Oz + 01=0)z
Here, with the mesh edges, the za can also be seen as the composition of z and za through coefficient 0, referred to as slice composition. The unified forms of Eq.(1 (12), Eq. and Eq.(5) benefit from the unified aggregation mesh edges in (1).
C. Metrics
We employ different metrics for specific tasks, adhering to the evaluation approaches used related works.
Long-time Prediction: Relative I.2 In line with prior studies on long-time prediction tasks (Raissi et al., 2019; et al., 2020a; Wu al., 2024), we the relative L2 assess performance. Given the input physical quantities and the predictions û, the relative 1.2 is computed as:
|u
ûl
Relative L2
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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
Autoregressive Simulation: RMSE Consistent with works focused autoregressive simulation tasks (Pfaff al., 2021; Yu et al., 2024; Li et al., 2019), Root Mean Square Error (RMSE) the evaluation metric. Given the input physical quantities and the predictions û, RMSE is calculated as:
RMSE
N
D. Datasets
extensively evaluate our model on two key tasks across seven datasets. These datasets encompass varying levels of complexity in the number of solids, diversity in solid materials, and variety applied loads. The summary of datasets is recorded in Table 1.
D.1. Public Datasets
Deforming Plate (Pfaff al., 2021) This dataset consists of 3D dynamic simulation deformable plate being pressed by rigid solid (Figure 5), with total of 1,200 samples. The deformable plate is made of hyperelastic material, and the target physical quantities include the geometry and inner stress of the plate. Typically used for autoregressive tasks, this dataset spans 400 time steps. In our study, we evaluate the long terg prediction performance of Unisoma. The first time step is used as input, and we predict the results for the time step corresponding to the largest rigid solid movement (approximately step 340). Each sample contains an average of 1,271 points. Following the strategy outlined in the original paper, we use 1,000 samples for training, 100 for validation, and 100 for testing.
(a) Before deformation
(b) After deformation
Figure 5. The deforming plate scenario.
Cavity Grasping (Linkerhägner et al., 2023) This dataset contains a 3D dynamic simulation of deformable cavity grasped by a rigid gripper (Figure 6), with total of 840 samples. The rigid gripper is treated two rigid solids, its two jaws move in different directions. The deformable cone-shaped cavities randomly generated with radii ranging from 87.5 to 50. The materials of the cavities are elastic, with Poisson's 's ratios cyclically assigned from {-0.9, 0.0, 0.49). Typically used for autoregressive tasks, this dataset spans 105 time steps. In our work, we evaluate both long-term prediction and autoregressive simulation tasks. For the long-term prediction task, we use the first time step as input and predict the outcome of the time step corresponding to the largest rigid solid movement (the last step). Each sample contains 1,386 points. Following the strategy the original paper, 600 samples are used for training, 120 for validation, and 120 for testing.
Figure 6. The cavity grasping scenario.
Tissue Manipulation (Linkerhägner et al., 2023) This dataset simulates the 3D dynamics tissue deformation caused by a rigid gripper, a scenario often encountered in treated two rigid solids, as its two jaws exhibit distinct movement patterns. The tissue material elastic, with Young's
surgery. It contains 840 samples in total. The rigid gripper is
15

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
modulus sampled from {10000, 80000, 30000). This dataset spans 105 time steps and is commonly used for autoregressive tasks. In work, we evaluate both long-term prediction and autoregressive simulation tasks. For long-term prediction, the first time step is used input, and the outcome of the step with the largest rigid solid movement (the last step) is predicted. Each sample contains 363 points. Following the original paper, we allocate 600 samples for training, 120 for validation, and 1 120 for testing.
Gripping Points
Figure 7. The tissue manipulation scenario.
Rice Grip (Li al., 2019) This dataset involves the 3D dynamic simulation of sticky rice being gripped by two rigid grippers (Figure 8). The rice is modeled an elasto-plastic material. Two parallel grippers, represented cuboids, are initialized at random positions and orientations. We discretize each of them into 180 mesh points. During each trajectory, the grippers move closer together before returning to their original positions. The task involves learning the physical interactions between the grippers and the rice, as well as the deformation within the rice. The dataset spans 41 time steps and is typically used for autoregressive tasks. our experiments, we evaluate the autoregressive simulation task on it. Each sample contains an average of 1,271 points, As for the experiment, we use 1200 samples in total and the numbers of samples used for training, validation and test 1000, 100 and 100, respectively.
>
Figure 8. The rice grip scenario.
D.2. Created Datasets
All public datasets mentioned before involve few solids and simple dynamics. To diversify the datasets and improve the complexity, we curate three multi-solid datasets with industrial inspiration for evaluation. These datasets are calculated by ABAQUS software (Abaqus, 2011).
150
35-
100
75
20-
5-
(a)
(b)
Figure 9. (a) the engineering stress-strain curve of the aluminum alloy; (b) The engineering stress-strain curve of the polyethylene rubher.
Bilateral Stamping Inspired by the metal stamping technique, we create a dataset related to bilateral stamping. As illustrated in Figure 10, this scenario involves stamping a hollow metal workpiece using two rigid stamping dies to form the desired target shapes. To maintain the integrity of the metal's cross-sectional shape during the process, rubber insert
16

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Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
squeezed into the hollow space while stamping. Both the metal and the rubber modeled elasto-plastic materials, with their material parameters derived from real aluminum alloy and polyethylene rubber. The Poisson's ratios for these materials are 0.37 and 0.27, while their Young's moduli are 69,000 MPa and 1,306.26 MPa, respectively. Their engineering stress-strain curves are depicted in Figure 9. In this dataset, two filleted cylinders serve as the stamping dies, and their positions and movement distances are randomly generated. The shape parameters of the solids are shown in Figure 11. All sample parameters are uniformly sampled from predefined intervals to ensure dataset diversity. This dataset is designed the long-time prediction task, collecting the system's s initial and final states. The target physical quantities include the geometry, inner stress, and plastic equivalent strain (PEEQ) of both the metal and the rubber. On average, each sample consists of 13,714 points. We generate total of 1, 1,200 samples, of which 1,000 used for training, 100 for validation, and 100 for testing.
(a) Before deformation
(b) After deformation
Figure 10. The bilateral stamping scenario.
[200,
[2. H
[30, 80]
[0 [30, 201
[20,
[L.5,
[35, 80]
I
[120,
150]
[8.
12]
(a) deformable metal
(b) rigid d die
Figure 11. The shape parameters (unit: mm) of the deformable metal and rigid dies in bilateral stamping dataset. The rubber shape is deprived from metal.
Unilateral Stamping Building on the concept of metal stamping. develop dataset focused multi-point unilateral stamping. As shown Figure 12, this scenario simulates the deformation of hollow metal workpiece using multiple rigid stamping dies-16 16 dynamic dies and static die in this dataset-to achieve the target shapes. To preserve the cross-sectional integrity of the hollow metal during stamping, rubber insert is pressed into the cavity. The material properties of the metal and rubber are consistent with those used in the bilateral stamping dataset. The shape parameters of the solids are detailed in Figure 13, with all sample parameters uniformly sampled within specified ranges to ensure diversity. This dataset is tailored for long-time prediction tasks, capturing both the initial and final states of the system. The target physical quantities include the geometry, internal stress, and plastic equivalent strain (PEEQ) of the metal and rubber. Each sample comprises an average of 49,386 points. A total of 1, 1,200 samples are generated, distributed as 1,000 for training, 100 for validation, and 100 for testing.
Cavity Extruding Inspired by the robotic gripping (Zhang et al., 2020), we enhance the complexity of the cavity grasping dataset (Linkerhägner et al., 2023). As illustrated in Figure 14, this scenario simulates the deformation of multi-layer cavity using four rigid grippers. The grippers' positions and movement distances are randomly generated. The cavity is composed of three deformable layers: the outer and middle layers are elasto-plastic materials, while the inner layer an elastic material.
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Unisoma: A Unified Transformer-basee Solver for Multi-Solid Systems
(a) Before deformation
(b) After deformation
Figure 12. The unilateral stamping scenario.
Da. 103
[5. 32
[10,
1%
(a) deformable metal and static die
(b) dynamic die
Figure 13.
The shape parameters (unit: mm) of the deformable metal and rigid dies in unilateral stamping datasct. The rubber shape deprived from the metal.
The stress-strain curves for the elasto-plastic layers follow those shown in Figure 9, with the Young's 's modulus uniformly sampled from [65000, 75000) MPa and [35000, 45000) MPa, and the Poisson's 's ratio uniformly sampled from [0.3, 0.45) and [0.3, 0.45]. For the elastic innermost layer, the Young's 's modulus is uniformly sampled from [30000, 70000], and the Poisson's 's ratio is uniformly sampled from [0.1, 0.45]. This significantly increases the complexity of the data distribution. The shape parameters the solids are provided in Figure 15, and all sample parameters are uniformly sampled within specified intervals to ensure dataset diversity. This dataset is designed for autoregressive simulation tasks and includes 121 steps from the initial state to the final state. For each step, the target physical quantities include the geometry, inner stress, and plastic equivalent strain (PEEQ) of the two plasto-plastic layers, as well as the geometry and inner stress the clastic layer. Each sample contains 4,800 points, which is significantly larger than existing autoregressive simulation datasets. In total, we generated 1,200 samples, of which 1,000 are used for training, 100 for validation, and 100 for testing.
t
Figure 14. The cavity extruding scenario.
E. Implementation Details
Implementations As shown in Table 7, Unisoma and all baseline models are trained and tested using the same training strategy. We utilize relative 1.2 as loss function for long-time prediction task and MSE (Mean Square Error) for autoregressive
18

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Unisoma: A Unified Transformer-base Solver for Multi-Solid Systems
[100, 170]
15,
[10, 12]
[10,
[75. 851
10
20
[100, 170]
(a) deformable Tayers
(b) rigid die
Figure 15. The shape parameters (unit: rum) of the deformable layers and rigid dies in cavity extruding dataset.
Table Training and Model Configurations of Unisoma. The definition of batch size differs between the two tasks. For long-time prediction, the batch size refers the number of samples in a batch. For autoregressive simulation, only one sample is processed during each forward and backward pass, and the batch size corresponds to number of time steps the sample. Since GPU memory usage varies across different models, the batch sizes of baseline models in the autoregressive simulation task are dynamically adjusted to avoid GPU memory overflow while maintaining performance.
TRAINING CONFIGURATION
MODEL CONRIURRATINN
TASKS
DATASETS
LOss
EPOCHS
LR
BATCH
PROCESSORS CHANNELS SLICES
DEFORMING PLATE
CAVITY GRASPING
RELATIVE L.2
200 3 10
20
LONG-TIME
TISSUE MANIPULATION
20
PREDICTION
BILATERAL STAMPING
RELATIVE L2 100
3 10
2
128
32
UNILATERAL STAMPING
CAVITY GRASPING
104
AUTOREGRESSIVE SIMULATION
TISSUE MANIPULATION
104
MSE
100
3 x 10
RICE GRIP
CAVITY EXTRUDING
120
simulation task. For multiple physical quantities, we add each loss with equal weights. For Unisoma, we set the number of processors, hidden feature channels, and slices to 2, 128, and 32, respectively, across all experiments. To ensure fair comparisons, we first approximate the parameter count of all baseline models to match that of Unisoma and then adjust their parameters to minimize overfitting and achieve better performance. All experiments are conducted on a single RTX 3090 GPU (24GB memory) and repeated three times. Additionally, when adjusting the parameters of the baselines, we take maximum GPU memory usage into account incur "Out of memory" error. This is important for practical applications. Due to the number of modules adaptable the number of solids and their contacts, for all experiments, Unisoma performs well under single set of network parameters, which demonstrates its robustness and generalizability across various tasks and datasets. This consistency eliminates the need for extensive parameter tuning when switching between tasks, making it more efficient and user-friendly for practical applications. Additionally, the ability of Unisoma to handle diverse scenarios with unified configuration highlights its capacity effectively model complex physical systems, where different dynamics and interactions are involved. This advantage becomes particularly significant when compared to bascline models, which often require task-specific parameter adjustments to achieve optimal performance.
constraint, which means the GPU memory usage of all models cannot
baselines employ implicit modeling and are primarily designed for Eulerian settings, they are not naturally suited for handling Lagrangian scenarios, such as multi-solid systems. Consequently, we preprocess the data to adapt it for use with these baselines. For different objects within the system, we first align the feature dimensions using padding, then concatenate along the length dimension to combine all objects into a single large Lagrangian sequence of points. For models that struggle handle the Lagrangian setting (Li al., 2023b; Wu al.,
Since these neural operators and transformer-
19

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Unisoma: A Unified Transformer-baset Solver for Multi-Solid Systems
2023; Li al., 2024b; 2023c), we map each point regular grid, transforming the data into Eulerian representation. Specifically, for a point sequence of size N x C, we discretize cubic space (ar, y: E [-1,1] into regular grid, with the number of discretization points along each axis set to VN. We then map each point's coordinates and physical quantities to the corresponding grid points. Excess grid cells are padded to align dimensions. Experiments show that this approach improves baseline performance to some extent but results a partial loss of information from the Lagrangian perspective, revealing limitations in addressing multi-solid problems effectively.
We implement baselines based on official and popular implementations. For autoregressive simulation task, we uniformly add noise with a mean of and a variance of 0.001 improve the error accumulation control during rollout. Because the results of MGN (Pfaff al., 2021) on tissue manipulation and cavity grasping datasets are well-explored in GGNS (Linkerhägner et al., 2023), we directly use the model and weights in their code repository. Additionally, the data preprocessing code for pooling is not provided in HCMT (Yu et al., 2024). Therefore, we attempted to implement our own version, which, however, encountered convergence issues in some scenarios. We provide key network hyperparameters and parameter numbers of baselines in Table 10. We refer the readers to original papers and code repositories for details.
Efficiency In this paper, we report GPU memory usage based on the operating system's measurements, which include any memory pre-allocated by the PyTorch caching allocator. This approach yields conservative yet realistic view of hardware requirements during training, since once memory is reserved, it is effectively unavailable for other processes. Even though the model may not be actively using all of the allocated space at each moment, including pre-allocation in the reported usage aligns more closely with the practical resource constraints encountered in real-world deployments. Furthermore, all models are evaluated under the default PyTorch memory usage strategy on single RTX 3090 GPU, ensuring a fair comparison across different architectures and methods.
F. Ablation Study
We include ablation studies in Table 8. In general, we observe that incorporating mesh edges benefits the final performance by enabling the model to capture more local features. However, as the number of mesh edges increases (especially when k: = 8), the performance declines. This is because an excessive number of edges introduces redundant information and additional noise, which negatively impacts the model's effectiveness. Furthermore, increasing the number edges significantly raises the computational cost, making the model less efficient. In principle, the optimal number of edges depends on the problem's scale and the number of mesh points. our experiments, & is easy-to-tune in the range of 5.
Besides, removing the loads (which are usually implicitly considered in existing models) will damage the model performance seriously, especially when the loads are more complex. This result further demonstrates the advantages of the explicit modeling in handeling complex multi-solid systems.
Table Ablation results. We experiment on two variants: the mesh edges and loads. We focus on long-tim time prediction task. The & denotes the number of neighbors used in constructing mesh edges with kNN method. For bilateral stamping with deformable solids, under the same target, the left column represents the results for the metal, while the right column represents the results for the rubber.
TISSUE MANIPULATION
BILATERAL STAMPING
ABLATIONS
GEOMETRY
GEOMETRY
STRESS
PEEQ
W/O EDGES
0.0269
0.0063
0.0059
0.0292
0.1097
0.2962
0.2429
A
0.0263
0.0058
0.0053
0.0289
0.1063
0.2843
0.2338
k 3
0.0265
0.0056
0.0051
0.0283
0.1058
0.2870
0.2281
A 4
0.0253
0.0057
0.0052
0.0278
0.1039
0.2817
0.2265
0.0257
0.0057
0.0054
0.0282
0.1049
0.2836
0.2269
A: =
0.0260
0.0056
0.0051
0.0279
0.1042
0.2842
0.2258
k: =
0.0254
0.0059
0.0051
0.0284
0.1046
0.2818
0.2294
k 8
0.0266
0.0062
0.0055
0.0293
0072
0.2905
0.2312
w/O LOADS
0.0346
0.0245
0.2353
0.0378
0.1949
0.5441
0.5720
20

---

Unisoma: A Unified Transformer-based Solver for Multi-Solid Systems
G. Out-of-distribution Generalization
Table 5, we have provided part results of the out-of-distribution experiment on unseen material parameters. We include the complete results Table impressive that Unisoma can still achieve the best performance the out-of-distrbbttion setting across all target physical quantities. This comes from special explicit modeling design, which enables Unisoma capture more foundational physical interactions and generalize unseen material parameters better.
Table 9. Complete results of the OOD generalization experiment the cavity extruding dataset. Relative 1.2 is recorded. "Outer", "Middle", and "Inner" to deformable solid the multi-layered cavity the outside the inside, respectively.
OUTER
MIDDLE
INNER
GEOMETRY
STRESS
PEEQ
GEOMETRY
STRESS
PEEQ
GEOMETRY
STRESS
GEO-FNO
0.0117
0.1036
0.1771
0.0177
0.0790
0.1171
0.0198
0.2526
GINO
0.0715
0.2003
0.7461
0.1165
0.1900
0.7268
0.1015
0.4645
GNO
0.0238
0.1452
0.2811
0.0444
0.1149
0.2914
0.0449
0.3727
LSM
0.0249
0.1194
0.3164
0.0436
0.0840
0.2633
0.0339
0.3111
GALERKIN
0,0931
0.1996
0.7899
0.1333
0.7641
0,1398
0.1398
0,5165
FACTFORMER
0.0202
0.1116
0.2289
0.0269
0.0738
0.1355
0.0308
0.2739
OFORMER
0.0134
0.1064
0.1930
0.0204
0.0722
0.1276
0.0229
0.2596
ONO
0.0213
0.1089
0.2088
0.0357
0.0792
0.1642
0.0372
0.2511
TRANSOLVER
0.0162
0.1066
0.2069
0.0334
0.0765
0.1521
0,0336
0,2643
GRAPHSAGE
0.0211
0.1434
0.2594
0.0339
0.1126
0.1971
0.0416
0.3571
MGN
0.0178
0.1413
0.2359
0.0280
0.1113
0.1788
0.0385
0.3526
UNISOMA
0.0077
0.0994
0.1527
0.0157
0.0706
0.1125
0.0179
0.2348
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Unisoma: A Unified Transformer-bascr Solver for Multi-Solid Systems
Table 10. Key hyperparameters and parameter numbers of models.
DEFORMING
CAVITY GRASPING MANIPULATION
TISSUE
BILATERAL UNILATERAL STAMPING
PLATE
STAMPING
MODES
[8, 8. 8]
[8, 8.88) 8]
[8. 8, 8] 28
[8, 8, 8]
[10. 10. 10J
GEOFNO
HIDDENS
24
28
36
36
PARAMETER (M)
1.30
1.77
1.77
2.92
5.47
MODES
[8, 8. 8]
[8, 8, 8]
[8. 8, 8]
[8, 8. 8]
[8. 8, 8]
HIDDENS
8
8
8
12
18
GINO
LATENT GEOMETRY
8
20
20
8
8
PARAMETER (M)
1.41
1.41
1.41
2.70
5.61
HIDDENS
96
96
96
64
64
KERNEL WIDTH
128
128
128
128
128
GNO
DEPTH
3
3
3
3
PARAMETER (M)
1.14
1.23
1.23
0.56
0.56
HIDDENS
8
8
8
16
16
TOKENS
8
8
8
8
8
LSM
BASIS
12
12
12
12
12
PARAMETER (M)
1.49
1.49
1.49
5.94
5.94
HIDDENS
128
128
128
192
256
FEEDFORWARD
512
512
512
512
512
GALERKIN
LAYERS
8
8
8
10
PARAMETER (M)
1.20
1.60
1.60
2.80
5.35
HIDDENS
128
128
128
256
256
HEAD DIM
64
64
64
64
128
FACTFORMER
KERNEL MULTIPLIER
3
4
4
PARAMETER (M)
1.19
1.58
1.58
3.16
6.32
HIDDENS
128
128
128
128
128
OFORMER
DEPTH
2
2
2
4
3
PARAMETER (M)
1.48
1.48
1.48
2.63
2.06
HIDDENS
128
128
128
256
256
LAYERS
4
5
5
4
4
ONO
MLP RATIO
4
2
4
PARAMETER (M)
1.31
1.65
1.65
3.27
5.13
HIDDENS
128
128
128
128
128
LAYERS
12
12
12
12
16
TRANSOLVER
MLP RATIO
1
4
4
SLICE
32
32
32
32
PARAMETER (M)
1.44
1.44
1.44
3.81
5.07
HIDDENS
128
128
128
128
128
LAYERS
2
2
2
3
4
MGN
PASSING STEPS
15
15
20
PARAMETER (M)
2.32
2.32
2.32
2.88
4.51
HIDDENS
256
256
256
256
512
GRAPHSAGE
LAYERS
8
8
8
12
10
PARAMETER (M)
1.18
1.18
1.18
2.10
5.46
UNISOMA
PARAMETER (M)
0.88
1.42
1.42
2.82
5.67
22


Reviews:
[{"review_id": "Tje3SMm4xF", "reviewer": "Reviewer_ksEb", "summary": "This paper presents the Unisoma focusing on the PDE solving of multi-solid systems. Different from previous methods, Unisoma proposes to embed the solid type (rigid or deformable) and load information into the model for explicit modeling of multi-solid interactions. Technically, Unisoma employs the slice operation proposed by Transolver to learn the physical states of solid information and external load. Additionally, the correlation between different solid units is calculated by the attention mechanism. Finally, attention is applied to compositional features with explicitly embedded solid information. Unisoma performs well in extensive benchmarks with good efficiency and generalizability.", "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": "Sincerely thanks for insightful comments.\n\n> About the convolution and local information in Transolver\n\nFrom the paper and official code of Transolver, the conv is applied only to **structured meshes or uniform grids** (Section 3.1 in Transolver); for **irregular meshes**—which are the focus of us—Transolver uses linear layers. Moreover, in Transolver, when conv is used, it is for generating deep features and slice weights at point level. The projection into slices is still by globally spatial aggregation over all points, diluting local relations. In contrast, as shown in Table 8, our **Edge Augmented Physics-Aware Tokens incorporate neighbor information via mesh edges into the slice projection**, preserving local relations. We will revise the sentence to: \n“However, when tackling irregular meshes, they only consider the point-level features and spatially aggregate points on the whole domain when transforming the mesh points into the slice domain....”\n\n> Efficiency comparison with Transolver\n\nAs explained in Appendix E, we ensured fair model capacity by **aligning the total parameter counts** of all baselines (except OOM error), with Unisoma. Efficiency was compared under similar capacity, making the evaluation meaningful. While both Unisoma and Transolver use slice, they differ in real computation. Transolver slices and deslices at **every layer** and applies FFN() **after deslicing on full mesh points**, leading to high memory usage as point count grows.  Unisoma slices/deslices only once and performs **most FFN() within the slice space**, greatly reducing memory.\n\nAs explained in Line 406-412, Transolver treats all points as a single large tensor of size $N \\times C$, applying slicing and FFN() directly, which leads to high memory usage. In contrast, Unisoma processes each object separately with **smaller tensors $N_i \\times C$, where $\\sum N_i = N$**. This splits large matrix operations into smaller ones, reducing memory overhead. Frameworks like PyTorch allocate memory for intermediate activations, so large tensors increase peak usage. \n\nWe include kNN time at efficiency test, using a KDTree algorithm with O(N log N) complexity. Since kNN is computed per object individually, not over all points, its cost remains small compared to the model’s runtime which is related to the number of all points.\n\n>  Fairness of comparison\n\nAs described in Appendix E, we made our best effort to ensure fairness .We aligned the parameter count and carefully tuned each baseline to achieve better accuracy. Importantly, **all models were provided with the same input information**, including solid types and load data. For domain-wise models, we treated **all deformable solids, rigid solids, and loads** as mesh points and concatenated them into a single input sequence. For graph-wise models,  we generated mesh edges using the **same kNN parameters** as Unisoma, ensuring consistency. \n\nWe provide more results below (batch size: 1 for deforming plate and 50 for others). It is worth noting that **the advantages of Unisoma become more evident as the number of solids and points increases**, due to its modular explicit modeling.\n\n|||deforming plate|||cavity grasping|||tissue manipulation||\n|:-:|:-:|:-:|:-:|-|:-:|:-:|:--:|:-:|:-:|\n||Param(M)|Time(s)|Mem(G)|Param(M)|Time(s)|Mem(G)|Param(M)|Time(s)|Mem(G)|\n|GINO|1.41|49.86|2.47|1.41|4.37|12.29|1.41|1.89|4.52|\n|GNO|1.14|29.81|7.64|1.23|8.44|23.45|1.23|1.73|5.53|\n|OFormer|1.48|48.55|0.92|1.48|4.16|7.41|1.48|1.14|1.80|\n|ONO|1.31|50.96|0.59|1.65|2.24|3.36|1.65|1.13|0.98|\n|Transolver|1.44|67.87|0.72|1.44|3.08|3.75|1.44|1.79|1.05|\n|Unisoma|0.92|50.20|0.41|1.40|1.92|0.86|1.40|1.45|0.63|\n\n> \"implicit\" and \"explicit\"\n\nWe first confirm that all models received the **same inputs**, except for edges, which were used only where supported. The distinction between \"explicit\" and \"emplicit\" we defined lies **not in the input itself**, but in **how the model structure leverages and organizes that information**. For example, Transolver concatenates all objects as a single domain-level sequence input and learns interactions implicitly via attention. The model does not explicitly structure the pairwise physical relations (e.g., contact constraints or force) that drive deformation.\n\nIn contrast, **Unisoma adopts an explicit modeling paradigm**, where physical interactions are structurally represented in the model architecture: the **contact modules** handle potential contacts, the **adaptive interaction allocation** computes equivalent load and constraint, and the **deformation triplet** encodes their influence on deformation. This structured design decomposes dynamics into controllable components. We will clarify this distinction more clearly in the revision.\n\n> Writing issues\n\nThanks again for insightful suggestions and we will revise accordingly.\n\nFinally, as stated in the *Software and Data*  (Line 448), we will open data and code upon acceptance to ensure the reproducibility  and advance the field.", "claims_and_evidence": "About the statement of Transolver \"However, they only consider the point-level features and spatially aggregate points on the whole domain. This leads to the loss of local relationships of mesh points which are important to model local interactions.\", I think it is incorrect as in Transolver's official paper, the slice operation can also be conducted based on convolution, which can consider the local information.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "(1)\tAbout the efficiency comparison with Transolver, I think it is unfair as the Transolver has much more layers and MLP ratio than Unisoma, as listed in Table 9. Actually, since Unisoma adopts the same slicing operation proposed by Transolver and additionally captures much more relations and solid information, it is theoretically impossible that Unisoma is more efficient than Transolver. Besides, I think the kNN clustering operations in Line 178 can be very time-consuming. I do not think the current efficiency statistics consider the clustering step.\n\n(2)\tAbout the fairness of comparison, I am wondering if Transolver or other baselines receive the same additional information as Unisoma, e.g. the solid types or loads. Actually, in the formalization of Transolver’s paper, it can also receive the optional physics information as the input. I think the authors should give different methods the same context of input.", "relation_to_broader_scientific_literature": "This paper is based on the design of Transolver but gives an insightful design tailored to the multi-solid system.", "essential_references_not_discussed": "I think this paper gives a comprehensive discussion of related works.", "other_strengths_and_weaknesses": "## Strengths\n\n1.\tThis paper presents a reasonable method tailored to the multi-solid systems.\n2.\tThe authors provide comprehensive experiments to verify the model’s efficiency.\n3.\tThis paper is well-written and gives a complete review of previous works.\n\n## Weaknesses\n1.\tAbout the categorization of “implicit” and “explicit” modeling.\n\nActually, since the additional solid information is point-wise, I think all the baselines can receive the same inputs as Unisoma. For example, the authors can just concatenate all the physics information along the channel-wise and project them into deep features. In this way, Transolver or other baselines can also make an explicit modeling of multi-solid systems, since the feature has been attached with “explicit” information.\n\nThus, I think the current discussion about previous methods is kind of inappropriate.\n\n\n2.\tAbout the efficiency comparison with Transolver.\nAs I stated before, I think the current efficiency results are unfair. Please give a detailed explanation. Also, I would like to see the efficiency comparison on other datasets.\n\n3.\tWhether different methods use the same physics information as inputs or not? please give a detailed clarification. If not, please input the same physics information to other baselines to enable a fair comparison.\n\n4.\tWriting issues.\n-\tRethinking the categorization of “implicit” and “explicit” modeling.\n-\tPlease give a citation to Transolver at the beginning of Section 3.2. This suggestion is not only for ensuring a clear discussion of previous work but also for making the slice operation easier to understand.\n-\tI think the authors should give the definition of some physics concepts, e.g. equivalent load or contact constraints at the beginning of related work, which will make this paper easier to understand.\n-\tI would suggest the authors use the subscript type of $ N_d, N_f, N_r$, as the superscript is easy to mix with exponential numbers.\n\nI think this paper provides a practical method for multi-solid modeling. However, as I have some concerns about the correctness of experiments, I cannot give a positive score. I would like to see the authors’ response.", "other_comments_or_suggestions": "Please see above.", "questions_for_authors": "N/A", "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": "EoZC2AMFvo", "reviewer": "Reviewer_g8HG", "summary": "This paper focuses on multi-solid tasks and proposes Transformer-based model to deal with the interactions between objects.\nTo better handle the interactions, the paper proposes to explicitly model the external forces and contact interactions, whose hidden representations are combined with the objects’ embeddings to predict the physical quantities. Experiments show that the proposed method outperforms baselines on various domains.\n\n## Update after rebuttal\nI appreciate the authors’ efforts in addressing my concerns, and the lastest explanation for \"Theoretical Claims\" is reasonable to me. Therefore, I have updated my score to 3.\n\nAdditionally, if the paper is accepted, it would be beneficial to include experiments demonstrating the performance of the corrected formulation with learnable parameters, accompanied by any necessary discussion.", "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": "Sincerely thanks for insightful comments.\n\n> Equations 4, 9\n\nThanks for pointing this out—Equations (4) and (9) do contain errors, and we apologize for the typos. We give a mathematically correct formulation here: \n$$\nz^\\xi_j=\\frac{\\sum_{i=1}^{N^\\alpha} w^\\alpha_{i,j}x^\\alpha_i + \\sum_{i=1}^{N^\\beta} w^\\beta_{i,j}x^\\beta_i}{\\sum_{i=1}^{N^\\alpha}w_{i,j}^\\alpha +\\sum_{i=1}^{N^\\beta}w_{i,j}^\\beta}=\\frac{(\\sum_{i=1}^{N^\\alpha}w^\\alpha_{i,j})z_j^{\\alpha} + \\sum_{i=1}^{N^\\beta} (w^\\beta_{i,j})z_j^{\\beta}}{\\sum_{i=1}^{N^\\alpha}w_{i,j}^\\alpha + \\sum_{i=1}^{N^\\beta}w_{i,j}^\\beta}=\\theta z_j^{\\alpha}+(1-\\theta)z_j^{\\beta}\n$$\nWe make $\\theta$ learnable. We test the correct form but it yielded **no noticeable performance gain** over direct element-wise addition. Hence, we chose **direct element-wise addition** for slice composition due to its simplicity, efficiency, and performance.\n\nThe **slice composition** is a simple yet effective fusion strategy that works well in practice. Importantly, it allows each solid to maintain its **own slice projection parameters**, rather than concatenating the two solids and projecting them jointly within each contact module (without parameter sharing across modules). We experiment with this more complex approach, but it results in **no significant improvement**, despite parameters increasing. \n\nWe will correct Equations (4) and (9) and clarify this design in the revision. Once again, we sincerely apologize for it and thanks for the careful observation.\n\n> Selection of baselines\n\nWe first emphasize the tasks. As claimed in the **bold text and corresponding citations in Section 3**, our tasks contain:\n\n- **Long-time prediction** follows Transolver, where the model directly predicts the target state from the given state and loads, usually spanning long time and skipping intermediate steps (e.g., using the 1st frame to predict the 105th frame in Cavity Grasping).  It can be formulated as: $P(\\hat{x}(t+T)|x{(t)})$, where $T$ spans many time steps. This task needs **long-range dependence** and is important for fast inference in many industrial scenarios..\n\n- **Autoregressive simulation** aligns with MGN, focusing on step-by-step rollout.  It can be formulated as: $P(\\hat{x}(t+1)|x(t))$, $P(\\hat{x}(t+2)|\\hat{x}(t+1))$, …, $P(\\hat{x}(t+T)|\\hat{x}(t+T-1))$. It focus more on the rollout trajectory and is suitable for scenarios where intermediate states are essential. \n\nWe select baselines based on their **recency and relevance to the tasks**. For long-time prediction, we compare Unisoma with **prevalent domain-wise and GNN-based models**. Domain-wise methods are better suited for global inference, while GNNs like MGN suffer from limited receptive fields, though we still include them for completeness. For autoregressive simulation, the task is mainly addressed by GNNs (e.g., MGN, HCMT, HOOD), and domain-wise models have rarely been extended to it. Thus, we focus on the most competitive baselines. For the efficiency comparison (Table 5), we selected models that are highly efficient or widely used in operator learning. GINO and GNO represent domain-wise neural operator models, while OFormer, ONO, and Transolver are recent efficient linear attention-based solvers.\n\nWe provide full efficiency comparison here. Notably, due to the difference in input shapes between Euler [B, X, Y, Z, C] and Lagrange [B, (X*Y\\*Z), C] views, the former is generally more memory-friendly for the same number of points (regular grid), but less suitable for solid system. We use a view mapping for part of baselines (Line 992-999). The main paper compares Lagrange-compatible models and we include Euler-only models (tagged “E”) here. Despite this disadvantage, **Unisoma maintains high memory efficiency as point count increases**.\n\n|||Bilateral||| Unilateral||\n|-|-|-|-|-|-|-|\n||Param|Time|Mem|Param|Time|Mem|\n|GeoFNO(E)|2.92| 32.28|0.75|5.47|58.73|1.38|\n|LSM(E)|5.94|43.23|2.77|5.94|225.85|19.49|\n|Galerkin|2.80|104.24|3.89|5.35|570.38|20.65|\n|Factformer(E)|3.16|33.15|1.61|6.32|126.96|14.21|\n|GraphSAGE|2.10|46.36|1.83|5.46|304.89|8.92|\n|MGN|2.88|119.68|13.93|4.51|373.49|23.30|\n|Unisoma|2.85|70.96|0.93|5.21|152.55|1.03|\n\n> RiceGrip Dataset\n\nThe DPI-Net repo includes the data generation code scripts. We generate the data on Ubuntu 18.04 with GTX 1080 (CUDA 9.1).\n\n> Long-time prediction and autoregressive simulation\n\nAs clarified before in *Selection of baselines*, long-time prediction aims to directly infer the target without intermediate steps, while autoregressive simulation rolls out step-by-step (105 steps for Cavity Grasping/Tissue Manipulation, 120 for Cavity Extruding). These simulation steps align with prior works (e.g., HOOD, HCMT), but **our cases involve more solids** (Cavity Extruding), making them more challenging and less explored.\n\n> Video results\n\nWe provide videos at https://anonymous.4open.science/r/unisoma_icml_2025-3DEF\n\n> Discussion of TIE\n\nWe acknowledge the relevance of TIE and will include a discussion.", "claims_and_evidence": "The claims are generally clear.", "methods_and_evaluation_criteria": "The methods and criteria make sense.", "theoretical_claims": "Equation 4 and 9 seem questionable. Since the author claims that $\\sum w_{i,j}^{\\alpha} \\approx \\sum w_{i,j}^{\\beta} \\approx 1$, a simple case to verify Equation 4 is that: if we just choose $\\sum w_{i,j}^{\\alpha} = \\sum w_{i,j}^{\\beta} = 1$, then the first row in Equation 4 becomes $\\frac{\\sum w^{\\alpha} \\mathbf{x}^{\\alpha}}{\\sum w_{i,j}^{\\alpha}} + \\frac{\\sum w^{\\beta} \\mathbf{x}^{\\beta}}{\\sum w_{i,j}^{\\beta}} = \\sum w^{\\alpha} \\mathbf{x}^{\\alpha}+\\sum w^{\\beta} \\mathbf{x}^{\\beta}$, while the second row of Equation 4 becomes $\\frac{\\sum w^{\\alpha} \\mathbf{x}^{\\alpha}+\\sum w^{\\beta} \\mathbf{x}^{\\beta}}{\\sum w_{i,j}^{\\alpha}+\\sum w_{i,j}^{\\beta}}  = 0.5(\\sum w^{\\alpha} \\mathbf{x}^{\\alpha}+\\sum w^{\\beta} \\mathbf{x}^{\\beta})$. Obviously, these two equations vary a lot and cannot be connected by $\\approx$ in Equation 4. The same applies to Equation 9. Could the author provide further explanations or corrections?", "experimental_designs_or_analyses": "1. I notice that in most of the quantitative results, such as Table 1, there are more baselines. However, for autoregressive simulation task (Table 3) and the efficiency comparisons (Table 5), fewer baselines are compared and the baselines are not the same as those in Table 1. For example, new baselines like HOOD and HCMT appear only in Table 3 while missing in Table 1. Could the author provide more details about how to choose these baselines and the complete tables of quantitative comparisons for all baselines?\n2. In the appendix at L787-803, the author claims that they “use 1200 samples” in the RiceGrip domain. However, in the original repo of DPI-Net [1], only 5 samples are publicly available. How the 1200 samples are obtained?\n3. The task of “long-time prediction” seems less convincing. From my understanding, the author may try to demonstrate that the model has robust performance in long-term predictions, which is also mentioned in MGN [2]. However, the settings  of “long-time prediction” as mentioned in Section 4.2 seem to predict the last frame given the initial frames, which are different from the experiments in MGN. On the other hand, the settings of “autoregressive simulation task” are much closer to the settings in MGN, where the total number of frames should be large enough. Since long-term prediction is an important challenge in simulation, more results should be provided. For example, results of longer trajectories should be provided. Notice that RiceGrip only has 41 frames for each sequence, it is not considered long enough from my opinion.\n4. While the video results may be optional, they are extremely important to evaluate the dynamic results of simulation, since this is a task predicting dynamics instead of static scenes. The problems, such as long-term predictions, can be easily observed in the video results. The same applies to the improvement.  However, this paper does not provide video results, making the experiments less convincing. \n\n[1]. Li, et al. Learning particle dynamics for manipulating rigid bodies, deformable objects, and fluids. ICLR2019.\n[2]. Pfaff, et al. Learning mesh-based simulation with graph networks. ICLR2021.", "relation_to_broader_scientific_literature": "N/A.", "essential_references_not_discussed": "The author should discuss TIE [1], which is also a Transformer-based model focusing on simulation and includes performance on multi-solid systems.\n\n[1]. Shao, et al. Transformer with Implicit Edges for Particle-based Physics Simulation, ECCV2022.", "other_strengths_and_weaknesses": "Please refer to \"Questions For Authors\".", "other_comments_or_suggestions": "Please refer to \"Questions For Authors\".", "questions_for_authors": "Here I summarize all the concerns:\n1. Questions about the Equation 4 and 9. (Theoretical Claims)\n2. Incomplete baselines comparisons. (Experimental Designs Or Analyses Q1)\n3. Questions about RiceGrip data. (Experimental Designs Or Analyses Q2)\n4. Autoregressive simulation task with longer trajectories as “Long-time predictions” could be more convincing. (Experimental Designs Or Analyses Q3)\n5. Missing video results. (Experimental Designs Or Analyses Q4)\n6. Missing reference. (Essential References Not Discussed)\n\nWhile I may be positive to this work, I hope the author could carefully address my concerns.", "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": "nVacZJ79fk", "reviewer": "Reviewer_M6EV", "summary": "This paper presents a transformer-based framework for explicitly modeling multi-solid interactions. The approach differs from implicit approaches that merge solids into a unified PDE or use graph-based message passing, and presents an explicit modeling one that structures interactions through a deformation triplet of a deformable solid, an equivalent load, and an equivalent contact constraint. It employs contact modules, an adaptive interaction allocation mechanism, and a deformable module to capture and process deformations. The model is evaluated on seven datasets and two multi-solid tasks, demonstrating improvements over existing deep learning methods in long-time prediction and autoregressive simulation.", "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": "Sincerely thanks for insightful comments.\n\n> Weakness: enforcing physical correctness\n\nWe first emphasize that the “explicit modeling” we defined **lies not in the explicit PDE constraints used in PINNs or hybrid models**, but rather in **how the model structure leverages and organizes input information**. In PINNs and hybrid models (e.g., NCLaw), PDEs are added to the loss to guide training, often relying on strong assumptions such that explicit PDEs exist—e.g., NCLaw assumes elastodynamic behavior. In **multi-solid systems**, it is difficult to define a single global and well-studied PDE (like N-S equation in CFD) to describe the entire system accurately. Instead, the behavior is usually governed by multiple **local relations**, such as contact penalties, load applications and deformations. Directly embedding them into the loss leads to **complex, multi-term objectives** that are hard to optimize and often limited in applicability. This remains an **open challenge** in the field.\n\nTherefore, we take a **purely data-driven approach** and propose to model the physics **through the model architecture**. Our framework explicitly organizes data using structured modules to capture the underlying physics. This design allows effective learning across **diverse materials, object counts, and task types**, with consistently strong results. We hope this view can complement PDE-based approaches, and we look forward to future work that combines both in a unified framework.\n\n> similar explicit modeling ideas\n\nWhile NCLaw is a valuable hybrid model combining neural networks with traditional solvers, our method is **purely data-driven**, learning physical priors **through architectural biases**. They belong to different classes, as NCLaw depends on **classical PDE solvers**.\n\n>  Line 178-185\n\nThe x denotes the deep features of an input object. For example, given a rigid solid $u^r_i \\in \\mathbb{R}^{N_i^r \\times C_r}$, we project it using a linear layer: x = Linear($u^r_i)$, where $x \\in \\mathbb{R}^{N_i^r \\times C}$. This is applied to each object individually, with a unified feature dimension $C$. We will clarify this more clearly in the revision. The edge set $E$ is constructed by k-nearest neighbors on mesh points of each object individually, i.e., $E = \\text{kNN}(u^r_i)$, where each point connects to its $k$ nearest neighbors. As shown in Appendix F, we find $k = 3 \\sim 5$ yields better performance.\n\n> Slicing\n\nAs shown in Eq. (1), slice weights are computed on deep features $x$ via a linear layer and softmax, and edge weights are derived from connected point weights. The slice is then formed by aggregating point and edge features. Since $x$ comes from mesh points, this applies directly to general meshes.\n\nAs noted in Line 201, slices group points with similar physical states, enabling attention to capture **physically consistent interactions** (Remark 3.1 in Transolver). We will emphasize this more clearly in the revision. When composing a slice from two potentially contacting solids, **contact becomes important interactions**, while unrelated interactions are suppressed. This focus improves attention modeling and enhances contact capture. Our results confirm that this explicit structure significantly boosts accuracy.\n\n> Equations 2, 6\n\nQ, K, and V are computed using standard Transformer linear projections: $Q = W_q(g), K = W_k(g), V = W_v(g)$, with learnable matrices $W_q, W_k, W_v$. In Eq. (2) and (6), $x$ is the **composed slice** from contacting solids or deformation triplets. As mentioned in the previous response, attention in the slice domain enables the model to learn **physically consistent interactions**, containing contact and deformation.\n\n> Training and Loss\n\nWe apologize for the brevity due to space constraints. As shown in Appendix C, our loss functions follow standard setups in related works (e.g., relative L2 for long-time prediction, RMSE for autoregressive simulation). As noted in the first response and Lines 076–089, Unisoma does not use PDE-based losses but **learns physical interactions structurally**:  the contact module, adaptive interaction allocation, and deformation triplet, all built on slice composition. These modules are trained end-to-end, and their physical roles are learned through architecture design.\n\n> Physical Meaning of the Contact Module\n\nUnlike PINNs or hybrid models that use PDEs as explicit constraints, we treat **contact interactions as learnable physical relationships** encoded in **high-dimensional features**. The model learns to minimize loss by focusing on key interactions, especially contact. We verify its effectiveness through improved accuracy across multiple datasets, particularly in complex multi-contact scenarios, where baselines that model interactions holistically tend to suffer performance degradation due to the holistic interaction mixture. Our structured, explicit modeling helps avoid such issues by organizing information around physically meaningful groupings.", "claims_and_evidence": "Overall, the paper presents strong empirical evidence supporting its claims, particularly regarding Unisoma's accuracy, efficiency, and ability to handle multi-solid interactions.", "methods_and_evaluation_criteria": "I believe the methods and evaluation criteria are genreally well-designed for the problem. The chosen benchmarks, tasks, and metrics effectively demonstrate Unisoma’s strengths in multi-solid simulation across different scenerios.", "theoretical_claims": "The paper does not contain formal mathematical proofs for theoretical claims.", "experimental_designs_or_analyses": "I think the experimental designs and analyses are sound. The paper compares Unisoma against strong baselines across multiple datasets using relevant tasks and metrics. It also includes out-of-distribution testing and efficiency comparisons.", "relation_to_broader_scientific_literature": "The paper builds on implicit modeling methods like PINNs and Neural Operators, as well as graph-based approaches using message passing such as MGN. Unlike these, this paper adopts explicit modeling, aligning with traditional FEM while leveraging Transformer-based PDE solvers for efficiency.", "essential_references_not_discussed": "The paper claims to be the first at explicit modeling for multi-solid systems, but similar ideas have been explored before. \"NCLaw: Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics\" (ICML 2023) also integrates explicit modeling by enforcing known PDE structures while learning constitutive models, ensuring physical correctness and generalizability.", "other_strengths_and_weaknesses": "The paper is weak in explaining its key method and is difficult to read in its current form. Several sections lack clarity, making it challenging to understand the approach. The explanation of the slicing algorithm and its connection to contact modeling is particularly vague, and the use of symbols and hyperparameters without proper definitions adds to the confusion. The training procedure and loss functions are not well-documented in the main text, making it unclear how supervision ensures physically meaningful constraints. Additionally, while the paper claims to use explicit modeling, it does not clarify how each module enforces physical correctness, particularly for contact constraints. While the experimental results apepar to be strong, these issues obscure the main contributions of the paper and make it unnecessarily complex.", "other_comments_or_suggestions": "See above comments", "questions_for_authors": "1. Line 178-185: The explanation of k-NN, edge sets, and deep features is unclear. What exactly are the x values referred to as \"deep features\"? What does C represent? How is the edge set E computed, and what is k in this context? How is it chosen?\n\n2. Slicing Algorithm: Figure 3 does not clearly explain how slicing is performed. How does the slicing process apply to a general mesh? Why is the slice domain relevant to contact modeling? How does slice composition help distinguish contact interactions from other types of interactions?\n\n3. Equations 2 and 6: How are Q, K, and V computed in these equations? What is their role in the model, and how does the attention mechanism specifically capture physical interactions?\n\n4. Training and Loss Functions: The loss functions are vaguely described in the appendix. Can the authors provide a clearer breakdown of the loss functions used for each task? How does the loss ensure that each explicit module (deformation, load and contact) learns its intended physical meaning?\n\n5. Physical Meaning of the Contact Module: How do the authors verify that the contact module correctly handles contacts? Does the contact module account for nonlinear contact forces, friction, or material-dependent constraints, or does it treat all contacts as uniform interactions?", "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": "7yhkUaDsPZ", "reviewer": "Reviewer_skRm", "summary": "The paper explicitly models the contact constraints and loads in multi-solid systems, using a Transformer-based framework. \n- The system contains three types of objects, deformable solids, rigid solids, and forces. Instead of treating each point as a token, the paper proposes to incorporate the mesh edges into embedding and proposes edge-augmented tokens.\n- Then, stacked processors are used to model physical interactions among solids. The processor separately consider and contact constraints, the loads, and the effects on each deformable solid. An independent deformation module is used for each deformable solid.\n- The outputs of the processor are mapped back to the original domain by weighted broadcast, to get the predicted outcomes. \n\nThe proposed methods is evaluated on two tasks, long-time prediction and autoregressive simulation.\n- The experiments use 7 datasets, including different systems containing few solids or multiple solids in 3D spaces. 4 of the datasets are public datasets for autoregressive task. The authors construct 3 additional datasets.\n- The proposed method is compared with ten baselines, and achieves state-of-the-art performance across the benchmarks.\n\n\n\nupdate after rebuttal\nThank the authors for the responses. I decide to keep my original score", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Sincerely thanks for insightful comments.\n\n> How to get all the solids pairs that are likely to contact?\n\nIn most cases, such as the stamping and grasping scenarios discussed in the paper, the solid pairs that are likely to come into contact are **known a priori** based on the deterministic nature of the physical setup.\n\nFor systems where contact is less certain, as mentioned in Line 368-375, we include **solid pairs with a high likelihood of contact** in the contact module. The **adaptive interaction allocation** mechanism then controls the extent to which each contact influences the deformation of a solid, effectively weighting more relevant interactions. This makes the process **flexible and robust** across both well-defined and uncertain contact settings.\n\n> Given a specific system, how to get the \"loads\"\n\nAs a supplement to lines 127–130 in the paper, we clarify that for a moving solid, We treat the displacement $d = u(t+1) - u(t)$ as a **load** applied over that time interval—commonly referred to as a **displacement load** [1]. Its positions at time $t$ and $t+1$ are denoted as $u(t)\\in R^{N\\times 3}$ and $u(t+1)\\in R^{N\\times 3}$, respectively. We represent each load in a **Lagrangian description** as the concatenation $\\text{concat}(u(t), d)\\in R^{N\\times 6}$, encoding both its origin position and next movement. When the moving solid comes into contact with others, this displacement load is transferred as a **force** onto the contacting objects, influencing their deformation.\n\n> Is it easy or hard to generalize to different time steps in the autoregressive task?\n\nSimilar to MGN[2], our framework learns to predict **the next state given the current state** in an autoregressive manner. This step-wise prediction scheme does not explicitly encode time intervals, so generalization across different time steps depends on the **temporal distribution of the training data** and the model’s capacity to learn the underlying dynamics.\n\nTo evaluate the generalizability across time steps, we designed an experiment using the **Cavity Grasping** dataset:\n1. We directly test Unisoma, trained on original time step size (600 samples, 100 epochs), on data with **doubled time intervals**.\n2. We **fine-tune** (20 epochs) the same model using a small amount of doubled-step data (120 samples), then test on data with doubled time intervals.\n3. We perform **full training** (600 samples, 100 epochs) on the doubled-interval data and test accordingly.\n\nIn all cases, all other parameters and test data remain consistent)\n\n||1.Directly test|2.Fine-tuning|3.Full training|\n|-|-|-|-|\n|Rollout-all RMSE($10^{-3}$)|13.43|11.06|9.68|\n\nThese results reveal several insights:\n\n- The **drop in performance (higher RMSE)** when directly applying the model to doubled-step data indicates **limited generalization** when time step statistics shift significantly.\n- However, **a small amount of fine-tuning** brings considerable improvement, suggesting that the model retains useful representations of the dynamics that are **transferable** across step sizes.\n- **Full retraining** yields the best performance, as expected, since the model can directly adapt its dynamics modeling to the new time scale.\n\n> About the complexity of Unisoma\n\nWhile attention mechanisms indeed have **quadratic time complexity** with respect to the number of tokens, our attention operations in Eq. (2) and Eq. (6) are performed over the slices, not directly on the original mesh points. Each slice is corresponding to an **Edge Augmented Physics-Aware Tokens**. Usually, the number of slices is significantly smaller than the number mesh points.\n\nLet the number of input mesh points be $N$, which are converted to $M$ slice tokens via Eq. (1) and $M \\ll N$. This step has a complexity of $\\mathcal{O}(MNC)$. The attention in Eq. (2) and Eq. (6) is then applied to sequences of length $M$, with a complexity of $\\mathcal{O}(M^2C)$ and is irrelevant to the mesh points $N$. Moreover, the number of modules per layer (contact modules, deformation module, etc) is typically small and the cost is negligible compared to the embedding step. Therefore, the **overall complexity is approximately $\\mathcal{O}(MNC + M^2C)$**. This means that as the input sequence length $N$ increases, the computational cost of our model scales **almost linearly** ($M \\ll N$ and $M$ is usually fixed), making it significantly more efficient than standard attention over full mesh sequences.\n\nThank you for your positive recognition of our work. We have carefully addressed the concerns you raised and provided additional explanations and experimental results to support our claims. We look forward to your response and feedback.\n\n[1] Mau S T. Introduction to structural analysis: displacement and force methods[M]. Crc Press, 2012.\n\n[2] Pfaff T, Fortunato M, Sanchez-Gonzalez A, et al. Learning mesh-based simulation with graph networks[C]//International conference on learning representations. 2020.", "claims_and_evidence": "yes.", "methods_and_evaluation_criteria": "yes.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "yes.", "relation_to_broader_scientific_literature": "Compared to the existing works, this paper explicitly models the multi-solid systems using transformer-based frameworks instead of implicit models.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "The explicit modeling of contact constraints, loads, and deformable modules using transformer-based models seems novel to me. Additionally, the paper presents good experimental results, comparing against multiple baselines across a relatively large set of tasks.", "other_comments_or_suggestions": "I just have some additional questions listed below.", "questions_for_authors": "- How do you get all the solids pairs that are likely to contact when calculating contact constraint?\n- As mentioned in the paper, the system's input includes deformable solids, rigid solids, and loads. Given a specific system, how do you get the \"loads\" input objects?\n- In the autoregressive task, the framework directly learns to predict the next step. Is it easy or hard to generalize to different time steps?\n- The attention module has a quadratic time complexity regarding the number of tokens. As the number of input tokens increases, computation can become costly. Does the proposed method (which applies attention on the sliced inputs) face a similar issue? Could you provide a brief complexity analysis?", "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": "AdRI97Zqi9", "reviewer": "Reviewer_YnTU", "summary": "This paper proposes an explicit modeling approach for solving multi-solid problems, incorporating factors influencing solid deformation through specialized modules. The authors design a novel transformer-based architecture to achieve this, demonstrating state-of-the-art performance across multiple datasets.", "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": "Sincerely thanks for insightful comments.\n\n> The experimental evaluation could be strengthened by including scaling studies to demonstrate the method's scalability.\n\nThanks for the insightful suggestion. We conducted a series of scaling experiments to analyze the model’s behavior under varying configurations. Specifically, we evaluated performance across different numbers of slices, processors, channels, and training samples. The default settings used in the paper is: 32 for slice number, 2 for processor number, 128 for channel number and 1000 training samples. We only adjust one variable in one experiment and keep other variables unchanged. The relative L2 recorded in the table is the average accuracy of all physical quantities.\n\nFrom the results, we observe that increasing the number of slices and channels leads to improved performance up to a certain threshold (approximately slices ≥ 128, channels ≥ 192), beyond which the gains become marginal. This suggests that our model is parameter-efficient, achieving strong performance without requiring excessive capacity. Interestingly, the model shows little sensitivity to depth, with performance remaining largely stable as the number of processors increases. We attribute this to the model's inherently wide architecture, where each layer consists of multiple parallel attention-based modules. This design reduces the model's reliance on depth and makes it less prone to overfitting with increasing layers. Additionally, the relatively flat performance curve across depth may indicate that the model is reaching saturation under the current data regime.\n\n\n|Slice Number|16| 32     | 64     | 96     | 128    | 160    | 192    | 224    | 256    |\n|-|-|-|-|-|-|-|-| ---- | ------ |\n| Relative L2 | 0.1140 | 0.1084 | 0.1078 | 0.1087 | 0.1069 | 0.1063 | 0.1066 | 0.1077 | 0.1073 |\n| Params (M)   | 2.83   | 2.85   | 2.88   | 2.91   | 2.95   | 3.00   | 3.04   | 3.09   | 3.15   |\n\n| Processor Number | 1      | 2      | 3      | 4      | 5      | 6      | 7      | 8      |\n| - | -| - | - | - | - | - | - | - |\n| Relative L2  | 0.1087 | 0.1084 | 0.1087 | 0.1080 | 0.1077 | 0.1069 | 0.1081 | 0.1074 |\n| Params (M)   | 1.82   | 2.85   | 3.87   | 4.89   | 5.92   | 6.94   | 7.96   | 8.98   |\n\n| Channel Number | 64     | 96 | 128 | 160    | 192    | 224    | 256    | 320    | 384    | 448    | 512    |\n|-|-|-|-|-|-|-|-|-|-|-|-|\n| Relative L2| 0.1095 | 0.1084 | 0.1084 | 0.1072 | 0.1070 | 0.1087 | 0.1081 | 0.1124 | 0.1112 | 0.1106 | 0.1109 |\n| Params (M)     | 0.73   | 1.61   | 2.85   | 4.43   | 6.36   | 8.63   | 11.26  | 17.55  | 25.24  | 34.32  | 44.80  |\n\nTo further investigate this, we conducted data scaling experiments. The results clearly show that model performance consistently improves as the number of training samples increases. Even at the largest scale we tested (1000 samples), the performance continues to rise, indicating that the model has not yet saturated and remains highly data-efficient.\n\n|Training Samples|200|400|600|800|1000|\n|-|-|-|-|-|-|\n| Relative L2   | 0.1387 | 0.1263 | 0.1210 | 0.1103 | 0.1084 |\n\nOverall, these findings highlight the robustness and scalability of our model. It delivers competitive accuracy with moderate parameter counts, maintains stability across architectural depths, and continues to benefit from additional data—demonstrating strong practical potential in real-world applications.\n\n> Visualizations of the learned Edge Augmented Physics-Aware Tokens would help verify whether the model effectively captures the underlying physical information in the datasets.\n\nTo evaluate the effectiveness of the **Edge Augmented Physics-Aware Tokens**, we visualize the slice weights of two deformable solids in the **Bilateral Stamping** scenario. The comparison is available at https://anonymous.4open.science/r/unisoma_icml_2025-3DEF\n\nAs shown in the figures, a **horizontal comparison** reveals that **different slices attend to different spatial regions after projecting from the original mesh space**: some focus on the pressed area under the rigid solid, others on the central stretching region, and some on the fixed ends. This indicates that the tokens successfully group points with similar physical states and differentiate between slices, enabling the model to capture diverse underlying physical behaviors.\n\nMore importantly, in the **vertical comparison** between orderly corresponding slices of the two deformable solids, we observe that **they tend to focus on similar regions**—areas that are not only of high relevance to each solid individually but are also **likely to come into contact**. This alignment further supports the effectiveness of Edge Augmented Physics-Aware Tokens in capturing meaningful, structured physical interactions.\n\nThank you for your positive recognition of our work. We have carefully addressed the concerns you raised and provided additional explanations and experimental results to support our claims. We look forward to your response and feedback.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "This paper does not present any theoretical claims.", "experimental_designs_or_analyses": "The experimental evaluation could be strengthened by including scaling studies to demonstrate the method's scalability. Additionally, visualizations of the learned Edge Augmented Physics-Aware Tokens would help verify whether the model effectively captures the underlying physical information in the datasets.", "relation_to_broader_scientific_literature": "While this paper focuses specifically on multi-solid scenarios, its methodological contributions may provide valuable insights for the broader field of PDE solving.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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: This paper presents a transformer-based framework for explicitly modeling multi-solid interaction. Four reviewers recommended accepting this work. AC agrees that this work is interesting and deserves to be published on ICML 2O25. The reviewers did raise some valuable concerns that should be addressed in the final camera-ready version of the paper.
Decision: Accept (poster)