UniMate: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Authors: Wangzhi Zhan
Jianpeng Chen
Dongqi Fu
Dawei Zhou
Venue: ICML 2025 poster (2025)
Keywords: metamaterial
topology generation
property prediction
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We propose a unified model for mechanical metamaterial generation, prediction and conditioning.

Abstract: Metamaterials are artificial materials that are designed to meet unseen properties in nature, such as ultra-stiffness and negative materials indices. In mechanical metamaterial design, three key modalities are typically involved, i.e., 3D topology, density condition, and mechanical property. Real-world complex application scenarios place the demanding requirements on machine learning models to consider all three modalities together. However, a comprehensive literature review indicates that most existing works only consider two modalities, e.g., predicting mechanical properties given the 3D topology or generating 3D topology given the required properties. Therefore, there is still a significant gap for the state-of-the-art machine learning models capturing the whole. Hence, we propose a unified model named UniMate, which consists of a modality alignment module and a synergetic diffusion generation module. Experiments indicate that UniMate outperforms the other baseline models in topology generation task, property prediction task, and condition confirmation task by up to 80.2%, 5.1%, and 50.2%, respectively. We open-source our proposed UniMate model and corresponding results at https://github.com/wzhan24/UniMate.

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Wangzhi Zhan' Jianpeng Chen
Dongqi Fu
Dawei Zhou'
Abstract
Metamaterials are artificial materials that are de- signed to meet unseen properties in nature, such as ultra-stiffness and negative materials indices. In mechanical metamaterial design, three key modalities typically involved, i.e., 3D topol- ogy, density condition, and mechanical property. Real-world complex application scenarios place the demanding requirements on machine learning models to consider all three modalities together. However, a comprehensive literature review in- dicates that most existing works only consider two modalities, c.g., predicting mechanical prop- erties given the 3D topology or generating 3D topology given the required properties. There- fore, there is still significant gap for the state- of-the-art machine learning models capturing the whole. Hence, we propose a unified model named UNIMATE, which consists of modality align- ment module and synergetic diffusion gener- ation module. Experiments indicate that UNI- MATE outperforms the other baseline models in topology generation task, property prediction task, and condition confirmation task by up to 80.2%. 5.1 5.1%, and 50.2%, respectively. We open- source our proposed UNIMATE model and corre- sponding results at https://github. com/ wzhan24 wzham24/UniMatte iMnae
Trinity Correlation in Metamaterial Design
Example
Topology
Property
Topology
Poleson's Rado
Density
Young's Modako
Medules
Density ity
Property
Needar Shea heste
Hodeaen a latto
a a. (.2......2....
Lsw
High
Figure 1. Trinity Representation of Metamaterials.
their engineered structures rather than the chemical compo- sition. These materials are revolutionizing various fields, offering novel solutions for challenges in engineering, man- ufacturing, and materials science (Barchiesi al., 2019). Metamaterials can have many possible important merits that classic materials do not have, like negative Poisson's ratio (Wang et al., 2020a) (favored for soft device), energy absorption behavior (Yuan et al., 2019) (favored for cush- ioning device), tunability (Sheng & Varadan, 2007) (favored for device with customized features), and extreme strength- to- to-weight ratio (Wang et al., 2020b) (favored for ultra light and strong device). With the unique and superior merits of metamaterials, they have become optimal choice in many engineering fields like energy storage, biomedical, acoustics, photonics, and thermal management (Surjadi et al., 2019).
1. Introduction
Metamaterials are synthetic materials with unique proper- ties, which are typically rarely observed natural mate- rials (Engheta Ziolkowski, 2006). To be specific, me- chanical metamaterials designed to achieve unusual mechanical properties through
Metamaterial design has a higher degree of freedom com- pared to traditional material design due to the possibility of manipulating and engineering the properties the sub- wavelength level. In traditional material design, owing to natural laws, only small amount of components can be tailored, like the percentage of each element, the phase of materials, etc. However, metamaterials can have artifi- cially programmed structures as long as certain manufactur- ing requirements (e.g.. periodicity) are satisfied (Sinha Mukhopadhyay, 2023). Therefore, the design tasks are mul-
specific class of metamaterials
'Department of Computer Science, Virginia Polytechnic Insti- tute and State University, Virginia, US *Meta AI. Correspondence to: Dawei Zhou <zhoud ed.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Table 1. Comprehensive Review of Model's Ability for Metamaterial
MODEL
TOPOLOGY GENERATION PROPERTY PREDICTION CONDITION CONFIRMATION
SCHNET (SCHÜTT ET AL., 2017)
FTCP (REN AL., 2020)
COND-DFC- VAE (COURT ET AL., 2020)
PAINN(SCHÜTT AL., 2021)
CDVAB (XIE ET AL., 2022)
CG-SCHNET (GEBAUER AI.,, 2022)
MCGCNN (MEYER ET AL., 2022)
MACE (BATATIA ET AI.,, 2022)
EOUIOORMER(LIAO & SMIDT, 2022)
EDM (HOOGEBOOM ET AL., 2022)
SPHERENET (LIU AL., 2022)
DIFFCSP(IAO ET AL., 2023)
UNITRUSS (ZHENG AI.,, 2023A)
VISNG ET AL., 2024)
SYMAT (Luo ET AL.,, 2024B)
EQUICSP (LIN ET AL., 2024)
COND-CDVAE (LUO ET AL., 2024A)
VE (GREGA
2024)
COMFORMER (YAN ET AL., 2024)
UNIMATE (OurS)
tifaceted (Zheng et al., 2023a; Bastek et al., 2022b; Maurizi al., 2022). e.g., giving part of the structure and generating the rest according to some desired property, giving part of the properties and generating a suitable structure and the rest of the property, giving the structures and predicting the possible properties, and giving the structures and properties and selecting the optimal conditions during the generation. Mastering the above tasks challenging and requires comprehensive knowledge and consideration of all possible aspects. To the best of knowledge, there is exist- ing AI work that models all metamaterial design aspects together, as shown in Table 1, and the detailed review and why we need the comprehensive modeling are expressed below.
design involves three modalities with different formats and distributions. Hence it is difficult utilize all such informa- tion; (C2) Task Diversity: given the complexity of data, any part of the data can be missing and requires design effort. Therefore there can be various design tasks; (C3) Lack of Benchmark: there is lack of suitable benchmark that cov- ers the diverse tasks, including a dataset providing varied topologies, density conditions and properties, and suitable metries to evaluate the model performance. However, as shown in Table 1, existing works on material design
satisfied, models like FTCP (Ren et al., 2020), SyMat (Luo et al., 2024b), EquiCSP (Lin et al., 2024) are solely targeted for topology generation. One relatively "versatile" model is Cond-DFC-VAL (Court al., 2020), which can perform both structure generation and property prediction. However, Cond-DFe DFC-VAE can only take structural information as in- put when performing property prediction tasks, such that when adopting it to the condition selection, the input be handled is biased, i.e., only topological information is taken.
Formally, three key aspects (i.e., data modalities) compose the entire mechanical metamaterial design or generation, shown in Figure 1, i.e. 3D topology, density condition, and mechanical property (Yu et al., 2018). Knowing any two of these three aspects, we expect to derive the other one. For example, given the restrained density condition and the target property (c.g., stiffness), in this paper, we study how to empower generative models to design an effective structural organization among many possible 3D topologies, which task can be named as Topology Generation, the other two tasks are named Condition Confirmation and Property Prediction, their mathematical illustrations are expressed in Section 2.2.
Motivated by the above analysis, we aim to propose a uni- fied model that is versatile enough and can tackle three data modalities together in mechanical metamaterial design. Our proposed UNIMATE consists of two novel components, modality alignment module and synergetic generation module. The modality alignment module compresses the three different modalities into shared latent space and uses tripartite optimal transport to align the latent tokens from each modality. The synergetic generation module receives the latent tokens (some are unknown) and completes the unknown tokens through score-based diffusion model.
The mechanical metamaterial community is eager for the comprehensive modeling method to address existing chal- lenges: (C1) Data Complexity: mechanical metamaterial
2

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Finally the denoised tokens from the diffusion model are reverted by decoder(s) into the raw design space. In prin- ciple, the modality alignment module helps to narrow the large gap among the three modalities, which significantly alleviates challenge CI; the synergetic generation mod- ule, owing to its intrinsic flexible nature, is well-suited for address challenge C2. We propose new dataset and new metries, which help to solve challenge C3.
Young's 's modulus, shear modulus, and Poisson's 'ss ratio (Wort- & Evans, 1965). Hence, we formally define mechanical metamaterial as follows.
Metamaterial Trinity Representation (MTR) represents a complete metamaterial representation, denoted as M = (T, p. p), where M is a tuple of 3D topology T, density condition p. and mechanical properties p.
This work has the following main contributions:
100
Problem Formalization: We propose and formalize general design task for mechanical metamaterial de- sign. The task defined in this work can be generalized to many specific important tasks, including topology generation, property prediction, and condition confir- mation.
Lattice
Equipment
Unified Model: We propose a unified model to address the general design task. This unified model can per- form diverse kinds of tasks mechanical metamaterial design.
Substrate M Molecule
Unlt
Benchmarking and Experimentation: We propose suitable benchmark for the metamaterial design tasks and compare model with six baselines. The result shows the superiority of our model.
Figure 2. Hierarchical Composition Mechanical Metamaterial.
3D Topology is represented by T (L, X, A), where L = 14 12 [3] contains the three lattice vectors indicating the axes along which the unit cell is stacked; X E RBx3 is the 3D coordinates matrix of nodes, and A E R=** is the adjacency matrix without self-loop, where n denotes the node numbers, with A.,j if ith node connects jth node.
2. Preliminary
Here, we briefly and formally introduce the background of metamaterial and relevant main tasks.
La dw Relative Density Condition denoted as p= Jg dw where
We use regular letters to denote scalars (e.g., nt), boldface lowercase letters to denote vectors (e.g.. vi), and boldface uppercase letters to denote matrices (e.g., Wi), italic upper- case letters to denote sets (e.g., L).
Ω and Ω stands for the unit cell zone and solid zone within the unit cell, and dw is a small volume; in other words, density means the ratio of solid matter within the unit cell.
Mechanical Properties is denoted p, vector indicating the mechanical attributes of materials, such as Young's 's mod- ulus, shear modulus, Poisson's ratio (Wortman & Evans, 1965), etc.
2.1. Background of Mechanical Metamaterial
As shown disentangled in Figure 2, mechanical metamateri- als are basically substrate material accumulated according to synthetic spatial structures, i.c., lattice structures. To avoid complicated design, lattice structures are normally set to be periodic that the entire body of metamaterial is made of repeating smallest cell, called unit cell. Therefore, the definition of the topology of metamaterial can be expressed as the topology of a unit cell (essentially 3D graph) and three lattice vectors indicating how the unit cell is repeated. Given such topology information, we can customize how "densely" substrate material accumulated along each edge (i.e.. relative density' condition). When 3D topology and relative density conditions determined, the mechanical properties of the metamaterial can be estimated, including
The definition of MTR allows us to formalize the problem of designing mechanical metamaterial as deriving complete MTR.
2.2. Main Tasks
Given the background in Section 2. 1, to derive a complete MTR, we focus on three basic tasks with different unknown components of the MTR concerned.
Conditional Topology Generation. Given restrained rel-
ative density p and desired property we aim to generate 3D topology T that meets certain geometric requirements (periodicity and symmetry) and matches the given property under the given density.
In this paper, we "relative density" and "density" inter- changeably
3

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Modality Alignment Module
Synergetic Generation Module
Initial Tokens
Code- book
Rounded Tokens
Add noise
Topology
TT
MIHA
Dr
Recon. Topology
Feed forward
Density
E
Add Norrm
D
Density
Recon.
Property
LMTR
Dr
Recon. Topology
Token rounding
Diffusion layers
Figure 3. Pipeline Illustration of UNIMATE. Three modalities fed into the model, crossing modulcs, and generating reconstructed data for training. In the inference phase, the targeted modality data generated from noise.
Property Prediction. Given the topology T and density condition P. we aim to predict the corresponding property p, iee, whether and how well a specific topology can perform under a particular density.
one for modality alignment and the other for synergetic generation as shown in Figure 3. The modality alignment module maps the raw information into a shared latent space and aligns the latent tokens from each modality. The syn- ergetic generation module takes known latent tokens as context and generates the unknown tokens.
Condition Confirmation. Given topology and property p. aim confirm the corresponding density condition p. This task helps the researchers to know how light heavy the designed material can be, with the given topology and given property. This task holds particular significance in designing materials that are light yet exhibit high stiffness, as well as in the development of lightweight cushioning materials.
Next, will elaborate the technical details of meta- material modality alignment and metamaterial synergetic generation.
3.2. Metamaterial Modality Alignment
In brief, the modality alignment module is designed to align the three modalities, thus facilitating the generation process and addressing the challenge C1.
3. Methodology
In this section, we introduce the details of the proposed UNIMATE model that is designed to handle diverse tasks in a unified framework for metamaterial design. We begin with an overview the model in Section 3.1, which consists of two main modules, i.e., Metamaterial Modality Alignment in Section 3.2 and Metamaterial Synergetic Generation in Section 3.3. Section 3.4 explains the training and inference procedures of our model.
Since continuous spaces are more difficult to align, inspired by inspired by VQ-VAE (Van Den Oord al., 2017), we first map the raw data into discrete latent space. Then, use three pairs of encoders and decoders to map the data from raw format into the latent space. These tokens are then mapped into a shared discrete latent space by comparing with codebook, which is essentially a set prototype tokens { where z; E R&d k the codebook size, and d the token dimension.
3.1. Overview
The three pairs of encoders & and decoders D correspond- ing to the three different modalities for topology T. density p, and property are denoted as (T) {Ep. D,), and { Dp). For example, e is a graph convolutional net- work (GCN) defined as
There three key challenges in mechanical metamaterial design, i.e., data complexity, task diversity, and lack of benchmark. To demonstrate this, in this paper, we first prepare a benchmark to solve the third challenge together, and the details are given in Appendix A.
To be specific, for addressing the first challenges, propose model with specially designed modules, i.e.,
X = E A) GCN (X,A),
X GCN A) 1018- (L)

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
where X = x and v I S L denotes the number of layers. W denotes the /th layer's parameters, o denotes Sigmoid activation function, and X is the latent embedding consisting of latent tokens Zt
to align the three modalities. Optimal transport (Sinkhom, 1967) (OT) provides way align two different distribu- tions. To apply OT in our task, we generalize it into tripartite OT (TOT), which considers three modalities. The process of implementing TOT is to lower the tripartite Wasserstein dis- tance (TWD) by optimizing the latent tokens. This optimiza- tion is then realized by optimizing the encoders mentioned in Section 3.1. The TWD, dw. defined as
After the latent tokens (e.g., Z.)) obtained, We introduce codebook to "round" the latent tokens, i.e., find the nearest prototype token in the codebook and substitute the latent tokens with the prototype token. The subscription "round" (e.g., X round) below means rounded token(s).
inf E B...v.eAta.B..)) [d(z, y. 2)], dw (...S.))AAto....)
(7)
Then, DT is introduced to decode the rounded tokens, con- sisting of several transformer layers with final linear layer:
where a, 9 and are the marginal distribution of x; y and 2, and c(I, y, z) is the tripartite cost defined as
== DD (X round) TransLayer" ww... round WT, (2)
c(2, y, 2) CosSim(2. v) + CosSim(y, =) + CosSim 2).
(8)
where each TransTayer' is a transformer layer using multi- head self-attention, and WT= gdx3 is to calculate the vertex coordinates. Consequently, the reconstructed the adjacency matrix À can be obtained by
Minimizing the dw in Equation (7) by optimizing the dis- tribution of the three modalities x. y and provides way to align the three modalities. To achieve this, we propose a tripartite Sinkhorn algorithm (TSA), which is a gener- ation of (Sinkhorn, 1967). The details can be found in Appendix B.2. Briefly speaking, the vanilla Sinkhorn al- gorithm (Sinkhorn, 1967) involves two distributions, such that to process the three modalities, we need to generalize the vanilla Sinkhorn algorithm into the tripartite Sinkhorn algorithm by generalizing the transport plan iteration pro- cess and adding the preprocessing (calculating the token frequency first, i.c., lines to 10 in Alg. 1).
A CosSim(ár, at):
where CosSim denotes cosine similarity.
In addition to coordinates X and adjacency matrix A, to derive the MTR, the lattice vector L can be calculated ac- cording to the method mentioned in Appendix A.2.
Similarly, Ep and DD are each a multilayer perceptron (MLP) to process density p and its latent representation p:
Alg. gives the detailed implementation of our proposed TSA. Firstly, the marginal distribution of X round Pround and Pround is calculated with respect to the latent token series {zi} (lines 2 to 10 in Alg. 1). According to TSA, the transport plan (which is essentially a 3D tensor) can be expressed as TransPlan = u @ V O w & M, where u, v and w are three bases and M is a damping matrix that is determined by the latent token series. Therefore, only u, v, and w need be calculated. Finally, lines 12 to 17 in Alg. I calculate the three bases in iterative way.
p= Ep(p,, and أ = Dp((Proudd
(4)
Also, Ep and Dp are two series of MLPs, each MLP for one dimension of p. The encoder maps the property p into a series of tokens P, while the decoder does the reverse operation.
should be noted that the output of the encoders, i.e., X, p, P, are not directly input into the decoders, but af- ter "rounding" them into the nearest tokens in the codebook Z. X, P, P are concatenated into latent MTR (LMTR):
With respect to the alignment operation, to optimize the model, the alignment loss can be defined as
M = [X: P).
Lutign = =a X1| + X X xround |1 + ||A
All
M can also be reformatted series of tokens: M (mg, m.....,, m.,) where each m E Rd is latent token, h is the LMTR token number, d is the token dimension (also named latent dimension). For each token m in M, the rounding process is performed as:
+aplllp- - ~ll + 11 Pround II)
(9)
+ +ap 1l1 + +1 P Pround|))
awdd.
where at, Cpps Op and O'w are four hyperparameters.
Meonnd Zj: where j = min [|m zell.
(6)
After minimizing this alignment loss, we can obtain a 3D tensor that indicates the collaborative distribution of the three (latent) representations. Therefore, in the generation stage, instead of randomly choosing one starting point from pure noise, we can choose from the elements with higher probability as the starting point. This operation can decrease the discrepancy between the starting point and ending point
then, we get [ round: Pround: P r-
The above three different modalities are being mapped into a shared discrete space comprising a series of tokens by the encoders and the codebook. A suitable method is needed
5

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
of the diffusion process. Hence, it will be simpler gen- erate proper outputs, which increase the robustness of the generation process.
generation as synergetic generation. Finally, the generation loss is defined as
Legnn || |M mund
Mrl.-
(13)
3.3. Metamaterial Synergetic Generation
3.4. Optimization and Inference
To address challenge C2, the synergetic generation module is proposed for diverse tasks during metamaterial design.
The training process is follows: (1) the encoders project the raw MTR into latent space; (2) the codebook is used to round the latent tokens; (3) TOT is used to align different modalities; (4) the decoders are used to reconstruct the raw MTR; (5) add Gaussian noise to one random type of tokens (e.g., topology tokens) to obtained noised LMTR; (6) the noised LMTR is input into the diffusion model for denoising (i.e., the synergetic generation process). Note that steps 4 and 5 do not rely each other, they are two parallel operations. The training loss is
We implement the synergetic generation process as a partially frozen diffusion process, since a diffusion model (Song& & Ermon, 2019) can take inputs with flexible shape if the backbone is a flexible model, while the frozen operation can provide context for the diffusion process.
The input to the synergetic generation module is the rounded LMTR M round, which is series of tokens. Part of the input tokens are added with noise and are considered as unknown tokens. The indices of the known tokens and unknown tokens are denoted as Iun and Iun, respectively, with Iun U Jun = I = {0, 1.....h - 1}, where h is the number of all tokens. The known and unknown tokens can be denoted as M round (Ik)) and Mruna (Tun). The score-based diffusion process (Song & Ermon, 2019) can be expressed as series of denoising steps, with each step being:
L1 ValignLalige Agendigens
(14)
which basically considers the MTR reconstruction error, the token rounding error, the modality align error and the error between generated tokens and initial latent tokens.
When using the model for testing or inference, it is assumed that part of the raw MTR is provided (although the model can also do unconditional generation to provide random de- sign). Then, the given part goes through the corresponding encoder(s), and some latent tokens are provided accordingly. Then, the unknown tokens are initialized with the condi- tional probability according to the transport plan to provide high-quality initialization that is aligned with the input condition. The diffusion model then generates the com- pleted M round, which then is reverted to the raw design space by the decoders.
M. = M + ai 22222 ((M + Va.Zi.
(10)
where t E 1,... T). T is the time step number, ag is a scalar declining with the diffusion process (we use the same setting of a; as in (Song& & Ermon, 2019)), @ is a transformer backbone, and Z, Gaussian noise with the same shape as M. Mo is set to be M round-
To keep the given context tokens intact, the transformer backbone performs partially frozen diffusiion In other words, for partially frozen diffusion, all the tokens go through each layer of the transformer and yield an output token series. Then, the formerly known tokens in the output are replaced with their initial values. Such partially frozen processing can be expressed formally as:
4. Experiments
4.1. Experiment Setup
Datasets. Existing works lack a suitable dataset covering all three tasks in Section 2.2. To address this, we construct a new dataset based on (Lumpe & Stankovic, 2021) by select- ing samples, assigning multiple density conditions each topology, and computing mechanical properties via finite element simulation. Further details are in Appendix A.1.
(M,, Iun) = (Tam) (Im))
(11)
Then, Equation (10) becomes
Baselines. As detailed in Section 5, existing methods related to our work fall into generation-oriented and prediction- oriented models. For generation, we use CDVAE (Xie et al., 2022) and SyMat (Luo al., 2024b) as baselines, both designed for periodic crystal structure generation. For pre- diction, we include Equiformer (Liao & Smidt, 2022), ViS- Net (Wang et al., 2024), MACE-ve (Grega et al., 2024), and uni (Zheng et al., 2023a), which predict (meta) ) properties from topology. Additionally, we adapt Tnuss for generation using its reconstruction capability.
M. M 0i OM, Ium) JonZe. 2
(12)
Owing to such a partially frozen operation and the intrinsic mechanism of the transformer, our model can take series of tokens with arbitrary series length (except for memory limit's ' sake), while an arbitrary subset of the tokens can be set as unknown. The known tokens provide context information for other tokens, especially in the attention operation of the transformer. Therefore, we name such a
6

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Table Effectiveness Comparison.
TOPO. GEN. TASK
PROP. PRED. TASK
COND. CONFIRM. TASK
MODEL
Iqua
Feond
NRMSEPp
NRMSEec
(x10 1
(x10-²,
(x10
(x10
CDVAE (XIE AL.,, 2022)
19.23
32.71
N/A
N/A
EQUIFORMER (LIAO SMIDT, 2022)
N/A
N/A
5.31
38 38.05
VISNET (WANG ET AL., 2024)
N/A
N/A
3.12
10.43
SYMAT (LUO ET AL., 2024B)
16.94
33.37
N/A
N/A
UNITRUSS (ZHENG ET AL.., 2023A)
19.43
33.77
2.71
8.89
MACE+VE (GREGA AI.,, 2024)
N/A
N/A
2.57
9.09
UNIMATE (OuRS)
2.74
2.44
4.43
4.2. Effectiveness Analysis
Trainina Time per Batch (s)
SyMat
To evaluate the effectiveness of our model, compare its performance with the baseline models on the three tasks mentioned in Section 2.2. For topology generation, we con- sider the two generation-oriented baselines. For property prediction and condition confirmation, we consider the four prediction-oriemet baselines. Existing models are not well suited for condition confirmation since they are not designed input 3D topology and mechanical properties simultane- ously. Therefore, models by forcing them to predict density for the condition confirmation task. The metrics we use to assess the genera- tion result are Fqua and Feoud- Specifically, Fqua measures topology quality, including symmetry and periodicity, while F Fooud measures how closely the result matches the ground truth topology. Details regarding these metrics can be found in Appendix A.2. For the property prediction and condition confirmation tasks, we simply calculate the normalized root mean square error (NRMSE) between the model's output and the ground truth value.
6
Equiformer
UniMetaMate
2
G
2000
Batch Size
revise the three prediction-oriented
Figure 4. Comparison of Time Efficiency.
ger an out-of-memory error with the GPU near the stopping point of each line. Our model, on the other hand, triggers no eтгог even after the batch size of 10000. Therefore, our model has far higher space efficiency than other models considered.
4.4. Parameter Sensitivity
We also study the parameter sensitivity of model by tuning the latent token dimension d and number of to- kens within the codebook n. The resulting generation met- ric Fqua is shown in Figure 5, with other metrics Ap- pendix C.2. From Figure 5, it can be seen that our model performs better roughly with larger latent token dimension and larger codebook size.
As listed in Table 2, our model outperforms other baselines in all three tasks. the topology generation task, property prediction task, and condition confirmation task, our model outperforms the second-best model by 80.2%, 5.1%, 11,, and 50.2%, respectively. This result verifies the superiority of our model in terms of effectiveness.
4.3. Time and Space Efficiency
To compare the time efficiency of our model with other models, we train each model on our dataset with varied batch sizes and record the average time required for processing each batch. The result is shown in Figure 4. According to Figure 4, the batch processing time for each model is roughly in linear relation to the batch size. In terms of the slope of each line in Figure 4, our model has medium-level slope, implying a medium-level time efficiency. Additional results for other baselines can be found in Appendix C.I.
0.07
3/0
0.0
0.25
0.44
18ftm dimeniion
42
number of tokens
0.03
However, it should noted that three lines in Figure 4 stop at smaller batch sizes. The reason is that these models trig-
Figure Fqua under different parameters.

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
4.5. Ablation Study
to 0.5). Our model can yield topologies as a transiting series along with the transiting properties, as illustrated in Figure 6.
Table 2 shows that our model performs well in capturing the cross-modality correlation. For the ablation study, cases used to figure out how much the diffusion generation module (with/without the partially frozen diffusion tech- nique), the encoder-decoder compression module, and the alignment operation contribute to our model's correlation- capturing ability.
Figure 6 shows that our model suggests octet truss topology in the HSL.D-targeted task, while octet truss is known to be a promising candidate for high-stiffness topology (Song et al., 2019). Our model can also generate some novel intermediate topologies that are not in the training dataset, which implies its potential to approximate the intermediate transition within a given distribution and to propose novel metamaterial candidates with wanted properties.
Table 3. Ablation Study.
(4)
Feond (4)
(4)
(LM
CASE
0.0531
0.1136
0,0472
0,0757
Sancture Treniiion
CASE 2
0.0295
0.0812
0.0263
0,0510
CASE 3
0.0405
0.0813
0.0322
0.0423
CASE 4
0.0274
0.0781
0.0244
0.0443
Ablation case I: (diffusiin model alone) Instead of gener- ation in the LMTR space, the diffusion model can also be used to directly generate samples in the raw MTR space (al- though a linear layer is needed to bridge the gap between the input space dimension and transformer layers' processing dimension).
Low Siffees
Medur Stilnees
High Saleass
Proport Torstoo
Ablation case 2: (mapping into latent space + diffusion model) In this case, we still use the encoder-decoder module to compress the original MTR into latent MTR and then im- plement the diffusion module without alignment operation.
Figure 6. Case Study on HSLD metamaterial.
5. Related Work
Material Structure Generation. Diffusion models (Yang et al., 2023) and variational autoencoders (VAEs)(Kingma, 2013; Chen et al., 2025a) are widely used for material struc- ture generation. SyMat(Luo et al., 2024b), EquiCSP (Lin et al., 2024), and DiffCSP (Jiao et al., 2023) employ diffu- sion models for crystal topology generation, with SyMat incorporating an additional VAE for lattice vector and atom type generation. These models focus unconditional gen- eration, lacking property awareness. CDVAE (Xie et al., 2022) integrates a diffusion model with a VAE, encoding structures into a latent space for conditional generation. Cond-CDVAE (Luo et al., 2024a) extends CDVAE with conditional Langevin sampling. For molecular generation, eG- (Gebauer al., 2022) and EDM (Hoogeboom et al., 2022) are conditional models: eG-SchNet leverages SchNet (Schütt et al., 2017) for latent space encoding, while EDM processes conditions akin to diffusion time steps. Cond-DFC- VAE (Court et al., 2020) combines crystal gen- eration with property prediction but relies on an external predictor.
Ablation case 3: (vanilla diffusion) In this case, we simply remove the partially frozen diffusion technique from the full model. Specifically, this means that all tokens, including the context tokens that should not be changed, are changed through each layer in the diffusion backbone.
Ablation case 4: (full model) This case corresponds to the full model, including design space unification, latent space alignment, and synergetic generation.
Table 3 lists the result of the ablation study, From Table 3 it can be seen that the unification and the alignment operation are both beneficial to our model's performance. The unifica- tion process boosts the performance by 37.5% in average, and the TOT alignment provides 7.8% boost upon case 2. By comparing case 3 and case 4 can be seen that the partially frozen diffusion technique provides 13.9% performance boost in average, in contrast to vanilla diffusion.
4.6. Case Study
To illustrate the application of our model, here we provide case study on topology generation with high stiffness and low density (HSLD). We train our model on dataset of metamaterials exhibiting better HSLD filtered from our original dataset to emphasize the wanted HSLD attribute. We restrict the density condition to a low value (c.g.. 0.3), and we tune the wanted stiffness within a range (e.g.. 0.1
Material Property Prediction. MACE (Batatia et al., 2022) introduces high-order message-passing passing network for graph- based materials, with MACF-ve (Grega et al., 2024) enhanc- ing it through additional components and training modifica- tions. ComFormer (Yan et al., 2024) applies transformers to tokenized crystalline structures. ViSNet (Wang et al., 2024) utilizes graph attention for feature fusion. SphereNet (Liu
8

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
al., 2022) employs 3D spherical representations for mes- sage passing. mCGCNN (Meyer et al., 2022) processes em- bedded node and edge features via CGCNN (Xie & Gross- man, 2018). Equiformer (Liao & Smidt, 2022) adapts trans- formers for graph-based learning with equivariant attributes. SchNet (Schütt et al., 2017) utilizes continuous filters for molecular convolutions, while PaiNN (Schütt et al., 2021) alternates between two types of equivariant message-passing blocks. uniTrrus (Zheng al., 2023a), VAE-based model for metamaterial property prediction, is not inherently de- signed for material generation.
Barroso-Luque, L.,, Shuaibi, M., Fu, X., Wood, B. M.,
Dzamba, M., Gao, M., Rizvi, A., Zitnick, C. L., and Ulissi, Z. W. Open materials 2024 (omat24) inorganic materials dataset and models, 2024. URL https: / wwwrriv..rggabs//.. 12771.
Bastek, J.-H. and Kochmann. D. M. Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models. Nature Machine Intelligence, 5(12): 1466-1475, 2023.
Bastek, J.-H., Kumar, S., Telgen, B., Glaesener, R. N., and Kochmann, D. M. Inverting the structure-property map of truss metamaterials by deep learning. Proceedings of the National Academy of Sciences, 119())c211150511) 2022a.
6. Conclusion
We formalize the diverse mechanical metamaterial design tasks into a general task. Based on the general task, we pro- pose unified model, UNIMATE, which addresses the data complexity challenge and task diversity challenges. Experi- ments show that UNIMATE outperforms other baseline mod- els by significant margin. We also prepare a benchmark to support the training and evaluation of metamaterial design models, which addresses the lack of benchmark challenge. In summary, this paper contributes to problem formaliza- tion, model, and benchmark development for mechanical metamaterial design.
Bastek, J.-H., Kumar, S., Telgen, B., Glaesener, R. N., and Kochmann, D. M. Inverting the structure-propent map of truss metamaterials by deep learning. Proceedings of the National Academy of Sciences, 119())=211150511) 2022b.
Batatia, I., Kovacs, D. P., Simm, G. N. C., Ortner, C., and Csanyi, G. MACE: Higher order equivariant mes- sage passing neural networks for fast and accurate force fields. In Oh, A. H., Agarwal, A., Belgrave, D., and Cho, K. (eds.), Advances in Neural Information Processing Systems, 2022. URL https: / openreview /ntiteeiieeee////ee net/ forum? Periidd--p PSggg----g-------. 20.
Acknowledgements
We thank the anonymous reviewers for their constructive comments. This work is supported by the National Sci- ence Foundation under Award No. IIS-2339989 and No. 2406439, DARPA under contract No. HR001 12490370 and No. HR001 124S0013, U.S. Department of Homeland Security under Grant Award No. 17STCIN00001008-00 Amazon-Virgna Virginia Tech Initiative for Efficient and Robust Machine Learning, Amazon AWS, Google, Cisco, 4-VA, Commonwealth Cyber Initiative, National Surface Trans- portation Safety Center for Excellence, and Virginia Tech. The views and conclusions are those of the authors and should not be interpreted as representing the official policies of the funding agencies or the government.
Chen, J., Ling, Y., Xu, J., Ren, Y., Huang, S., Pu, X., Hao, Z., Yu, P. S., and He, L. Variational graph generator for multiview graph clustering. IEEE Transactions Neural Networks and Learning Systems, pp. 1-14, 2025a
Chen, J., Zhan, W., Wang, H., Jia, Z., Gan, J., Zhang, J., Qi, J., Chen, T., Huang, L., Chen, M., et al. Metamat- bench: Integrating heterogeneous data, computational tools, and visual interface for metamaterial discovery. arXiv preprint arXiv:2505.20222 2025b.
Court, C. J., Yildiri,, B., Jain, A., and Cole, J. M. 3-d inorganic crystal structure generation and property pre- diction via representation learning. Journal of Chemical Information and Modeling, 60( (0(100:4518-44 4535, 2020.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Du, Y., Wang, Y., Huang, Y., Li, C.,, Zhu, Y., Xie, T., Duan, C., Gregoire, J., and Gomes, C. P. M 2 hub: Unlocking the potential of machine learning for materials discovery. Advances in Neural Information Processing Systems, 36: 77359-77378, 2023.
References
Dunn, A., Wang. Q., Ganose, A., Dopp, D., and Jain, A. Benchmarking materials property prediction methods: the matbench test set and automatminer reference algo- rithm. npj Computational Materials, 6( 6(1):138, 2020.
Barchiesi, E., Spagnuolo, M., and Placidi, L. Mechani- cal metamaterials: a state of the art. Mathematics and Mechanics of Solids, 24(1):212-234, 2019.

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Engheta, N. and Ziolkowski, R. W. Metamaterials: physics and engineering explorations. John Wiley & Sons, 2006.
Jiao, R., Huang, W., Lin, P., Han, J., Chen, P., Lu, Y., and Liu, Y. Crystal structure prediction by joint equivariant diffusion on lattices and fractional coordinates. In Work- shop on "Machine Learming for Materials" " ICLR 2023, 2023. URL https:: /oppneeeeiie .et. forum? id-VPByphdu241
Fu, D., Zhu, Y., Tong, H., Weldemariam, K., Bhardwaj. O., and He, J. Generating fine-grained causality in climate time series data for forecasting and anomaly detection. CoRR, 2024.
Kingma, D. P. Auto-eneoding variational bayes. arXiv
Fu, D., Zhu, Y., Liu, 7.,, Zheng, L.,, Lin, X., Li, Z.,, Fang. L., Tieu, K., Bhardwaj. O., Weldemariam, K., Tong, H., Hamann, H. F., and He, J. Climatebench-m: A multi- modal climate data benchmark with a simple generative method. CoRR, 2025.
preprint 1r1iv 2013.
Kipf, T. N. and Welling, M. Variational graph auto-encoders. arXiv preprint arXiv: 0e211111100000 2016.
Lee, S., Zhang, 2., and Gu, G. X. Generative machine learning algorithm for lattice structures with superior mechanical properties. Materials Horizons, 9(3)9552 960, 2022.
Gebauer, N., Gastegger, M., and Schütt, K. Symmetry- adapted generation of 3d point sets for the targeted dis- covery of molecules. Advances in neural information processing systems, 32, 2019.
Liao, Y.-.. and Smidt, T. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs. arXiv preprint arXiv:2206. 11t.:2206..99990 11990, 2022.
Gebauer, N. W. Gastegger, M., Hessmann, S. S., Müller, K.-R., and Schütt, K. T. Inverse design of 3d molecular structures with conditional generative neural networks. Nature communications, 13(1):973, 2022.
Lin, P., Chen, P., Jiao, R., Mo, Q., Jianhuan, C., Huang, W., Liu, Y., Huang, D., and Lu, Y. Equivariant diffusion for crystal structure prediction. In Forty-first International Conference on Machine Learning, 2024.
Grega, I., Batatia, I., Csanyi, G., Karlapati, S., and Desh- pande, V. Energy-conserving equivariant GNN for elas- ticity of lattice architected metamaterials. In ICLR, 2024. URL https: / http//pperrviieeeeee net. forum? Id=smy4DsÜbBo.
Liu, S., Li, Y., Li, Z., Zheng, Z., Duan, C., Ma, Z.- M. Yaghi, O., Anandkumar, A., Borgs, C., Chayes, J., et al. Symmetry-informe geometric representation for molecules, proteins, and crystalline materials. Advances neural information processing systems, 36, 2024.
Gruver, N., Sriram, A., Madotto, A., Wilson, A. G., Zitnick, L.,, and Ulissi, Z. W. Fine-tuned language models generate stable inorganic materials as text. In The Twelfth International Conference on Learning Representations, 2024. URL https: / http///pppteeeee/////////////eep wwtp:/iwwwwweeeeeeeeeeeeeeeeeeee / forum? id-vN9 9pfqqol
Liu, Y., Wang, L., Liu, M., Lin, Y., Zhang, X., Oztekin, B., and Ji, S. Spherical message passing for 3d molec- ular graphs. In International Conference on Learning Representations, 2022. URL net
He, X., Fu, D... Tong, H., Maciejewski, R., and He, J. Tem- poral heterogeneous graph generation with privacy, utility, and efficiency. In ICLR, 2025.
Lumpe, and Stankovic, T. Exploring the property space of periodic cellular structures based on crystal networks. Proceedings of the National Academy of Sciences, 118 (7)xe2003504113 2021.
Hoogeboom, E., Satorras, V. G., Vignae, C., and Welling, M. Equivariant diffusion for molecule generation in 3d. In ICML, pp. 8867-8887, PMLR, 2022.
Luo, X., Wang, Z., Gao, P., Lv, J., Wang, Y., Chen, C., and Ma, Y. Deep learning generative model for crystal structure prediction. npj Computational Materials, 10(1): 254, 2024a.
Indurkar, P. P., Karlapati, S., Shaikeea, A. J. D., and Desh- pande, V. S. Predicting deformation mechanisms in architected metamaterials using gnn. arXiv preprint arXiv:2202.09427, 2022.
Luo, Y., Liu, C., and Ji, S. Towards symmetry-aware gener- ation of periodic materials. Advances in Neural Informa- tion Processing Systems, 36, 2024b.
Jia, Z., Gong, H., Liu, S., Zhang, J., and Zhang, Q. Designing dinen-dimeniionl lattice structures with anticipated properties through deep learning method. Materials & Design, 244:113139, 2024. ISSN 0264-
Maurizi, M., Gao, C., and Berto, F. Predicting stress, strain and deformation fields in materials and structures with graph neural networks. Scientific reports, 12(1):21834 2022.
1275. doi: htpp://biinng/00001jjjmmmete
URL
httpe://www.eciencedircccc
10

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Meyer, P. P., Bonatti, C., Tancogne-Deeean, T., and Mohr, D. Graph-based metamaterials: Deep learning of structure- property relations. Materials & Design, 223:111175, 2022.
Sinha, P. and Mukhopadhyay, T. Programmable multi- physical mechanics of mechanical metamaterials. Mate- rials Science and Engineering: R: Reports, 155:100745, 2023.
Mistal, Herrnnez-Garria,, A., Volokhova, A., Duval, A. A., Bengio, Y., Sharma, D., Carrier, P. L., Koziarski, M., and Schmidt, V. Crystal-GFN: sampling materials with desirable properties and constraints. In Al for Accel- erated Materials Design NeurlPS 2023 Workshop, 2023. URL https: httpss//pppeeeeeeeeeeeeeeee net/ forum? jdFO
Sinkhorn, R. Diagonal equivalence to matrices with pre- scribed row and column sums. The American Mathemati- cal Monthly, (4(4):402-404) 1967.
Song, J., Zhou, W., Wang, Y., Fan, R., Wang, Y., Chen, J., Lu, Y., and Li, L. Octet-truss cellular materials for improved mechanical properties and specific energy ab- sorption. Materials & Design, 173:107773, 2019.
O'keeffe, M., Peskov, M. A., Ramsden, S. J., and Yaghi, M. The reticular chemistry structure resource (resr) database of, and symbols for, crystal nets. Accounts of chemical research, 41(12):172211789, 2008.
Song, Y. and Ermon, S. Generative modeling by estimating gradients of the data distribution. Advances neural information processing systems, 32, 2019.
Surjadi, J. U.. Gao, L., Du, H., Li, X., Xiong, X., Fang, N. X., and Lu, Y. Mechanical metamaterials and their en- gineering applications. Advanced Engineering Materials, 21 2103):1808864 2019.
Qi, J., Jia, Z., Liu, M., Zhan, W., Zhang, J., Wen, X., Gan, J., Chen, J., Liu, Q., Ma, M. D., et al. Meta- scientist: A human-ai synergistic framework for auto- mated mechanical metamaterial design. arXiv preprint arXiv:2412. 16270, 2024.
Van Den Oord, A., Vinyals, O., et al. Neural discrete rep- resentation learning. Advances in neural information processing systems, 30, 2017.
Ramsden, S., Robins, V., and Hyde, S. Three-dimensions euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples. Acta Crystallographica Section A: Foundations of Crystallography, 61-2)881--08, 2009.
Vogel, G., Sortino, P., and Weber, J. M. Graph-to-string variational autoencoder for synthetic polymer design. In Al for Accelerated Materials Design-NeurIPS 2023 Work- shop, 2023.
Ren, 7., Noh, J., Tian, S., Oviedo, F., Xing, G., Liang, Q., Aberle, A., Liu, Y., Li, Q.. Jayavelu, S., et al. Inverse design of crystals using generalized invertible crystallo- graphic representation. arXiv preprint arXiv:2005.0766 3(6):7, 2020.
Wang, H., Zhang, Y., Lin, W., and Qin, Q.-H. A novel wvo-dimensional mechanical metamaterial with negative poisson's 'ss ratio. Computational Materials Science, 171: 109232, 2020a.
Schüitt, K., Kindermans, P.-J., Sauceda Felix, H. E., Chmiela, S., Tkatchenko, A., and Müller, K.-R. Schnet: A continuous- filter convolutional neural network for model- ing quantum interactions. Advances neural information processing systems, 30, 2017.
Wang. Q., Li, Z., Zhang, Y., Cui, S., Yang. Z., and Lu, Z. Ultra-low density architectured metamaterial with superior mechanical properties and energy absorption capability. Composites Part Engineering, 202:108379, 2020b.
Schütt, K., Unke, O., and Gastegger, M. Equivariant mes- sage passing for the prediction of tensorial properties and molecular spectra. In International Conference on Machine Learning, pp. 9377-9388. PMLR, 2021.
Wang, Y., Wang. T., Li, S., He, X., Li, M., Wang, Z., Zheng, N., Shao, B., and Liu, T.-Y. Enhancing geometric repre- sentations for molecules with equivariant vector-scalar interactive message passing. Nature Communications, 15 (1):313, 2024.
Sheng, Z. and Varadan, V. V. Tuning the effective properties of metamaterials by changing the substrate properties. Journal of applied physics, 101(1), 2007.
Wortman, J. and Evans, Young's 's modulus, shear modulus, and poisson's 's ratio in silicon and germanium. Journal of applied physics, 36(1):153-156 1965.
Silveirinha, M. G. Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters. Physical Review B Condensed Matter and Materials Physics, 75(11): 115104, 2007.
Xie, т. and Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable predic- tion of material properties. Physical review letters, 120 114):145301, 2018.

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Xie, T., Fu, X., Ganea, O.-E., Barzilay, R., and Jaakkola, T. S. Crystal diffusion variational autoencoder for pe- riodic material generation. In International Confer- ence on Learning Representations, 2022. URL https: / //openreviees net. net//orumm forum? -
Yan, K., Fu, C.,, Qian, X., Qian, X., and Ji, S. Complete and efficient graph transformers for crystal material property prediction. arXiv preprint arXiv:2403. /40. 2024.
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., and Yang, M.-H. Diffusion models: A comprehensive survey of methods and applications. ACM Computing Surveys, 56(4):1-39, 2023.
Yu, X., Zhou, J., Liang, H., Jiang, Z., and Wu, L. Mechani- cal metamaterials associated with stiffness, rigidity and compressibility: A brief review. Progress in Materials Science, 94:1 14-173, 2018.
Yuan, S., Chua, K., and Zhou, K. 3d-printed mechanical metamaterials with high energy absorption. Advanced Materials Technologies, 4(3):1800419, 2019.
Zheng. L., Karapiperis, K., Kumar, S., and Kochmann, D. M. Unifying the design space and optimizing linear and non- linear truss metamaterials by generative modeling. Nature Communications, 14(1):7563, 2023a.
Zheng. L., Karapiperis, K., Kumar, S., and Kochmann, D. M. Unifying the design space and optimizing linear and non- linear truss metamaterials by generative modeling. Nature Communications, 14(1):7563, 2023b.
Zheng, L., Zhou, D., Tong, H., Xu, J., Zhu, Y., and He, Fairgen: Towards fair graph generation. In 2024 IEEE 40th International Conference on Data Engineering (ICDE), pp. 2285-2299 IEEE, 2024.
Zhou, D., Zheng, L., Xu, J., and He, J. Misc-gan: A multi- scale generative model for graphs. Frontiers big Data, 2:3, 2019.
Zhou, D., Zheng, L., Han, J., and He, J. A data-driven graph generative model for temporal interaction networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp- 401-411, 2020.
12

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
A. Benchmarking Details
A.1. Dataset
Our data set is built on the basis the dataset proposed Lumpe & Stankovic (2021). That dataset contains 17087 topologies, out of which we randomly select 500 topologies with no more than 20 vertices. Figure A. illustrates some of the selected topologies.
100
1,0
1.0
0.6
0.1
0.4
0.1
0.2
0.0
0.0
0.0
0.6
0.0
0.2
0.6
0.2
0.9
0.8
0.8
1.0
1.0
1.0
1.0
10
10
0.8
0.8
0.8
0.1
0.2
0.2
0.2
0.0
0.0
0.0
0.0
0.2
0.4
0.6
0.6
1.0
0.8
0.6
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
0.4
0.0
0.2
04
0.6
1.0
0.6
0.4
Figure A.I. Example Topologies.
Given these 500 topologies, we randomly assign 3 different edge radius for each topology. Given a topology, edge radius and relative density be deferred based each other according
d nr
mT loqu:
(15)
where d is relative density, r is edge radius lequ is the equivalent total edge length:
Logu Σ Linner, + Σ Utace. + 4
1 Σ Leame,is
(16)
where the "inner", "face" and "frame" subscripts indicate edges that are within the unit cell, on the surface, or on the outer frame (connecting two near corner vertices). Therefore, edge radius and relative density are equivalent, given topology. For each topology denossy pair, mesh the 3D structure into small cubic voxels and then apply homogenization simulation (Silveirinha, 2007) to calculate the homogenized mechanical properties of the structure. After such operation, we have 1500 data points (500 topology and densities for each topology, with matching properties). Based on these 1500 data points, we apply data augmentation by rotating the topology and the properties with the same random rotating angle. Each data point rotated times, so in total, our dataset contains 15000 samples. Our dataset is also provided in the project repository https: : //ghthbb....
thub. com/ www.n22ha /U. Maaee.
13

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
A.2. Metries
We different metrics for evaluating a model's performance in the three tasks: topology generation, property prediction and condition confirmation.
A.2.1. TOPOLOGY GENERATION
For the generation result assessment, we design two metrics, Fqua and Feond. Fqua is used to measure the quality of a 3D topology, defined as:
Fqua 2l'sym4'o Fpor
(17)
Faym + Fper"
where Faym measures the central symmetry and Fper measures the periodicity. We denote the vertex coordinates as X [ 21...., Lgg. where R is 3D coordinates of vertex, q is the number of vertices within unit cell. F Faym
defined
Faym (X)
Σ
min
2. +
cli
(18)
2
where e is the centroid coordinates of all the vertices:
Σ
æi-
(19)
Since all the topologies in our dataset are of cubic shape, define periodicity metric based on cubic shape. We find the eight corners out of all the vertices by:
X corner = (m: in Icorner); with Tcorner argsort( x) 8).
(20)
Tenrner corresponds the indices of the eight vertices that are farthest from the centroid. After finding the eight corners, set them into four pairs, each pair containing one corner and another corner that is farthest from it. Figure A.2 illustrates the four pairs (e.g., corner and I'). One random corner is set as anchor corner (e.g., corner 1). For the other 3 pairs (2, 2', 3, 3' and 4, 4"), we consider the one closer to the anchor as "positive" corner. For example, in the 2-2' pair, corner is closer to corner 1.
2
1
4'
3
3'
4
1'
2'
Figure A.2. Illustration of Eight Corners.
After matching all eight corners with their relative status (i.e., the topological relation with respect to the anchor), the vector of each outer frame edge can be derived, e.g.:
e 02 21
(21)
where 1 is the edge vector between corner and corner 1, with other ei-j similarly defined. Then, three average axes can be defined as:
4 (e2-1 1 er-3 + ev-2 ey-1))
(22)
14

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
12 (e3-1 to---t tz)
(23)
(24)
With the above definition, Fpr can be defined as:
Fee 12 (1lez-1 - 11ll ||ew
1
|ley 411 + 1|ev-2 - 11ll
(25)
+ |le3-1 12/l + |lev-2 12 + lle - [2|l + ller-33 - 1211
+ ||ee-11 1a/l + 1|e2-33 Lal + lly - tall + ||ep-1' - -tal)).
Foond is defined as:
Feond Xgt) = 1 Σ min | a; - Dgt.ll
(26)
where Xgt is the ground truth vertex coordinates, with Zgt.. being the jth ground truth vertex's coordinates. Therefore, Foond measures how close a generated result is to the ground truth structure that matches the given condition perfectly.
A.2.2. PROPERTY PREDICTION AND CONDITION CONFIRMATION
For property prediction and condition confirmation tasks, we use NRMSE as the metric, respectively as NRMSEpp and NRMSF defined as:
1
NRMSEpp maxx, ; mins, (Petis.s)
lp. Pa.l?
(27)
NRMSE max, (Pgt.s) - (Pgt.)
Σ (p: - Pet.i)
(28)
where agt.i is the ith ground truth property (with Pgt.i. being its jith component) in the dataset, P, is the output value from the model for the ith property, Pgt.i is the ground truth density the dataset, P; the output value from the model for the ith density and N is the dataset size.
B. Experimental Details
B.1. Implementation Details
We conduct all the experiments on a Linux platform with an NVIDIA A100 GPU (80GB version). We use the same dataset division ratio 70% for training, 15% for validation and 15% for testing) for each model. We run the training process until each model reaches its best performance, although other models generally require more epochs to converge than our model.
The topology encoder ε and decoder DT are each three-layer GCN, with the latent dimension same as the codebook token dimension (e.g.. 128). The property encoder Ep and decoder Dp. and the density encoder &, and decoder DD, are each three-layer MLP, with the latent dimension same the codebook token dimension.
B.2. Algorithm Details
Alg. shows the detailed Tripartite Sinkhorn Algorithm.
C. Additional Experimental Results
C.1. Time and Space Efficiency
Table C.1 gives additional results of time and space efficiency analysis results for baselines not mentioned Section 4.3. Each element Table C.1 indicates processing time for one batch (seconds per batch) with different size, and "OOM" indicates out of memory error.
15

---

UNIMATE: A Unified Model for Mechanical Metamaterial Generation, Property Prediction, and Condition Confirmation
Algorithm Tripartite Sinkhorn Algorithm
1: Input: (X round (oon B-1
i= (rou
B-1 { (ounddd i=
B-1
2: Fr, Fp. Fp On //F represents the frequency of tokens, B is batch size, is small positive number
3: u, U, w 1
4: for = =0 1011 do
5: for j == by // b is the number of tokens X do
6: F F + if z E round. // rroudd, is the jth token in T round
7: end for
8: Update FD and Fp
9: end for
10: Normalize PT, Fpr Fp
C {Ct,,k Ct.skk c(ze, 2j: 2x)). M exp(-C/c)
12: for i = 0 N do
13:
G v W f-
Fr
10001I1 M
Fp
//O is Hadamard product, @ is Kronecker product
14: if u, all converge then
15: break
16: end if
17:
end for
TransPlan u N & M
dw (C, Traanslaan) MK, is Frobenius inner product
20:
Return TransPlan, dwy
Table C.I. Additional Time and Space Analysis Results.
MODEL BATCH SIZE
200
1000
2000
3000
5000
7000
10000
CDVAE (XIE ET AL., 2022)
0.2416
0,4372
0OM
OOM
OOM
0OM
OOM
VISNET (WANG AL., 2024)
0.0285
0.0366
0.0575
0.0772
0.1085
0.1473
0.1931
UNITRUSS (ZHENG ET AL., 2023A)
0.0108
0.01385
0.015
0.01435
0.01406
0.01441
0.0199
C.2. Parameter Sensitivity
Table C.2 provides additional results for parameter sensitivity analysis. Each element in Table C.2 shows the Foond (NRMSEpp [NRMSE& metrics under given latent dimension and number of tokens (i.e., codebook size).
Table C.2. Additional Parameter Sensitivity Analysis Results.
LAT. DIM. NUM. OF TOKEN
32
64
128
32
0. 0.0836,,00032440.000
(.769)00.02880004433
0. 0..042 0.0818\..0282V 0.0202.0.043 00.047
64
0003099 0..004
1.0813),0003000000000
128
0.0795 0.0795....28660.00499
0.0398,0.033110.0000
16


Reviews:
[{"review_id": "ZtYNiMZA8g", "reviewer": "Reviewer_44SV", "summary": "The paper introduces UNIMATE, a unified model for mechanical metamaterial design that simultaneously addresses three key aspects: 3D topology, density condition, and mechanical property. Unlike previous approaches that typically consider only two modalities, UNIMATE integrates all three through a modality alignment module—which compresses and aligns diverse design information into a shared latent space—and a synergetic diffusion generation module that completes missing design tokens via a score-based diffusion model. Experimental results demonstrate that UNIMATE significantly outperforms existing models in topology generation, property prediction, and condition confirmation tasks, offering promising improvements and establishing a new benchmark for comprehensive metamaterial design.", "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": "Thanks for the reviewer's time and inspiring comments. Here, we summarize the major points from the reviewer and our rebuttal as follows:\n\n**Q1:** The proposed model is evaluated on a new dataset and custom metrics.\n\n**A1:** Our work focuses on developing a unified model capable of handling diverse metamaterial design tasks as a whole. This is a relatively underexplored area, and to the best of our knowledge, there has not been a comprehensive effort to address it so far. As a result, no suitable dataset currently exists for this task. Therefore, we propose our own dataset tailored to this problem. Additionally, we introduce custom evaluation metrics to assess the quality of generated structures by referring to previous material works. Specifically, the two metrics—Fqua and Fcond—are inspired by previous works [1–4]. References [1,2] emphasize the importance of symmetry and periodicity in metamaterial design, which directly motivates our Fqua metric that quantifies the degree of symmetry and periodicity in the generated topologies. Similarly, references [3,4] propose comparison techniques for assessing topology similarity and generation coverage. Building on these ideas, our Fcond metric evaluates how well the generated structures match the target conditions by comparing their topological features.\n\n*[1]Abu-Mualla, Mohammad, and Jida Huang. \"A Dataset Generation Framework for Symmetry-Induced Mechanical Metamaterials.\" Journal of Mechanical Design 147.4 (2025).*\n\n*[2]Bitzer, Andreas, et al. \"Lattice modes mediate radiative coupling in metamaterial arrays.\" Optics Express 17.24 (2009): 22108-22113.*\n\n*[3]Tian Xie, Xiang Fu, Octavian-Eugen Ganea, Regina Barzilay, and Tommi S. Jaakkola. 2022. Crystal Diffusion Variational Autoencoder for Periodic Material Generation. In ICLR.*\n\n*[4]Nils ER Zimmermann and Anubhav Jain. 2020. Local structure order parameters and site fingerprints for quantification of coordination environment and crystal structure similarity. RSC advances 10, 10 (2020), 6063–6081.*\n\n **Q2:** The baselines are not originally designed for the tasks in our work.\n\n **A2:** As mentioned in A1, this is a relatively new task, making it difficult to find baselines that are directly aligned with our objective. This challenge is further amplified by the fact that metamaterial design remains largely underexplored within the computer science community. Nevertheless, we conducted extensive experiments by tuning the hyperparameters of existing baseline methods to better assess their potential performance on our task. Here, we focus on the second best model, uniTruss, as an example. uniTruss has two main hyperparameters, learning rate (LR) and latent dimension. As shown in the following, we report the results on four LR and five latent dimensions. Results show that for both property prediction and density confirmation, the best LR is 5e-4 and the best latent dimension under this LR is 32.\n\n(Property Prediction Task) Tuning learning rate, **bold** denotes the results reported in original manuscript.\n| LR | 1e-3 | **5e-4** | 1e-4 | 5e-5 |\n| -------- | -------- | -------- | -------- | -------- |\n| NRMSE_pp |  0.0285    |   **0.0271**  |  0.0420   |  0.0440   |\n| NRMSE_cc |  0.0899    |   **0.0889**  |  0.106   |  0.115   |\n\n(Property Prediction Task) Tuning latent dimension for property prediction\n| Latent dim | 16 | 32 | 64 | 128 | 256 |\n| ---------- |----|----|----|-----|-----|\n| NRMSE_pp | 0.0328 | **0.0271** | 0.0329 | 0.0298 | 0.0472 |\n\n(Density Confirmation Task) Tuning latent dimension for density confirmation\n| Latent dim | 16 | 32 | 64 | 128 | 256 |\n| ---------- |----|----|----|-----|-----|\n| NRMSE_cc | 0.0905 | **0.0889** | 0.0909 | 0.0899 | 0.891 |\n\n\n\n**Q3:** More experimental details about dataset construction and network design might help readers to reproduce the method.\n\n**A3:** We appreciate the suggestion to include more details about dataset construction and network design. We will incorporate the following information into the appendix to improve the clarity and completeness of the paper.\n\n* For dataset construction, we use homogenization simulation to calculate the property of topologies. We will add details about how homogenization works. Also, we will add details about density selection, e.g., the range from which the edge radius is selected.\n* For network design, we will add more details about each component of our model, including the layer number of the GCN encoder, the MLP encoder, the transformer backbone, the latent dimension of each component, etc. We hope this will help increase the clarity and reproducibility of our work.", "claims_and_evidence": "1. The integrated approach is supported by experiments showing notable performance gains ( In the topology generation task, property\nprediction task, and condition confirmation task, our model outperforms the second-best model by 80.2%, 5.1%, and\n50.2%, respectively), but it relies on a new dataset and custom metrics. \n\n2. The unification and the alignment operation both benefit to the model’s performance. The unification process boosts the performance by 37.5% in average, and the TOT alignment provides 7.8% boost.", "methods_and_evaluation_criteria": "Yes, the quantitive metrics and the ablations are convincing.", "theoretical_claims": "I believe this is an application paper that employs multimodal alignment and generation for metamaterial synthesis, so it does not propose any specific theoretical novelty.", "experimental_designs_or_analyses": "1. The authors construct a new dataset (based on Lumpe & Stankovic, 2021) and introduce domain specific metrics (F_qua for topology quality and F_cond for topology matching) to evaluate their model. While these enable comprehensive testing, I would recommend that AC consider having domain experts in material design review its validity\n\n2. They compare UNIMATE against multiple models across three tasks (topology generation, property prediction, condition confirmation). For some tasks, especially condition confirmation, they adapt models not originally designed for that purpose (e.g., forcing property prediction models to predict density). This adaptation might introduce bias, so while the comparisons are informative, they might not fully reflect each baseline’s intended performance.", "relation_to_broader_scientific_literature": "None", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strength:\n1. Multimodal Alignment: Uses a multimodal codebook to map heterogeneous data (3D topology, density, and mechanical properties) into a unified latent space.\n2. Synergetic Generation: Introduces a partially frozen diffusion process with a transformer backbone that generates missing tokens while preserving the context of known tokens.\n3. Flexible and Unified Design: Efficiently bridges diverse modalities, enabling robust metamaterial synthesis even with arbitrary missing inputs.\nWeakness:\n1. One potential weakness of the paper is the limitation in data scale. The approach relies on a newly constructed dataset, which could restrict the model's generalizability and robustness in real-world scenarios.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "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": "MCjjDIYYlz", "reviewer": "Reviewer_f6U7", "summary": "In this paper, the author proposed UNIMAE, a unified model that can tackle three tasks simultaneously, namely, the topology generation task, property prediction task, and condition confirmation task, by training a shared, aligned latent space using a novel TOT and frozen diffusion. In the three tasks, the UNIMAE shows 80.2%, 5.1%, and 50.2% higher performance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their time and constructive feedback. Please find our detailed responses and corresponding revisions below:\n\n**Q1:** One point in the work that \"it will be simpler...the generation process\" is not proven in later experiments.\n\n**A1:** Thank you for your detailed reviews. To address this issue, i.e., the effectiveness of using a codebook for diffusion starting point initialization, we conduct additional experiments (random noise initialization and mixed initialization), with detailed results reported in reply to Q3 from reviewer 79DH.\n\n**Q2:** Ablation for partially frozen diffusion is needed.\n\n**A2:** We appreciate your constructive suggestion. Accordingly, we compare the use of a partially frozen diffusion with a vanilla diffusion setup (i.e., where variables in all dimensions are denoised at each step). The results below verify the effectiveness of partially frozen diffusion. We will include the results in the final version of the manuscript.\n\n|                            | Fqua   | Fcond  | NRMSE_pp | NRMSE_cc |\n|----------------------------|--------|--------|----------|----------|\n| Vanilla Diffusion          | 0.0405 | 0.0813 | 0.0322   | 0.0423   |\n| Partially Frozen Diffusion | 0.0274 | 0.0781 | 0.0244   | 0.0443   |\n\n**Q3:** Further ablation studies are needed, such as the size of the code book and different soft-round strategies.\n\n**A3:** Thanks for the valuable suggestion. According to your suggestion, in addition to the varying codebook sizes experiments conducted in Section 4.4. Below, we provide more results using a soft-round strategy (midpoint interpolation between the initial value and the target rounding token), which shows the effectiveness of codebook rounding.\n\n|                   | Fqua   | Fcond  | NRMSE_pp | NRMSE_cc |\n|-------------------|--------|--------|----------|----------|\n| Soft Round | 0.0421 | 0.0787 | 0.0369   | 0.0449   |\n| Codebook Round | 0.0274 | 0.0781 | 0.0244   | 0.0443   |\n\n**Q4:** It is unclear why Fqua and Fcond are valid metrics for evaluating generation quality; generated topologies are not validated for effectiveness and manufacturability.\n\n**A4:** Thanks for the detailed reviews. Our evaluation metrics are inspired by and adapted from prior literature that adopted similar concepts. Specifically:\n* Fqua evaluates the quality of generated topologies by measuring their symmetry degree and periodicity.\n* Fcond evaluates how well the generated topologies satisfy the input conditions by comparing them with ground truth topologies that fully match those conditions.\n\nPlease also refer to Reviewer 44SV **A1** for more detailed illustrations. We will include these detailed illustrations in our final version.\n\nFurthermore, we have collaborated with a metamaterial lab to physically manufacture the generated topologies and test their properties. We plan to release a demonstration video showcasing this process after the double-blind review period concludes.\n\n**Q5:** Why the author only compares three baselines in time and memory efficiency analysis?\n\n**A5:** We appreciate your careful reviews. In the model efficiency analysis section, we present the results for three representative baselines for clarity and brevity. Below, we provide the complete set of results for all baselines, from which we can reach the same conclusion as stated in our manuscript, and we will include these in the appendix of the revised version.\n| Model\\Batch Size | 200    | 1000    | 2000   | 3000    | 5000    | 7000    | 10000   |\n|------------------|--------|---------|--------|---------|---------|---------|---------|\n| CDVAE            | 0.2416 | 0.43716 | OOM    | OOM     |       OOM  |   OOM      |       OOM  |\n| VisNet           | 0.0285 | 0.0366  | 0.0575 | 0.0772  | 0.1085  | 0.1473  | 0.1931  |\n| UniTruss         | 0.0108 | 0.01385 | 0.015  | 0.01435 | 0.01406 | 0.01441 | 0.0199  | \n\n**Q6:** In parameter sensitivity, why do the authors only use the metric Fqua?\n\n**A6:** We thank you for the constructive feedback. Accordingly, we include the parameter sensitivity analysis w.r.t. all metrics. In the sensitivity study, since we consider the topology generation task to be relatively hard among the three tasks, and Fqua is a typical indicator of the generation quality, we provide the results related to Fqua. Below, we provide results for the other evaluation metrics as well. These will be added to the appendix in the final version.\n\nParameter Sensitivity regarding other metrics (Fcond/NRMSE_pp/NRMSE_cc)\n| Lat. Dim\\Codebk Size | 32                   | 64                   | 128                  |\n|----------------------|----------------------|----------------------|----------------------|\n| 32                   | 0.0836/0.0324/0.0402 | 0.0769/0.0288/0.0436 | 0.0833/0.0294/0.0419 |\n| 64                   | 0.0779/0.0309/0.0404 | 0.0813/0.0300/0.0409 | 0.0783/0.0295/0.0442 |\n| 128                  | 0.0795/0.0286/0.0409 | 0.0798/0.0331/0.0430 | 0.0818/0.0282/0.0437 |", "claims_and_evidence": "Most of the claims made in this paper are supported by clear evidence.\n\n1.\tThe author claims that it is a unified model for different tasks in mechanical Metamaterial generation, property prediction, and condition confirmation, which is proven in Table 2\n\n2.\tThe author claims that the aligned latent space is helpful in handling challenge 1 (mechanical metamaterial design involves three modalities with different formats and distributions.). This is supported in the ablation study case 2\n\nHowever,\n1.\t I noticed that from line 267 to line 269, the author claims that “it will be simpler to generate proper outputs, which increase the robustness of the generation process.”, which is not proven in later experiment\n\n2.\tThe author claims that using frozen diffusion to handle the challenge 2 (given the complexity of data, any part of the data can be missing and requires design effort. Therefore there can be various design tasks). But no ablation study supports this.", "methods_and_evaluation_criteria": "The paper proposed a generalized Sinkhorn algorithm by generalizing the transport plan iteration process for the alignment of more than two latent distributions and also uses partially frozen diffusion to tackle potentially different tasks such as mechanical metamaterial generation, property prediction, and condition confirmation with the unified model. These make sense.\nAs for the evaluation metric, I have some concerns about the validation of the metamaterial generation. It is unclear why Fqua and Fcond are valid metrics for evaluating generation quality. Maybe a FEM simulation is needed to validate the generated metamaterial.", "theoretical_claims": "The main theoretical claims made by the author are \n\n1.\tThe round operation of latent tokens to the codebook, which is shown from equations 1 to 6.\n\n2.\tthe proposed generalized Sinkhorn algorithm by generalizing the transport plan iteration process, which is shown from 7 to 9 and algorithm 1 in the appendix.\n\n3.\tThe partially frozen diffusion, which is shown from equation 10 to 12.\n\nThey are all correct.", "experimental_designs_or_analyses": "Yes, I review all of the experiments.\nThe main experiment conducted by the author can be summarized as follows:\n\n1.\tThe main results are shown in Table 2, where different baselines for different tasks are compared with the proposed unified method.\n\n2.\tThe time and memory efficiency comparison between the proposed method and the baseline models. But not clear why the author only compares three baselines instead of all the baselines mentioned in Table 2.\n\n3.\tChoose the latent dimension and the code book size by the Fqua, but it is not clear why choose these two hyperparameters by only one evaluation metric.\n\n4.\tAblation study shows the effectiveness of the latent MTR alignment, but additional ablation study of partially frozen diffusion is needed.", "relation_to_broader_scientific_literature": "Typically, the feature alignment is done for two modalities. This paper proposes a method to align features from three modalities and is capable of generalizing to more modalities. This can be very useful in other applications.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tThis paper introduces a novel multi-modality alignment method that extends beyond conventional two-modality approaches.\n\n2.\tIt effectively integrates diffusion-based generation with selective token freezing, enabling flexible handling of multiple tasks.\n\nWeakness:\n\n1.\tFor topology generation, it is unclear why Fqua and Fcond are valid metrics for evaluating generation quality. A more detailed explanation is needed.\n\n2.\tAdditionally, the generated topology does not appear to have been validated for effectiveness and manufacturability using numerical simulations. Demonstrating its practical applicability is crucial.\n\n3.\tFurther ablation studies are needed such as the size of the code book, and different soft-round strategy", "other_comments_or_suggestions": "Further ablation studies are needed such as \n\n1.\tthe size of the code book and different soft-round strategies: I noticed that in parameter sensitivity, the author provides the Fqua VS. the codebook size and the latent dimension, but what about another evaluation metric? Why determine the hyperparameter with only one single evaluation metric?\n\n2.\tThe ablation study on partially frozen diffusion is missing.", "questions_for_authors": "See weakness", "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": "YojpSyCfHe", "reviewer": "Reviewer_79DH", "summary": "In this work, the authors propose a method that jointly solves topology generation, condition confirmation, and property prediction. Their approach consists of two stages: first, using three sets of encoders and decoders to embed different modality conditions into discrete latents; second, employing a diffusion/score-matching model to generate masked latents from unmasked ones. Experimental results demonstrate significant improvement over the baselines across all three tasks, with ablation studies validating the effectiveness of the proposed modules.\n\n## update after rebuttal\nThe rebuttal has addressed my minor concerns, and I will maintain my rating.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the reviewer’s time and constructive comments. We sincerely appreciate the opportunity to address the concerns and clarify our work.\n\n**Q1:** Similar concepts have been explored in an image-text work (i.e., Janus, which is mentioned by the reviewer).\n\n**A1:** We appreciate the reviewer for pointing out the paper Janus, a multimodal understanding and generation model. While we acknowledge that our work and Janus may share high-level conceptual similarities, there are significant distinctions between the two:\n* **Task Focus**: Janus is designed for general vision-and-language understanding tasks, whereas our model specifically targets metamaterial design, a domain with unique challenges and goals not addressed in Janus.\n* **Methodology**: Janus integrates image information by projecting it into the input space of a large language model (LLM), combining it with textual data for joint understanding. In contrast, our approach projects all three modalities into a shared embedding space that is not biased toward any single modality. This unified representation is better suited to the nature of metamaterial design problems.\n\n**Q2:** The reviewer was not sure whether any important baselines are missed in our work.\n\n**A2:** In our experiments, we have carefully selected representative baselines to provide a fair and comprehensive comparison. To the best of our knowledge, the models we included reflect the current state-of-the-art computer science works in the field of metamaterial design. Still, we are open to including additional baselines should we find any relative suggestions.\n\n**Q3**: For masked tokens, is the marginal distribution initialized directly as a Gaussian distribution?\n\n**A3:** We thank the reviewer for the question regarding token initialization. In our implementation, we initialize tokens based on the transport plan, assigning higher probabilities to tokens with higher values in the transport plan. In response to the reviewer's question, we explore several alternative initialization strategies: pure Gaussian noise, a mixture of transport plan-based initialization, and Gaussian noise. The results of these variants are presented below.\n\n|                   | Fqua   | Fcond  | NRMSE_pp | NRMSE_cc |\n|-------------------|--------|--------|----------|----------|\n| Rand. Noise Init. | 0.0291 | 0.0795 | 0.0304   | 0.0448   |\n| Mixture Init.     | 0.0525 | 0.0814 | 0.0313   | 0.0446   |\n| Trans. Plan Init. | 0.0274 | 0.0781 | 0.0244   | 0.0443   |", "claims_and_evidence": "- This work appears to be the first to propose a unified formulation capable of handling multiple tasks, as shown in Table 1.\n- The approach of embedding different modality inputs into a common latent domain for generation is innovative, though similar concepts have been explored recently in image-text applications [1].\n- The manuscript's thorough evaluation, benchmarking against existing baselines, and open-source models provide strong evidence for result reproducibility.\n\n[1] Janus: Decoupling Visual Encoding for Unified Multimodal Understanding and Generation", "methods_and_evaluation_criteria": "Overall, the proposed method has achieved significant improvement over all baselines on different tasks, including topology generation, property prediction, and condition confirmations. The ablation study has shown the effectiveness of joining encoding of latent spaces and alignment operation.", "theoretical_claims": "As far as I know, this work does not provide a strong theoretical claim on the proposed framework.", "experimental_designs_or_analyses": "The experiment setting looks reasonable to me, and the experiments show improvement on all tasks over existing baselines. However, since I am not a domain expert for mechanical design, I am not sure if some important baselines are missed in this case.", "relation_to_broader_scientific_literature": "Nil.", "essential_references_not_discussed": "Nil.", "other_strengths_and_weaknesses": "Nil.", "other_comments_or_suggestions": "Nil.", "questions_for_authors": "I have one minor question about the inference process, specifically regarding the computation of the transport plan. For masked tokens, is the marginal distribution initialized directly as a Gaussian distribution? This point could use some clarification in future revisions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The submission introduces a novel method that tackles multiple tasks at the same time: topology generation, property prediction, and condition confirmation.  Reviewers all liked the submission based on its technical contributions and experimental results.  Some concerns about the metrics were also addressed during the rebuttal phase.  The AC would like to follow the recommendation of reviewers and suggest acceptance.
Decision: Accept (poster)