FACET: A Fragment-Aware Conformer Ensemble Transformer
Authors: Duy Minh Ho Nguyen
Trung Quoc Nguyen
Ha Thi Hong Le
Mai Thanh Nhat Truong
TrungTin Nguyen
Nhat Ho
Khoa D Doan
Duy Duong-Tran
Li Shen
Daniel Sonntag
James Zou
Mathias Niepert
Hyojin Kim
Jonathan E Allen
Venue: ICLR 2026 Poster (2026)
Keywords: molecular properties prediction
3D conformers
graph transformer
2D-3D fusion
fragment aware module
Fused Gromov-Wasserstein distance
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Accurately predicting molecular properties requires effective integration of structural information from both 2D molecular graphs and their corresponding equilibrium conformer ensembles. In this work, we propose FACET, a scalable Structure-Aware Graph Transformer that efficiently aggregates features from multiple 3D conformers while incorporating fragment-level information from 2D graphs. Unlike prior methods that rely on static geometric solvers or rigid fusion strategies, our approach utilizes a differentiable graph transformer to theoretically approximate the computationally expensive Fused Gromov–Wasserstein (FGW), enabling dynamic and scalable fusion of 2D and 3D structural information. We further enhance this mechanism by injecting fragment-specific structural priors into the attention layers, enabling the model to capture fine-grained molecular details. This unified design scales to large datasets, handling up to 75,000 molecules and hundreds of thousands of conformers, and provides over a 6× speedup compared to geometry-aware FGW-based baselines. Our method also achieves state-of-the-art results in molecular property prediction, Boltzmann-weighted ensemble modeling, and reaction-level tasks, and is particularly effective on chemically diverse compounds, including organocatalysts and transition-metal complexes.

Published as a conference paper at ICLR 2026
FACET: A FRAGMENT-AWARE CONFORMER ENSEMBLE TRANSFORMER
Duy M. H. Nguyen 1,23 1223 Trung Q. Nguyen
Ha T. H. Le
Mai Thanh Nhat Truong*
TrungTin N
Nhat Ho
Khoa D Doan? Duy Duong-Tran'
Li Shen
Daniel Sonntag
James Zou
Mathias Nieperth Hyojin Kim
Jonathan Allen
Max Planck Research School for Intelligent Systems (IMPRS-IS)
University of Stuttgart
German Research Center for Artificial Intelligence (DFKI)
4 ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems
Schooo of Mathematical Sciences, Queensland University of Technology
University of Texas at Austin 'VinUniversity $University of Pennsylvania
'Oldnnuur University Stanford University LLawrence Livermore National Labs
ABSTRACT
Accurately predicting molecular properties requires effective integration struc- tural information from both 2D molecular graphs and their corresponding equilib- rium conformer ensembles. In this work, we propose FACET, a scalable Structure- Aware Graph Transformer that efficiently aggregates features from multiple 3D conformers while incorporating fragment-level information from 2D graphs. Un- like prior methods that rely on static geometric solvers rigid fusion strategies, our approach utilizes a differentiable graph transformer to theoretically approxi- (FGW), enabling
mate the computationally expensive Fused dynamic and scalable fusion of 2D and 3D structural information. We further enhance this mechanism by injecting fragment-speciff structural priors into the attention layers, enabling the model to capture fine-grained molecular details. This unified design scales to large datasets, handling up to 75,000 molecules and hun- dreds of thousands of conformers, and provides over a 6x speedup compared geometry-awan FGW-based baselines. Our method also achieves state-of-the-an ensemble model-
results in molecular property prediction, mootzzannn-weighaee ing, and reaction-leve tasks, and is particularly effective on chemically diverse compounds, including organocatalysts and transition-ne metal complexes. We pro- vide implementations at this link: Code implementation
INTRODUCTION
Machine learning has become powerful tool for predicting molecular properties, with wide-ranging
applications drug discovery and materials science (Choudhary et al. 2022, Fedik et al. 2022
Batatia et al. 2023 Most existing models rely either on 2D moleculat lar graphs which efficiently capture topologiaa connectivity (Xu et al.] [2018] Veličković et al.] 2018). or on 3D representations derived from single conformer (Schütt al.) 2017 Batzner et al. 2022 Batatia al., 2022). While 2D graphs are computationally efficient, they lack geometric information that is often critical for accurate property prediction. Incorporating 3D conformers helps address this by introducing spatial features such as bond lengths, and torsion angles. However, relying on single conformer still fails capture the intrinsic flexibility of molecular structures.
In reality, molecules dynamically sample a range of thermodynamically accessible conformations due bond rotations, vibrations, and environmental interactions (Ramsundar et al. 2019). As a result, many experimentally observable properties such as solubility and binding affinity depend on the full ensemble of conformers a molecule can adopt (Perola & Charifson 2004). Yet, fully modeling this distribution is computationally prohibitive, as generating and evaluating large numbers of conformers using quantum methods is costly (Medrano Sandonas et al. 2024). This has motivated hybrid models that combine the structural efficiency of 2D graphs with the geometric richness of small and representative subset 3D conformers. By jointly capturing topological and spatial variation, hybrid models offer scalable and expressive frameworks for molecular representation, enabling more accurate prediction of conformation-sone sensstive properties across a range chemical and biological tasks.

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Building this hybrid paradigm, recent methods have introduced hybrid models that integrate 2D molecular graphs with 3D conformer information to capture both topological and spatial features (Zhu et al., 2024b; Axelrod & Gomez- Bombarelli 2023). Despite the successes, these methods often assume conformers contribute equally or can be reweighted without considering deeper geo- metric context. In practice, only subset of conformers may be thermodynamically
functionally relevant, and naive aggregation overlooks their spatial relationships, such as alignment or structural similarity. Moreover, current strategies rarely leverage interactions between 2D structural priors and 3D conformational variability, hindering the formation of truly expressive representations.
To address this, structure-aware ensemble methods based on optimal transport, especially those using fused Gromov-Wasserstein Wasserstein (FGW) alignment, have shown promise Ma et al. ( Nguyen et al. 2024a). By aligning both feature and geometric spaces, these models better preserve spatial correspondences across conformers and enable expressive ensemble aggregation. However, such methods are computationally expensive and struggle to scale to large molecular datasets such as Drugs-75k Zhu et al. (202);; Axelrod & Homez-Bombarell) (2022), limiting their utility for high- throughput applications in generative biology.
To address scalability challenges in geometry-aware molecular modeling, we introduce a novel ap- proach that replaces expensive FGW alignment with efficient attention-based conformer aggrega- tion. By supervising the model with FGW distances during training. we learn a latent embedding space where conformer similarities reflect both topological and geometric structure. This enables fast, permutation-invant yond efficiency, we further enrich our model with fragment-level structural priors from 2D molecu- lar graphs, injecting chemically meaningful hierarchies into both message passing and 3D attention layers. This unified 2D-3D framework captures fine-grained spatial and topological interactions essential for applications such as molecular property prediction, virtual screening, and functional optimization. In summary, our key contributions are:
conformer integration suitable for large-scale generative pipelines. Be-
We propose a scalable, geometry-aware conformer aggregation framework, denoted as FACET, that replaces costly FGW alignment with trainable Graph Transformer, enabling efficient, deterministic attention-bann based inference. We further provide theoretical bounds on the approximation error relative to FGW distances.
We introduce unified 2D-3D representation learning approach that embeds fragment- level structural priors into both 2D message passing and 3D spatial self-attention, captur- ing multi-scale interactions between molecular topology and geometry.
Our method delivers over 6x faster aggregation than prior geometry-ave aware baselines and achieves fate-of-the-ar performance across six benchmarks, including molecular prop- erty prediction and Boltzmann-weightee ensemble tasks, demonstrating robustness across diverse molecular scenarios and dataset scales.
RELATED WORK
Conformer Ensemble Learning in Molecular Representations. Traditional molecular repre- sentations span connectivity fingerprints (Morgam) 1965). ID string encodings (Ahmad et al. [022) Wang et al. ) 2D topological graphs (Yang et al. 201 9a Rong et al. (020)) and 3D geometric graphs (Fang et al. 2021 /hou et al., 2023). 3D models typically rely a single conformer, over- looking the fact that molecules often adopt multiple low-energy conformations, which can serve as informative features, particularly in capturing thermodynamic properties. Hybrid approaches now combine 2D graphs with ensembles of 3D conformers (Zhu et al. 2024b Wang et al., 2024), ag- gregated via mean pooling, DeepSets (Zaheer et al.) 2017), or self-attention (Vaswani et al. 2017). More advanced geometry-aware methods based on Fused
-asso--Wassrstteinn (FGW) alignment (Ma et al. 2023 Nguyen al. 2024a) capture both feature and structural similarity across con- formers, but remain computationally costly and scale poorly to large datasets (e.g., Drugs-75k) or foundation models (Zhou et al. 2023] Chithrananda
al.) 2020). To address this, we propose scalable framework that learns latent embeddings of 3D conformers with graph transformers, in- tegrating geometry-aware signals inspired by FGW and hierarchical fragment-level features. This yields sermutationiivaaiane expressive, and efficient method.
Scalable Optimal Transport for Graph Learning. Learning-bar based approximations of Optimal Transport (OT) have emerged as efficient alternatives to traditional solvers. Early works introduced
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differentiable Sinkhorm distances with entropic regularization for stability and scalability (Cuturi 2013 Feydy et al. 2019. Genevay et al. 2018). Later methods improved efficiency via structural
assumptions e.g.. low-rank factorization (Scetbon et al. 2021] Cuturi et al. 2020) and spatial ge- ometry (Bachmann et al. (022 Solomon et al. | 2015). Meta-leaming approaches further accelerated
convergence by learning initialization schemes (Amos et al.) 2023). More recently, neural OT sur-
rogates trained directly on data have bypassed iterative solvers entirely (Courty et al. 2017 Tong
al. 2021
Haviv et al. (024).
However, prior works focus on standard OT and fail to extend to structure-aware variants like FGW, which jointly capture node attributes and graph topology. To address this, we introduce the first leamned approximation of FGW with a graph transformer, enabling scalable, geometry-awan con- former aggregation. By embedding fragment-level priors into both 2D and 3D encoders, our ap- proach supports multi-scale reasoning across topological and spatial hierarchies, effectively bridging molecular graphs with 3D conformational diversity.
Fragment-biases in Molecular GNN. Fragment-level substructures such rings, functional groups, and pharmacophores are key to molecular property prediction and drug design (Merlot al. 2003 Varnek al. 2005). Recent works have leveraged these motifs for scaffold-aware drug discovery (Lee et al. [024 Chan et al. 2024). self-supervised learning via fragment- based masking or contrastive tasks (Rong et al. 2020; Zhang al. [202]] Wen et al.) 2024). and GNN architectures that encode fragment-level inductive biases (Wang et al. 2025 Wollschläger et al. 2024). These methods show that fragments enhance generalization, interpretability. and data efficiency. Building on these insights, we explore complementary direction: integrating fragment-leve priors into hybrid 2D-3D ensemble models. In our approach, fragment hierarchies are embedded into both 2D meessage-passnng and 3D spatial attention layers, enabling multi-scale processing across molecular topology and geometry. This design improves conformer aggregation and yields more expressive, geometry-awan representations suited for conformation tasks.
3 FRAGMENT-AWAE CONFORMER ENSEMBLE TRANSFORMER
Notation. Let AN == (w B w W = I} denote the probability simplex, where 1N is the Q. 8r is the Dirac measure at z. We write [K] == {l...., K}
all-ones vector in R., For for K E N. and use (..)) to denote the Frobenius inner product. For a tensor L = (Lijkt) and matrix B
(Bι)), define the contraction L @ B (Eu BB))) A graph G (V, has N == [V| nodes and edges E C (f C V u # v). An attributed graph is given by G == (H, (H,A,w) where H E Nxd is the node feature matrix (with row H. for node v),
A E 2 encodes structure (e.g., adjacency or shortest-path distance matrix), and w E AN is node weight distribution.
Given two graphs Gj and G2 with N1 and No nodes, the Fused Gromov- Wasserstein (FGW) distance (Peyré et al. 2016 Titouan et al. 2019: 2020) is: FGW p.a. (91, G2) == millxel(ww) (oz) (11-a) M + a L(A,, A2) @ ग., T)) where II( W2) = { E IR XN2 : TIN; W1. π 1N,,
H H2)) is the distance between feature of node i in G1 and of node j in G2. L(A1, Ae)
w2) is the set of valid couplings, M Mi. jl
= [ Azl1, mz]l captures structural mismatch, and [0, 1] balances feature and structure alignment. Consider set of K graphs (9 ( the FGW barycenter graph & is the graph that has the smallest distances to other graphs the set: G == arg ming Ek= XgFGW 9&)
3.1 CONFORMER GENERATION
Following prior work, we generate molecular conformers using distance geometry methods that convert interatomic constraints such as bond lengths, angles, stereochemistry, and steric limits- into 3D coordinates (Hawkins) 2017). A lightweight force field refines the structures toward low- energy conformations. Compared to quantum methods like DFT, this approach is highly scalable and efficient for large datasets. As in prior studies (Raza et al., 2022 Nguyen et al. 2024b). RDKit (Landrum) 2016) for fast and reliable conformer generation.
3.2 FRAMEWORK OVERVIEW
We propose neural architecture with three main components (Fig. D.. First, 2D message pass- ing neural network (MPNN) captures molecular topology, while another 2D-MPNN operates on a fragment-level graph, which consists of pairwise edges between fragment nodes, to encode higher-
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2D MPNN
Aparogation
Fused Gromev-Wassenee Wosserstein
Fragment Graph
20 MPNN
Edga encoding
Adaptor
Mikeal
Adaptor
HD MPNN
Conformet Generation and Sumply
Boole
Tp
Motrol
Stractural
Input Graph
Aggregation
Fragment-awars Graph Transtormer
pD MPNN
Downstream Losa
and Eageeett
Figure I: FACET overview. The model takes 2D molecular graph and its fragment graph, en- coded by separate 2D-MPNNs and aligned via fragment- atom correspondence (dashed boxes). In parallel, multiple 3D conformers are sampled and encoded by shared 3D-MPNN (f). The re- sulting 2D/3D features and edge information are adapted and fused through a frozen fragment- aware graph attention module (Ta), pre-trained (right) using graph transformer supervised by Fused Ciromo--Wasserster (FGW) distances to preserve FGW geometry (Sec. 5.I). The fused, geometry-awan representations are then used for downstream prediction. "Fire" and "snowflake" icons denote trainable and frozen components, respectively.
order structural priors (Sec. 3.3 Their outputs are fused and refined through a lightweight adaptor module before entering pre-trained FGW-guided graph transformer (Sec. 3.4 For 3D informa- tion, a set of conformers is sampled from the input molecule, and 3D-MPNN extracts conformer embeddings (Sec. 3.4.1), which are also calibrated by an adaptor layer handle variability between 2D and 3D features. Then, conformer embeddings are fed into the graph transformer, where each node attends to all other nodes, taking into account the conformers graph structure and fragment- level information (Sec. 3.4.2). In essence, the graph transformer encodes conformer embeddings into another space where their pairwise Euclidean distance equal FGW distance (Sec. 3.4.3). Fi- nally, a permutation- and E(3) invaarant fusion module unifies the 2D and 3D features into single embedding for downstream tasks (Sec. 3.4.4).
3.3 GRAGMENT-ENNANEE
2D MOLECULAR GRAPH
Each molecule represented as 2D graph = (V, E), where nodes V correspond to atoms and edges E to covalent bonds. Atom features h
(0) E Rd encode properties like atom type and valence, while bonds (u, v) are annotated with features e(u, v) (Scarselli et al. 20088) Gilmer et al. 2017). We adopt a 2D message-passing neural network (MPNN) that updates node embeddings layer-wise:
h UPD' (h AGG'
AGG'(MMML3 (h h ev,u) u N(v))),
(1)
where M is a message function, AGG' sum aggregation, and UPD is identity or multilayer per- ception layers. We use Graph Attention Networks (GAT's) (Veličković et al. 2017), where messages are computed as:
M of Wh f-1 o.,, = softmax, (LeakyReLU (7 ("('")) wh))"))
(2)
After L layers, we obtain final atom-level features hy for each atom used for downstream tasks.
Fragment-Based Structural Augmentation. To enhance atomic representations with higher- order structural context, we construct a fragment-level graph from the input molecular graph G using ring-path decomposition (Kong al. 20222] Geng et al. 2023 Wollschläger al., 2024) to identify key substructures such as aromatic rings and functional groups (Fig. 5). Each fragment is treated node new graph frag (Viing Effy), where nodes correspond fragments and edges are induced from the connectivity in G. two fragments are connected if they share an atom or are directly bonded. In this work, we specifically follow the approach proposed in (Wollschlägen et al. 2024). as it offers good balance of simplicity and effectiveness for our use case.
We apply the same GAT formulation in Eq. (T) to the fragment graph to obtain fragment embeddings frng
{h fEViL. Then for each atom e that belongs to its fragment fu), we fuse their atom-level
representations h." (1) with (h] frag by:
(()) = (L) FFN (h()))
(3)
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where (-N is learnable feedforward network that projects fragment- context into the same space as atom features. Finally, we define a fragment- enhanced graph level representation that is computed by applying a readout function hæp READOUT ((b!)) v E Evev h
Intuitively, the dual-level encoding combining local atomic features and global fragment-level con- text as Eq.(3) allows the model to reason over both fine-grained and coarse-grained structures, enhancing the expressivity of the molecular representation.
3.4 LEARNING GRAPH TRANSFORMER FOR 3D MOLECULE AGGREGATIONS
S {r,, Z = where N denotes the number of atoms, r; E R corresponds to the 31) Cartesian coordinates of atom i, and Zr E N indicates its atomic number.
A molecular conformer is represented
3.4.1 3D conformer feature representation. For each conformer S, can define its graph Gs and compute its 3D feature embedding by using the geometric MPNN SchNet (Schüitt et al. 2017). though other -i3)-invaraant neural architectures can be readily substituted without modification (Table 2) We represent the matrix of atom-level features from the final message-passing layer L of SchNet as H, where each column H[u] corresponds to the feature vector h of atom v. We
3d. + byd Rd with Waa and byd are learnable vectors. Given a set of K conformers (Sx 1K R==1 we define Haa [k] = hap.s, as the feature embedding of the k-th conformer. The matrix Hgd E RKxd thus summarizes the feature representations of all conformers in the set.
then compute the vector representation of a conformer S 33g,5 Evev (Waa) h (L)
3.4.2 Fragment-awas Graph Transformer. Given the atom-wise feature matrix H for each conformer S, we aim to learn structure-encoded latent representations using Graph Transformer architectures (Ying al. 2021 Kreuzer et al., 2021 Luo et al. 2024 We adopt the architecture from Ying et al. (022)) due to its strong expressiveness on small molecular graphs, and further extend is atlention mechanism with fragment sub-structures (Fig 5). It important note that framework is flexible and can incorporate alternative transformer- based models.
In partieular, we compose N transformer layers (Vaswani al., 2017), each consisting of a self-attention mechanism followed by a position-wis feed-forward network. Given H [h Ih....., TT E Iand computed in Section 3.4.1 by a 3D-MPNN, where h, 3d,, E Rildd is the vector embedding of an atom v; with d dimensions. We compute self-attention, by linearly pro-
(I.)
jecting H into query (Q), key (K), and value (V) matrices using learned weights Wq, WK. Wv E Rdxd
H-HWQ Q: K HWK, = HWv, À QK /Va, Attention = softmax (oottaxxA)V.. (4)
Here, A denotes the attention score matrix representing pairwise similarities between tokens. For clarity, we present the single-head version; extending to multi-head attention is straightforward. Bias terms are omitted for brevity.
While the attention in Eq. 4 operates only on feature nodes, leveraging the structural information of the 3D conformer graph is essential. Following Ying al. (202I). we incorporate (i) centralit encod i nng which measures the importance of node in the graph via its degree, and (ii) spat ial encodi ing which captures the spatial relation between two nodes U; and fy in Gs using the short- est path distance (SPD) (Cormen et al. 2022 Balaban 1985). augmented with a learnable weight assigned to each edge along the SPD. Specifically, we incorporate (i) by:
(5) where: 27, 2T E Rd are learnable embedding vectors indexed by the indegree deg (v:) and out- degree dw;) of atom i; respectively. For (ii), the shortest-path distance (SPD) matrix is first computed, and these distances are used to retrieve the corresponding embeddings, which are then integrated into the attention mechanism to inject topology-awan structural bias. Assume Ai as the (i, -element of the Query-Key product matrix A. the condition (ii) extends Aij as:
h; h, + (w:) (wz)"
Äsj (h; W (h) W / So(m.,y)) + Cij.
(6)
where So(v. is a learnable scalar indexed by the SPD distance 6(v;, 2z)) and shared across layers; Gj B(xe, (w (wFn))) )), with Se., the feature of edge en in SPDij, w IRde its weight embed-
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ding, and dg the dimensionality of edge features, computed the difference between the feature embeddings of its incident nodes.
While the spatial encoding in 6 is implicated by the SPD, argue that this might inadequately capture chemically meaningful substructures (ablation Tab. D. This motivates us to extend at- tention scores in Eq. (6) using values derived from (iii) Fragment-lever
node features computed on 2D topology graph in Eq. 3 directly guiding attention toward structurally and func- tionally relevant regions such as rings, functional groups, or scaffolds. To this end, we compute an adjacency- matrix AG) using cosine distance over the final node embeddings h!!") Specifi- cally, for each pair of atoms (vt, U2) ) in the 2D molecular graph, we define
(6 h ((0))
A())
h 1 12- 12
which quantifies their directional dissimilarity the embedding space. Finally, we compute the attention score as:
Ai (h; Wh) (hxk"// so(u..)) + Cij A(G)
(8)
3.4.3
Learning to Approximate FGW distance. We denote (.) as the graph transformer
model whose attention operation is Eq.(8) our goal is to train (.(.) to map the feature representation of each conformer S into a latent space where the L2 distance between any pair St, Sy approximates their FGW distance an effective, yet computationally expensive, geometry-aware metric (Ma et al.) 2023 Nguyen et al. 2024a). To this end, given set of Ω {S of K generated conformers, we sample B conformers from S2 then compute their encoding features by Ta (H.) for each S; E B. These outputs are compared with their pair-wiss FGW distance to optimize the loss:
Lene Σ (H.) To H )
FGW PGWpa(((())
By minimizing the loss Lenes we update the parameters of the transformation module Ta(:) using gradient descent: O <- 0 - Lene- Once trained, freeze To and incorporate it back into the framework to compute a geomelry-aware representation across K conformers (Sx {$ as follows:
H = E ((T(H.))) where H denotes the aggregated structural embedding. Intuitively, H acts the feature embedding of the FGW barycenter in the latent space (see Notation begin of Sec D.. It represents the geometric mean of the input conformers, taking into account both their structural characteristics and features. However, the 3D conformer feature distribution, extracted by 3D-MPNN, used to train Lene (Eq, 99) may experience domain shift when co-trained with other components in the full framework (Sec. (344) due to the continuous updating of 3D-MPNN. To ad- dress this, we design adapter layers simple FFN layers to transform the input features in Eq. (9) aligning them to the seen distribution during training Tg.
3.4.4 Invariant Aggregation of 2D and 3D Representation. We integrate representations from the 2D molecular graph and multiple 3D conformers using both average pooling and GraphTrasfformer- based aggregation. The transformer captures rich spatial interactions while en- suring permutation invariance across conformers and H(3) equivariance, preserving robustness to 3D transformations. Given K conformers, using H as the Graph Transforrrr (GT)-aggregated atom features. We compute the global GT representation as: hęr = Σvev (WaT h + byT), where h H[v] and WGT, bęT are learnable parameters. We then define Hap and H&T be the matrices whose columns are, respectively, K copies of the 2D feature hap (Sec 3.3) and hØT representations. We fuse those representations with the 3D conformer features H3D to produce the final atom-we atom-wise embedding: Heomb = W2D H2D + WaD HAD + WGT HGT, where each W,, i (2D, 3D, GT) trainable projection matrix. The combined embedding Hoomb is fed into a final FFN layer to predict the target property (Sec K Appendix).
4 THEORETICAL BOUNDS FOR EMBEDDING NON-EUCLIDEAN FGW
Learning a Transformer To(.) to predict the FGW problem is closely related to multidimensional scaling (MDS) (Torgerson) 1952). Building on recent advances (Haviy et al. 2024 Sonthalia et al.) 2021). extend MDS theory derive bounds on the error of embedding non non-Euclidens distances, specifically Wasserstein and FGW, into a Euclidean space suitable for graph transformer integration. While computing FGW barycenters is costly, embedding enables efficient approximation via

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ESOL
FreeSolvv
BACE
Lipo
Enooding Distencs
0.5
0.6
0.571
Encoding Distance
pi00.004
Encoding Distance
1.0
pl:0.522
0.5
p=0.122
0.4
0.4-
0.6
0.2
12
0.2
0.1
0.0
0.0
0.0 0.1 0.2 0.3 0.4
0.5
0.6
0.0
0.2
0.4
0.8
5.8
1.0
0.0
0.1
0.4 0.5
OT Distance
Distance
OT Distance
Figure 2: Correlations between FGW distance and trained GraphTTrnsffrmer on four datasets in MoleculeNet benchmark. For each test molecule, we compute pairwise FGW distances between conformers and compare them with Euclidean distances between their Graph Transformer embed- dings. The correlation is reported, with the reference line y = x shown in blue.
averaging and decoding in latent space. Prior work (Haviv et al. 2024) validated this approach for Wasserstein distances; we generalize it FGW and provide theoretical justification, offering a scalable path for structure-aware graph alignment.
Cumulative Stress Optimization Problem via Pairwise FGW Distance Matrix. We define the pairwise FGW distance matrix D for set of K distributions as Dij == FGW p.a (SS,),, (S,))
for all i,j E [K], following Section 3.4 The empirical FGW barycenter is given by 9x arg mingep,, K ΣK FGWP (. ((S)), where (S2) denotes the space of attributed graphs with finite p-th order FGW distance.
To approximate this barycenter in embedding space, we require |Æex - eyl
FGW (9x. ((S))) == Dk.j for all j E [K], where εg Ex e, is the mean embedding and e; To (H.) is the learned representation. To assess how well the embeddings { C Rd preserve both pairwise FGW distances and barycenter structure, we define the cumulative stress:
mime,, ERd Exje(( (Ile; e - Dby)² + Eje(k] (IÆε - e,112 Dx.))² This objective encourages faithful reconstruction of both the distance structure and the barycenter alignment the leameed embedding space, as formalized in Theorem which is proved in [Appendix
Theorem 1. Let D denote the pairwise FGW, p.a distance matrix, and let thi o represent the eigendecomposition of the associated criterion matrix F DCDC, where C = IK 1 k is the centering matrix. The optimal stress value, denoted by S", is bounded as follows: L LSS S* Su where Σix, <0 X7. U == Eij(Ag; + Ag,)² L + C, Ag. = 1 E <o Ag Diy denotes the j-th component of the i-th eigenvector vn of F, and quantifies the approximation error between the empirical barycenter in the Euclidean embedding space and the one in the original spuce of undirected altributed gruphs.
v.y. Here,
5 EXPERIMENTS
5.1 IMPLEMENTATION DETAILS
General pipeline. Our training consists of three stages. Stage 1: We train the 2D and 3D MPNNs independently for 150 epochs and the learning rate of 1e-3 to extract features from 2D molecular graphs and 3D conformers, used to predict molecular properties by regression loss. These extracted features also serve as dataset to supervise the training of Graph Transformer for approximating the FGW distance. Stage 2: The Graph Transformer is trained separately to approximate the compu- tationally expensive FGW distance between pairs of conformers, using the learned representations from Stage 1. We use the architecture of Graphormer (Ying et al. 2021 with 12 attention layers, heads, and hidden size of 64 (372k parameters). is trained for 1000 epochs with learning rate of le 1e-5. Stage 3: We train the full model end-to-end with 2D/3D MPNNs and the Graph Transformer (300 epochs, learning rate 5e 4)) Details of the training scheme are provided in 6 Section of the Appendix. To mitigate feature shift during finetuning, MLP adaptors project conformers into 64-dim refined embeddings for both 2D and 3D features before the Graph Transformer.
5.2 APPROXIMATION OF FGW DISTANCE VIA GRAPH TRANSFORMER
Beyond theoretical estimation, we empirically evaluate how well the Graph Transformer approxi- mates FGW distances between conformers in Euclidean space. As shown in Figure 2 results on the MoleculeNet benchmarks reveal a strong correlation between learned embeddings and true FGW distances, validating the transformer's effectiveness in simulating costly FGW computations. While

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Table Number of samples each split on molec- ular property prediction, classification tasks, and re- action prediction for MoleculeNet and the MAR- CEL benchmark.
Table 2: FACET results on SchNet and VisNet.
Model
Lipo
ESOL
FreeSolv
BACE
CONAN (NisNett)
0..55 0.45
1.03 0.12
0.69
0.03
0.61 0.15
Datasel
Lipo ESOL FreeSoly BACE Drug-75k Kraken
CONAN-FGW
00.00 0.00
0.55 + 0.05
0.64 0.02
0.47 0.01
Train
2940
789
449
1039
52569
1086
112
64
7509
CONAN (SehNes)
(0.56::0.011
0.57 0.001
1.30
0.051
Test
840
227
129
303
15021
311
CONAN-FOW
0.42 0.02
0.53 4 0.02
1.07 2 0.008
0.55 0.02
FACET
0.42 0.01
0.04
0.03
4200)
1552
correlation varies slightly across datasets, the results consistently highlight the model's reliability as a fast FGW surrogate, especially as the number conformers in the aggregation increases.
Time (9)
ConAN-TGW
13291s
ConAN-FGW
FACET
2576s
5.16x faster
FACET 56s
6.54X faster
2000
10000 12000
14000
100
(a) Drugs-75K
(h) Kraken
Figure 3: Comparison of the one-epoch training time of CONAN-FGW (Nguyen 2024b and the proposed FACET on the Drugs-75K and Kraken datasets from the MARCEL benchmark.
5.3 SCALING FRAGMENT GEOMETRY-AWAL AGGREGATION
To validate the scalability of FACET model, based on a Graph Transformer for structure-aware aggregation, we compare it against ConAN-FGW (Nguyen et al. 2024a), a method computing FGW distances on-the-fly during training and inference. We evaluate two key aspects: (i) inference-time efficiency with varying numbers of conformers, and (ii) average training time per epoch at different dataset scales. For inference-time
measure the time required to generate output embeddings from K conformers (K E 5, 10, 15, 20) using a single GPU. Experiments are conducted on FreeSoly and BACE, which differ in node/edge distributions, assess performance across molecular graph complexities. In addition to ConAN-FGW, we further compare FACET against strong 3D>GGN baselines (c.g.. SchNet, VisNet, GemNet) assess efficiency and accuracy relative to established geometry-aware models. In the second setting, we compare the average per-epoch training time of FACET and ConAN-FGW on two datasets of different scales: Kraken (1,086 molecules) and Drugs-75k (52,569 molecules).
As shown Figures and [0] FACET scales linearly with the number of conformers and achieves a 5-6xx reduction in training time compared to ConAN-FGW (Figure This improvement is espe- cially important for large-scale training;
example, ConAN-FGW requires 1,107.58 GPU hours to train on Drugs-75K for 300 epochs, whereas FACET completes the same schedule in 214 hours, and in only 26.75 hours with 8 GPUs (vs. 138 hours for ConAN-FGW), Relative to other 3D GNN baselines, FACET provides
balance: it avoids the heavy cost of FGW align- ment while remaining competitive with and some cases more efficient than existing geometric GNNs. Further analysis of these scaling behaviors is provided in Sections D and
eccuracy-efficieee
115
Oranth FIG Liaga bmoigeelae
Suntma tecoree
NLPT en CWhi
RETT
167F1 29h)
300
200
100
20 12
23
23
(a) BACE
(b) Lipo
Figure 4: Inference running time comparison between FACET and other GNN-based meth- ods datasets, BACE (a) and Lipo (b). Results shown for both single-GPU and 4-GPU (multi-GPU) configurations. Reported runtimes represent the total time required to extract structural embeddings for all molecules in the of each dataset.
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Published as a conference paper at ICLR 2026
Table 3: Comparison of molecular property regression performance on the
MoleculeNet benchmark (MSE L)
The results of competing methods are FACET uses SchNet backbone.
adapted from Nguyen 2024b).
Model
Lipo
ESOL
FreeSolv
BACE
2D>GAT
1.387
0.206
2.288
0.017
8.564
1.345
1.844 + 0.33
D-MPNN
0.534 0.022
0.923 = 0.045
4.213 +: 0.068
0.723 +: 0.021
Attentive FP
0.520 0.001
0.771
0.026
4.197 + 0.199
PretrainGINN
0.545 0.003
1.210
0.005
6.392
GROVER lurge
0.676 0.012
0.798 0.018
5.162 0.047
ChemBERTa-2
0.639 0.006
0.795
0.033
1.858 + 0.029
ChemRL-GEM
0.486 0.008
0.706
0.061
3.924 0.436
MolfFormer
0.492 0.012
0.766
0.026
5.485
0.045
1.091 +: 0.021
ConfNet
1.360 0.038
2.115
0.484
1.329 + 0.042
UniMol
0.374 0.012
0.741
0.014
2.867 +: 0.186
Figure 5: RingsPaths decom- position BACE, splitinng molecules into rings, paths, and linkers. This reflects molecular topology and im- proves interpretability and generalization.
SchNet-scalar
0.704 0.032
0.672 = 0.027
1.688
0.158
0.723 1 0.100
SchNet-emb
0.589
0.635
0.057
0.692 + 0.028
ChemPrupSD
0.602 0.035
0.681
0.023
2.144
0.182
0.815 +: 0.170
CONAN
0.556 0.013
0,571
0.019
1,496 + 0.158
0.635 + 0.051
CONAN-FOW
0.422 0.016
0.022 0.529
1.068
0.083
0.549 + 0.016
FACET
0.424 = 0.009 0.516 0.044 0.967 0.082
0.495 + 0.034
5.4 STATE-OF-THE-ART PERFORMANCE COMPARISON ON MOLECULAR TASKS
Datasets. We evaluate molecular property regression on the MoleculeNet (Wu et al. 2018) and MARCEL (Zhu al. 2024a) benchmarks. MoleculeNet includes four datasets, ESOL, BACE, Lipo, and FreeSolv, with targets covering solubility, inhibitory concentration (pICso), lipophiliciity, and hydration free energy. MARCEL consists of Drugs-75K and Kraken, where the goal to predict the Boltzmann-averageee tum descriptors (IP, EA, X). while Kraken focuses on Sterimol features (Bs, L,, and their buried forms). The Boltzmann average is computed as weighted sum over conformer--pecific values ys with probabilities pi. All datasets follow the original random split settings, using the provided sampled conformers.
property (U) from sampled conformers. Drugs-75K uses quan-
Baselines. For the MoleculeNet benchmark (Wu et al., 2018). we compare FACET with a
wide range of baselines, including (i) 2D supervised methods (c.g.. GAT (eelčkkvic et al.
2018). D-MPNN (Yang et al. 2019a). AttentiveFP (Xiong et al., 2019)).
(ii pre-training
approaches (e.g., PretrainGNN (Hu et al. 2020b). GROVER (Rong al. 2020 ChemBERTa-
2* (Ahmad et al.) (022)) ChemRL-GEM (Fang et al. 2022). MolFormer (Ross et al.] 2022 )
(iii) 3D-conformer: based models (ContNet (Liu 2021),, UniMol (Zhou al. 2023).
SchNet (Schütt et al., 2017). ChemProp3D (Axelrod & Gómez-Bombarelii 2023), CONAN-
FGW (Nguyen al. (024b)). Training follows the setup in CONAN-FGW (Nguyen al. 2024b).
For the MARCEL benchmark Zhuu et al. 2024a). we compare FACET against 2D models 1s
(c.g., GIN (Xu et al. 2019). GI (Hu et al. (020a). ChemProp (Yang et al. GraphGPS (Rampäšek et al. 2022) )
2019b).
3D models (e.g., SchNet (Shhitt et al. 2017
DimeNet++ (Klicpera et al. (0020) GemNet (Gasteiger et al., (02T). PaiNN (Schütt et al. 2021)
ClofNet (Du et al.] (0022). LEFTNet (Du et al. 2023)), and ensemble strategies such as DeepSets-based ensemble (Zaheer et al. | 2017). self-attention /Vaswani et al., 2017), etc. All meth-
ods are evaluated under the same settings as described in the MARCEL benchmark.
5.4.1 RESULTS
MOLECULENET. As shown in Table FACET achieves state-of-the-arl performance on three molecular property regression tasks (ESOL, FreeSolv, BACE), with the lowest MSEs: 0.044, 0.967 0.082, and 0.495 0.115, respectively. Its consistent gains over ConAN-FGW indicate that, beyond geometry-aware aggregation, FACET's use of fragment substructures (Figure 5 enhances attention to localized chemical contexts. This demonstrates the advantage of combining 3D spatial information with chemically meaningful substructures for molecular property prediction.
MARCEL. In Table 4. we evaluate FACET on two backbones, SchNet and GemNet. FACET con- sistently boosts both, confirming the benefits of structure-awar aggregation and fragment-leved hi- erarchy. Unlike ConAN-FGW, which struggles to scale on the large MARCEL benchmark, FACET remains efficient and achieves near-SOTA performance across all targets, demonstrating robust ef- fectiveness in diverse molecular property prediction tasks.
ADDITIONAL ANALYSIS. this section, we analyze the key components of FACET through ablation studies. Specifically, we evaluate the impact of: (i) removing fragment structures from

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Published as a conference paper at ICLR 2026
Table 5: FACET ablation study. "wo Frag." Table 6: Performance (MSE) of different training means without using the fragment graph, "w/o strategies. "FACET (default) refers to training three Frag. in Trans." indicates without using the frag- steps with ablation studies for merging, and "FACET ment graph the graph transformer, and "w/o (w/o FGW)" refers to a version without using FGW to supervise the graph transformer.
Adapt." depicts not using adaptors to adjust fea-
tures from JD-GNN and 2D-GNN.
Settings
ESOL() (re)Sols())
BACE()
Lipe())
ConAN-PGWW
0:055 1.07 0.088 0.06 0.42 0.016
Dataset
FACET w/o Frag. w/o Frag. in Trans. w/o Adap.
FACET (detsul)
0.52 1.5200000 0.97 a 1.0022 0.50 a 0.115 0.12 a 0:00
FSOL
0,516
0.531
0.525
0.546
Merge all stepe:
0.57 a 0.023 1.26 a 0,044 0.59 a 0.022 0.53 a 0.03
FreeSolv
0.967
1.022
0.973
1.085
Merge steps 2-3:
0.51 a 0.014 0.87 a 0,102 0,50 e 0.085 0.44 + 0.014
Kraken
0.238
0.247
0.242
0.262
FACET twdo FGW) (055 a 0.053 0,98 0.007 0.53 0.024 0.45 0.000
both the 2D MPNN and the self-attention mechanism in the graph transformer (w/o Frag); (ii) using fragments only in the 2D MPNN but not in the graph transformer (w/o Frag in Trans.); and (iii) omitting the trainable adaptor (w/o Adap.) that aligns 3D conformer features with the graph transformer, which can lead to performance degradation due to domain shift during training. Furthermore, we also evaluate training strategies that (iv) merge all stages into a unique step, (v) retain stage 1, but merge steps 2-3, and finally (vi) FACET (w/o FGW) means without supervised Graph Transformer with FGW.
As shown in Table []] the ab- sence of (i) significantly reduces performance, making FACET comparable to ConAN-FGW but with better scalability. In- corporating fragments into both components (ii) provides further gains. while (iii) proves essen- tial for mitigating the domain shift introduced by changes in the 3D MPNN during training. The Table o presents results for settings (iv)-(vi), showing that learning a geometry-awant module explicitly regularized by FGW is important, which can- not be replaced by downstream loss alone. In Table we further present FACET's ' model param- eters compared with other GNN
Table 4: Comparison of molecular property regression perfor-
mance on the MARCEL benchmark (MAE 4). The results of
competing methods are adapted from Zhu et al. 2024a).
Category
Model
Drugs-5K
Kraken
IP
EA
L
BurB;
BurL
GIN
0.4354
0,4169
0.2260
0.3128
0,4003
0.1719
0.1200
GIN+VN
0.4361
0.4169
0.2267
0.3567
0.4344
0,2422
0.1741
2D models
ChemProp
0,4595
0,4417
0.2441
0.4850
0.5452
0.3002
0.1948
GripGGPS
0.4351
0.4085
0,2212
0.3450
0.4363
0.2066
0,1500
SchNet
0,4394
0,4207
0.2243
0.3293
0.5458
0.2295
0.1861
DimeNete+
0.4441
0.4233
0.2436
0.3510
0.4174
0,2097
0. 0,1526
3D models
GemNet
0,4069
0.3922
0.1970
0.2789
0.3754
0.1782
0,1635
PaiNN
0.4505
0.4495
0.2524
0.3443
0.4471
0.2395
0,1673
CloiNet
0,4393
0.4251
0,2378
0.4873
0.6417
0.2884
LEFTNet
0.4174
0.3964
0.2083
0.3072
0.4493
0.2176
0,1486
SchNet
0.4452
0.4232
0.2243
0.2704
0.4322
0.2024
0.1443
Ensemble
DimeNet++
0.4126
0,3944
0,2267
0.2630
0,3468
0,1783
0,1185
Strategy with
GemNet
0,4066
0,3910
0,2012
0.2113
0.3386
0,1589
0.0947
DeepSets
PaiNN
0,4466
0.4269
0,2294
0.2225
0.3619
0,1693
0.1324
CloiNet
0.4280
0.4033
0.2199
0.3228
0,4485
0,2178
0.1548
LEFTNet
0.4149
0.3953
0.2069
0.2644
0,3643
0.2017
0.1386
FACET
SchNNet
0.4235
0.3971
0.2155
0.2508
0.3982
0,1803
0.1245
GemNet
0.3891
0.3852
0.1970
0.2225
0.3402
0.1503
0.0952
baselines, indicating a balance between model size and performance that matches or outperforms much larger 3D-GNNs. More details on training time are discussed in Table TT Appendix.
Table 7: FACET vs 3D-GNN ensemble baselines on model
6 CONCLUSION
size, inference time, Mean Absolute Error (MAE) J.
We introduce FACET, a scalable frame- work that replaces costly FGW alignment with a Graph Transformer trained to ap- proximate FGW fusion between 2D frag- ments and 3D conformers. This approxi- mation enables efficient, end-to-end fusion of 2D and 3D structure, yielding strong gains across MoleculeNet and state-of-the- art performance on the large-scale MARCEL benchmark.
Model
Kraken (BurL)
Drugs (ip)
Param
Run. time (s)
MAE
Param
Run. time (s)
MAE
SchNet
215215
1.33
0.1443
210607
64.45
0.4452
PaiNN
1310209
2.36
0.1324
1305601
80.44
0.4466
CloiNet
605122
2.02
0.1548
600514
88.48
0.4280
LEFTNet 2722724
6.49
0.1386
2718116
138.28
0.4149
FACET
584065
3.17
0.1245
584065
130.68
0.4235
While FACET performs well on small, drug-like molecules, its evaluation is limited to standard benchmarks, leaving open questions about generalization to more complex regimes, including biomacromolecules with long-range dependencies, polymers and materials without stable con- formers, and multi-molecular systems such as pigtein-ligand interactions. Future directions include (i attention mechanisms capturing both local fragment-level and long-range structure, (ii) exten- sions to flexible input formats (e.g., voxel grids or material-specific graphs), and (iii) cross grashh or co-embedding strategies for intermolecular modeling. Broader evaluation datasets such as PDBbind (Liu et al. 2015) and PolyInfo (Otsuka et al. 201T) in future work would further assess FACET's applicability.
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ACKNOWLEDGMENT
This work partially performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AAC22-0NN277444. Lawrence Liv- ermore National Security, LLC. This work was funded by the Defense Threat Reduction Agency (DTRA), HDTRAI24204 (HK) and HDTRA
(IA). The project was also supported by Deutsche Forschungsgemeinscchatt (DFG, German Research Foundation) under Germany's Excel- lence Strategy EXC 2075 390740016, the DARPA ANSR program under award FA8750-23- 2-0004, the DARPA CODORD program under award HR00112590089, The authors thank the In- ternational Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Duy M. Nguyen. Duy M. H. Nguyen. Trung Q. Nguyen, Mai Thanh Nhat Truong, and Daniel Son- ntag are also supported by the No-IDLE project (BMBF, 01IW23002), the MASTER project (EU, 101093079), and the Endowed Chair of Applied Artificial Intelligence, Oldenburg University.
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SUPPLEMENTARY MATERIAL FOR "FACET: A FRAGMENT-AWARE CONFORMER ENSEMBLE TRANSFORMER
CONTENTS
Implementation Details
18
B Further Visualization Fragment Outputs
19
Analysis of Conformer Diversity
21
Additional Analysis of FACET's Scalability and Performance with More 3D Conform-
ers
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Inference Time when Increasing the Number 3D Conformers Each Molecule,
22
D.2
Average Training Time Epoch Function of Dataset Size.
23
D.3
Ablation Study on the Impact of Increasing the Number of 3D Conformers in FACET
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Comparison of Training Time between FACET and ConAN-FGW
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Performance of FACET and CoBAN-FGW on MARCEL benchmark
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Comparisons with SOTA methods in 2D (or 3D)
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Unified training pipeline
26
Limitations of FACET
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FACET
FACET Operates on a Predefined Set of 3D Conformers,
27
1.2
Limitations in Scope: Focus Small Molecules
27
J
Proof of Theorem 1
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Non-Euclidean Nature of Pairwise FGW Distance Matrix
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J.2
Lower Bounds on Embedding non non-Euclidean FGW Distances
29
J.3 Upper Bounds on Embedding of Pairwise Empirical FGW Barycenter Distances
J Proof of Lemmal
33
E(3) Invariant Property
34
A IMPLEMENTATION DETAILS
Our training pipeline includes three stages: the first stage, we train only the 2D and 3D MPNNs to learn corresponding features from 2D molecular graph and 3D conformers. The features in this stage also serve as a dataset for approximating Graph Transformer to the FGW distance. In next stage, the Graph Transformer is trained separately simulate the costly computation of FGW distance between two conformers using learned features from stage 1. In the last stage, Graph Transformer is integrated in single end-to-end training with 2D and 3D MPNNs. At this stage, only 2D and 3D MPNNs are trained. As a result of changing MPNNs during the last stage, shift in the distribution of the Graph Transformer input might occur. We solve this problem by
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adding an adaptor layer using an MLP both 3D and 2D features before feeding them to the Trasfrranrformer. For all experiments on the MoleculeNet and MARCEL benchmarks, we use the same number of conformers specified in their original settings.
In all stages, we use Adam as our optimizer. We train our model on an V100-GPUs cluster.
Stage 1. Learning 2D and 3D features. For each molecule, we define by H24-34 = Wnn Han +
HDD we then train for 150 epochs and set the learning rate to le to optimize target property tasks pred = 12a-3d yI2 where y be the ground-truth value and ÿ be our predicted value defined by:
K
D2a-3d 3d
W
Σ
Haa
aa(k)] +
(10)
K
k=1
with w and bd are learnable parameters and K is number of conformers.
Stage 2. Training Graph Transformer to approximate FGW distance. The Graph Transformer trained separately in the second stage to approximate the FGW distance by Euclidean embedding space. For the Graph Transformer architecture, we employ the same setting as Graphormer from Yiig et al. (2021). Specifically, number of attention layers, number of attention heads, and the hidden dimension of the transformer are set to 12, 8, and 64, respectively, which makes the total number of parameters of the Graph Transformer 372k. In attention, distance (SPD) between a pair of nodes. Following practical implementation in Ying et al. (2021), pre-compute SPD distance for each 3D molecule graph and load these values during training and inference. We set a learning rate of le 5 and train for 1000 epochs with the following loss function:
the shortest- paath
Lene Σ (H,) THH FGW, w. (C(.))
(11)
Stage 3. Training Fragment-aware Graph Transformer. the final stage, we freeze the trained Tranforresr TO(-) and use it compute aggregated features from 3D conformer embeddings generated by the 3DD-MPNN. To accommodate potential distribution shifts, we add lightweight FFN adaptor layers on top of both the 2D- and 3D-MPNNs used in TO(-), while continuing to update the MPNNs during training. The full model is trained for 300 epochs with reduced learning rate to optimize the training loss Ppeed
g
y
where
K H Hoomb [k]] + +b².
9 w
(12)
K
k=1
Hoomb is final atom-wiss wise embedding.
B FURTHER VISUALIZATION FRAGMENT OUTPUTS
Fragment Generation Algorithms. We use a structural fragmentation method based on Ring- Path algorithms (Kong et al. 2022 Geng et al. 2023 Wollschläger et al.) 2024) that decompose a molecular graph G (V, E). where V denotes atoms and E denotes covalent bonds, into a set of chemically interpretable fragments. The fragmentation process identifies set of ring fragments Fring S F using RDKit's 's cycle basis algorithm (SSSR), where each ring f, E Fring is encoded by its atom indices and size class.
Next, all bonds not part of any ring are grouped into acyclic path fragments Fpah S F, where each f Fptth is a linear chain of nodes, extracted via depth-first search under degree constraint. Each fragment f E F Fring U F U Fjunction is assigned a type ((f) E {0, 2} (representing ring, path, or junction) and type index d(f) {0, 1,.., K- within fixed vocabulary of size K. Fragments whose sizes exceed a predefined threshold kmax are mapped to the final index of their category preserve bounded dimensionality.
We define fragment-atom incidence matrix M E {0, VIXIT where Me if atom V belongs to fragment f. From this, derive fragment-level graph Gfrng (Viag Enag), where each node f F represents a molecular fragment and an edge (fr,f) E Emig is added two fragments share at least one atom directly bonded.
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Compared to traditional fragmentation algorithms like BRICS (Degen et al. 2008 BBB (Sommer et al. ). or MagNet (Hetzel et al. 2023), the RingPath algorithm offers a more topology-awe aware decomposition by explicitly capturing key structural motifs such rings, paths, and linkers. While BRICS and BBB often generate chemically meaningful fragments based on retrosynthetic rules, they may overlook contextual connectivity critical for graph-based learning. In contrast, RingPath preserves the relational structure between fragments, aligning closely with how molecules are built and understood in topological space-maae maainng it particularly beneficial for tasks requiring structural interpretability and generalization in graph neural networks. The advantages of RingPath have also been empirically validated in recent studies, demonstrating improved performance across various molecular property prediction benchmarks.
Visualization of Typical Extracted Fragment Graphs. Figures and illlutratte representative examples of fragment extraction using the RingPath algorithm on the Kraken and Drug-75k datasets. The top row displays the original 2D molecular structures, while the bottom row shows the corre- sponding RingPath decompositions. Each colored region highlights distinct structural fragment, such as a ring path, demonstrating the algorithm's 's ability segment complex molecules into chemically meaningful and interpretable components.
(a) Sample
(b) Sample 2
(c) Sample
Figure 6: RingsPaths decomposition three samples of the Kraken dataset. Top: 2D molecules;
bottom:
corresponding RingsPaths decomposition results.
(a) Sample
(b) Sample
(c) Sample
Figure 7: RingsPaths decomposition on three samples of the Drugs-75K dataset. Top: 2D molecules; bottom: corresponding RingsPaths decomposition results.
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Figure 8: Boxplots illustrating conformer diversity the four datasets. For each dataset, two boxplots are shown. Left (blue): distribution of molecules based on the average FGW distance between their conformers, reflecting conformer diversity. Right (green): distribution of the 20 molecules with the lowest conformer diversity, measured by the La distance between the mean embedding produced by our Graph Transformer and the nearest conformer embedding.
C ANALYSIS OF CONFORMER DIVERSITY
The diversity of conformers plays crucial role in learning effective molecular representations. When molecules have very similar conformers, the embeddings produced by the Graph Transformer may collapse into single, overly similar conformer representation. To assess whether this collapse occurs in our model, we conducted both quantitative and qualitative analyses.
Quantitatively, Figure shows
shows two boxplots summarizing conformer diversity for each dataset. For the first boxplot, we computed the average FGW distance between all pairs of conformers for each molecule in the test set. This captures how structurally diverse the conformers are. The results show that most molecules exhibit non-zero average FGW distances, indicating meaningful conformer variation; the Lipo dataset in particular contains molecules with highly diverse conformer sets.
For the second boxplot, we selected the 20 molecules with the lowest conformer diversity based the FGW distance. For each molecule, Graph Transformer, calculated their mean embedding, and subsequently measured the T-z distance between this mean embedding and the closest conformer-level embedding. It can be seen that even among these least-diverse molecules, the embeddings remain distinct: the average embedding does collapse into single conformer representation.
obtained the latent embeddings from our trained
To complement our quantitative analysis and provide a more intuitive view of conformer behavior, we applied t-SNE yan der Maaten & Hinton (2008) directly the embeddings produced by our Graph Transformer. For each molecule, we used both the mean embedding (obtained by averaging its conformer embeddings) and the individual conformer embeddings from each molecule in the test set to map into 2D domain. Rather than summarizing distances boxplots, this visualization allows us to inspect how embeddings are arranged in lower-dimensional space. We randomly selected two molecules with large FGW distances and two random others with small FGW distances and visualized their embeddings in Figure 9 In both cases, high and low conformer diversity, the mean embedding remains well separated from the individual conformer embeddings, and the conformers themselves occupy distinct regions in 2D space, confirming that the model preserves conformer variability without collapsing representations.
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Molecule average distance
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Figure 9: t-SNE visualization of graph transformer embeddings of four molecules in FreeSolv, in which molecules have a large average FGW distance between their conformers (left) and the other two molecules have a small average FGW distance (right).
D ADDITIONAL ANALYSIS OF FACET's SCALABILITY AND PERFORMANCE WITH MORE 3D CONFORMERS
In this section, we further analyze FACET's scalability on the following two factors:
D.1 INFERENCE TIME WHEN INCREASING THE NUMBER OF 3D CONFORMERS FOR EACII MOLECULE.
We compare FACET against versions of CONAN-FGW in running time extract structure- aware embedding aggregation with different input of 3D conformers. We use two variations of CONAN-FGW, including single GPU version and another relaxed solver that permits running Sinkhorn iterations on GPUs by matrix multiplication. thus supporting distributed multi-GGUS ac- celeration. The experiments are conducted on a single GPU using batch size of 32 molecules, each with different conformers ranging from 3, 5, 10, 15, and 20, and another experiment with four GPUs on the same batch size, i.e., molecules per GPU.
Figure TO indicates our observations across four datasets of MoleculeNet benchmark, where we re- port the required time to extract embedding aggregations for all molecules in the test set. We see that (i) FACET demonsirates excellent scalability where its runtime remains nearly constant regardless of the number of conformers, both in single-GPU and multi-GPU settings. contrast, ConAN-FGW shows poor scalability where runtime increases steeply with the number of conformers. While the multi-GPU usage improves runtime over single-GPu GPU, the growth trend remains significant, with runtimes still exceeding 30 seconds at 20 conformers (e.g., with ESOL dataset).
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Secondly, the nearly identical runtime of FACET across single- and multi-GPU settings, shown in the plot, can be attributed to its computational efficiency and the relatively small workload in this experiment. In such cases, the overhead introduced by multi-GPU parallelization such as inter-GPU communication and data synchronization can outweigh its potential speedup benefits. Therefore, argue that multi-GPU acceleration for FACET becomes advantageous only under substantially larger workloads, such as batch processing of thousands to millions of molecules or handling complex input representations that exceed the memory capacity of single GPU.
D.2 AVERAGE TRAINING TIME PER EPOCH A FUNCTION OF DATASET SIZF.
We analyze the scalability of FACET with respect to the number of training molecules. To this end, we report the average training time per epoch across four datasets from the MoleculeNet benchmark. Figure 11 compares the training time of FACET and ConAN-FGW on a single GPU, using batch size of 256 and conformers per molecule. As shown in the figure, FACET achieves 2.28x to 3.17x speedup over ConAN-FGW. Notably, this speedup is roughly proportional to the number of training molecules each dataset, as reported in Table
Coddi Segle 2N1
Romtin trecodt
BCTT uniiiallio
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(b) FreeSolv
Figure 10: Inference running time comparison between FACET and other GNN-based methods on two datasets, ESOL (left) and FreeSolv (right). Results are shown for both single-GPU and 4-GPU (multi-GPU) configurations. Reported runtimes represent the total time required to extract structural embeddings for all molecules test set each dataset.
Time (8)
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Figure 11: Comparison the one-epoch training time of CONAN-FGW (guuen et al. 2024b and the proposed FACET on four datasets from the MoleculeNet benchmark.
D.3 ABLATION STUDY ON THE IMPACT OF INCREASING THE NUMBER OF 3D CONFORMERS FACET
We provide below comprehensive ablation study on the impact of using increasing number of RDKit-generated conformers set 3, 5, 10, 15, 20, 50, 100 across four datasets (ESOL, FreeSolv, BACE, and Lipo). For completeness, note that using 100 conformers for the BACE and LIPO datasets exceeded GPU memory (OOM) capacity in our setup and therefore could not be evaluated.
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Table 8: Comparisons on performance with different numbers of conformers generated by RDKit,
"OOM" indicates out-of-memory.
Settings
3 conf.
5 conf. (default)
10 conf.
15 conf.
20 conf.
50 conf.
100 conf.
ESOL
0,539 + 0.06
0.516 # 0.04
0.501 + 0.02
0,511 + 0.03
0.546 = 0.02
0.529 0.040
0.530 + 0.037
FreeSolv
0.977 0.25
0.967 0.08
0,933 0.23
0,946 0.24
0.949 = 0.21
0,940 0.036
0.945 0.039
BACE
0,542 + 0.05
0.495 + 0.03
0,513 0.02
0.519 0.01
0.517 0.03
0.463 0,004
OOM
Lipo
0.45
0.424 = 0.01
0.444 0.02
0,447 0.08
0.445 0.01
0.440 0.010
OOM
Table 9: Comparisons on performance without conformers generated by RDKit.
Method
ESOL(I)
FreeSolv(I)
BACE(()
Lipo(()
FACET
0.516 # 0.04
0.967 = 0.08
0.495 0.03
0.424 0.01
w/o 3D conformers
0.546 # 0.03
1.197 = 0.09
0.584 + 0.03
0.543 # 0.02
As shown in Table g we observe a consistent trend across datasets: increasing the number of con- formers from 3 to 5 leads to improved regression performance (lower values indicate better results). However, beyond conformers, the performance tends to converge or slightly fluctuate, confirming that our geometry-aware embedding approach using the FGW distance provides stable and reliable approximations. This aligns with the theoretical expectation that the approximation error scales with O(1/K), where K is the number of conformers used.
When 3D conformers generated by RDKit are not used, our FACET model simplifies significantly. In this configuration, the model only receives 2D molecular graphs along with fragment-level in- formation, and key components such as the Graph Transformer are removed. Table 9 presents the performance comparison between the full FACET model and its 2D>onlly variant across four bench- mark datasets:
These results clearly demonstrate that incorporating 3D conformers, even those generated by RDKit, is critical to the expressiveness and performance of FACET. The full model consistently outperforms its 2D-only counterpart, highlighting the importance of 3D geometry in learning accurate molecular representations.
E COMPARISON OF TRAINING TIME BETWEEN FACET AND CONAN-FGW
To provide comprehensive comparison, conducted additional experiments to compare the train- ing time of FACET and ConAN-FGW, with the addition of SchNet, VisNet, and GemNet, a strong state-of-the-art 3D molecular model, two benchmark datasets: BACE (1,059 molecules) and LIPO (2,940 molecules). All models were trained for 200 epochs under the same settings. Since both FACET and ConAN-FGW are originally built the SchNet architecture, which is generally less expressive than GemNet, we also report the performance of FACET when upgraded to use Gem- Net as its backbone. From the results listed in Table I0 we have the following key observations:
FACET vs. ConAN-FGW: FACET consistently shows reduced training time compared to ConAN-FGW, though the degree of reduction varies by dataset size.
On BACE: the time savings are marginal due to the additional cost introduced by the Graph Transformer component in FACET, which is trained using the pre-computed FGW distances from the optimal transport solver.
-On LIPO: the training time reduction more substantial. This is because ConAN- FGW incurs a high computational cost from directly computing FGW distances be- tween sets of 3D conformers in every forward pass. In contrast, FACET leverages pre-learned geometry-aware embeddings, where the corresponding operation reduces to a lightweight matrix multiplication in the Graph Transformer.
FACET vs. GemNet: FACET represents a balanced trade-off between ConAN-FGW and GemNet in terms of training time. Despite using the simpler SchNet backbone, FACET achieves competitive, sometimes better, performance compared to GemNet, thanks to its geometry-aware aggregation via FGW-based embeddings. This efficiency stems from
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Table 10: Comparisons on performance in terms of MSE(L) and corresponding training time()).
Model
Metric
BACE
LIPO
MSE
0.51 + 0.07
0.45 0.01
GemNet
Time
2.04 hours
4.8 hours
Model Param
1.95M
MSE
0.46 = 0.03 2.47 hours
0.39 0.02
FACET (GemNet)
Time
6.4 hours
Model Param
2.25M
MSE
0.64 = 0.05
0.56 = 0.01
SchNet
Time
1.4 hours
2.24 hours
Model Param
273K
MSE
0.50 + 0.03
0.42 + 0.01
FACET (SchNet)
Time
2.3 hours
3.16 hours
Model Param
584K
MSE
0.61 0.15
0.55 0.45 4.27 hours
VisNet
Time
1.89 hours
Model Param
1.8M
placing costly pairwise conformer comparisons with latent-space transformer that cap- tures 3D geometric information in more scalable manner.
FACET (GemNet) vs. GemNet:
When both models share the same GemNet architecture, FACET outperforms GemNet in terms of predictive accuracy both datasets. We observe that (i) the additional training time incurred by FACET is relatively modest: approximately +21 % on BACE and +33% LIPO, and (ii) given the performance gains, this extra time remains acceptable in practical scenarios and demonstrates FACET's scalability and effec- tiveness.
FACET other GNN baselines: Although FACET introduces additional fusion compo- nents, the overhead relative to each backbone remains small. The observed increases in end-to-end training time moderate (e.g., GemNet: 2.04h - 2.47h on BACE; SchNet: 1.4h + 2.3h). Importantly, most of this extra time comes from the separate pre-training of the graph transformer Step 2, which takes roughly 0.6 hour. The parameter growth is also limited mostly in the graph transformer module (e.g.. GemNet: 1.95M 2.25M; SchNet: 273K - 584K). our experiments, these modest increases were consistently accompanied by improved predictive accuracy, suggesting practical trade-off between accuracy and cost.
F PERFORMANCE OF FACET AND CONAN-FGW ON MARCEL BENCHMARK
To provide meaningful comparison, we benchmarked FACET against ConAN-FGW on 10% of the Drug-75k dataset and the Kraken dataset, which serve representative subsets. The results (provided below) show that FACET performs competitively or outperforms ConAN-FGW, even un- der these reduced-scale settings, reinforcing the efficiency and effectiveness of our approach. The results are shown Tables 12 and [3
G COMPARISONS WITH SOTA METHODS IN 2D (OR 3D)
FACET is designed as a modular framework for enhancing molecular property prediction by inte- grating structure-awant aggregation that can be plugged into a variety of existing backbone architectures, whether 2D or 3D, thus offering complementary mechanism rather than alternative to these models.
multiple conformers. A central strength of this design is
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Table 11: Comparisons on performance with different standalone 3D architectures.
Model
BACE())
LIPO(L)
SchNet
0.64 0.05
0.56
FACET (SchNer)
0.50 +
0.42
GemNet
0.51
0.45
FACET (GemNet)
0.46
0.39
VisNet
0.15
0.55 6.550.04
FACET (VisNet)
0.47 = 0.01
0.48 + 0.01
Table 12: Comparisons of performance between FACET and ConAN-FGW on Kraken
L
BurL
B5
BurB5
ConAN-FGW (SchNet)
0.397
0.117
0.272
0.195
FACET (SchNet)
0.398
0.125
0.251
0.180
FACET improves standalone 3D architectures We integrated FACET with established 3D mod- els such as SchNet, GemNet, and VisNet, and consistently observed performance improvements across datasets. Table 11 demonstrates that FACET's geometry-aware aggregation multiple conformers complements even strong 3D baselines, validating its utility beyond what these models achieve on their own.
FACET enhances simple 2D MPNNs We also applied FACET lightweight 2D message- passing neural network and found that incorporating FACET's fragment-level structure-aware ag- gregation significantly improved performance. This result underscores the compatibility of FACET with 2D backbones and its ability to enhance models that do not explicitly process 3D information.
H UNIFIED TRAINING PIPELINE
We investigated the performance of the proposed method when combining all training steps into an end-to-end pipeline. Below, we summarize our findings step by step:
Step Pretraining 2D and 3D MPNNs: As suggested prior work like ConAN-FGW, we begin by pretraining the 2D and 3D MPNNs independently. This initial phase is critical ensure that the encoders, especially the 3D MPNN, converge stable and meaningful representation before introducing structure-awart aggregation. To test the necessity of this stage, we experimented with variant where all three stages were co-trained from scratch. The results showed substantially lower performance, confirming that Stage is crucial for learning rich, aligned, and stable representations.
Steps 2 and 3 Co-training Graph Transformer and Downstream Fine-tuning: While our default setup trains Step 2 (Graph Transformer with FGW supervision) and Step 3 (fine- tuning on molecular properties) sequentially, we explored an alternative setup where both steps co-trained. To manage the computational cost of FGW supervision, adopted an alternating strategy: after every five steps of property prediction optimization, we update the Graph Transformer to approximate FGW distances. This reduces the training burden compared to full FGW supervision at every iteration (as in ConAN-FGW).
Table 13: Comparisons of performance between FACET and ConAN-FGW on Drugs-7.5k
X
IP
EA
ConAN-FGW (SchNet)
0.374
0.541
0.587
FACET (SchNet)
0.365
0.535
0.552
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As shown in 14 without separately training Step the model got low performance, confirming that this stage helps the model ensure rich, aligned, and stable molecular representations before incorporating more advanced structure awareness. Secondly, four MoleculeNet datasets, co- training Steps 2 and 3 produced slightly improved performance over the default FACET setup. For example, on ESOL., performance improved from 0.505 to 0.516, and on FreeSolv, from 0.867 to 0.967. This improvement can be attributed to the model's ability to jointly adapt the 2D/3D encoders and the Graph Transformer, leading to more aligned, task-relevant representations. However, there is a tra This co-training strategy comes with an increased training cost, as FGW distances must still be computed periodically. As result, while training is slower than the default FACET setup, it remains significantly faster than ConAN-FGW, and achieves a strong balance between efficiency and predictive performance, especially on large-scale datasets like Drug-75k.
Table 14: Comparisons of performance (MSE o)of different training strategies.
ESOL(L)
FreeSolv([)
BACE(L)
Lipo(()
ConAN-FGW FACET
0.529 = 0.022
1.068 = 0.083
0.019
0.422 + 0.016
0.516 = 0.044
0.967 = 0.082
0.424 = 0,009
FACET (Merge all steps)
0.567 # 0.023
1.264 = 0.094
0.591 0.062
0.530 + 0.013
FACET (Merge steps 2-3)
0.505 # 0.014
0.867 = 0.102
0.497 = 0.035
0.44 0# 0.014
I LIMITATIONS OF FACET
1.1 FACET OPERATES ON A PREDEFINED SET OF 3D CONFORMERS.
Our method enables efficient geometry-aware aggregation without requiring expensive alignment procedures at inference time. While FACET demonstrates improved performance even with a small subset of conformers, the quality and representativeness of this subset can still influence downstream predictions. In particular, if the selected conformers are heavily biased or fail to capture key struc- tural variations, some aspects of molecular fiexibility may be underrepresented. Addressing this challenge through better conformer sampling strategies or task-aware selection mechanisms could further enhance model robustmess, especially for highly flexible molecules.
Future direction: A promising extension would be to develop end-to-end models that can learn to generate conformers dynamically during training, using gradient feedback from downstream predic- tion losses. Such differentiable conformer generation module could enable task-aware structural modeling, ensuring that the generated conformers are optimized not just for physical plausibility, but also for relevance to the predictive task at hand.
1.2 LIMITATIONS IN SCOPE: FOCUS ON SMALL MOLECULES
FACET has primarily been evaluated on standard molecular property prediction benchmarks such as those in MoleculeNet, which consist mostly of small, drug-like molecules. While this setup is well-suited for many pharmacological applications, it limits the assessment of FACET's general- izability more complex molecular systems. For example, biomacromolecules (e.g., peptides, proteins, nucleic acids) exhibit high flexibility, long-range dependencies, and hierarchical organi- zation that are not present in small molecules. Polymers and materials often involve much larger structures without well-defined conformers, challenging FACET's s reliance on discrete 3D inputs. Additionally, FACET currently models only single-molecule properties and has not been extended to multi-molecular interactions, such as protein-ligand binding.
Future direction: To broaden FACET's applicability, several promising future directions can be explored. First, incorporating efficient attention to capture both local fragment-level information and long-range structural dependencies is essential for handling large biomolecules. Second, adapt- ing FACET to support flexible input formats, such as voxel grids or material-specific graphs, would allow to process polymers and crystalline materials that lack stable conformers. Third, extend- ing FACET to jointly model molecular interactions through cross-grapy graph attention co-embedding mechanisms could open applications in drug docking and molecular complex prediction. Finally, applying and evaluating FACET on broader datasets, such as PDBbind (Liu et al.) 2015), PolyInfo
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(Otsuka al. 2011). CoRE-MOF 2019 (Chung et al. 2019), would provide a more comprehen-
sive understanding of its strengths and limitations across molecular domains.
J PROOF OF THEOREM
Recall that aim to establish the following novel theoretical bounds: Let D denote the pairwise FGW p.a distance matrix, and let {Ax, Uy u5)k=1 represent the eigendecomposition of the associated criterion matrix F = CCDC, where C IK
R x is the centering matrix. The optimal L S* U,, where
stress value, denoted by S", is bounded follows:
: Σ X K0
u = I(Ag + Ag) C
Agh 2
1
Σ A A Ak E [K].
1:A,CO
Here, Usi denotes the (-th component of the k-th cigenvector U2 of F, and C quantifies the ap- proximation error between the empirical barycenter the Euclidean embedding space and its counterpart in the original space of undirected attributed graphs. This is equivalent to that given == kelA] E RdxK our objective to derive lower and upper bounds for the following cumulative stress:
S = min S(e),
S(e) = Si(e) S2(e),
(13)
CERaxx
S = min eERaXA
Si(e), (i(e) Σ k.EE(K]
(lle: eil12 -Da)²
(14)
8
min eER4xK
So(e), S2(e) (E[K]
Σ (|ēx exl12 Dk.1)
(15)
To this end, we begin by specifying and formally defining the following important concepts in Appendix J.1
J.1 NON-EUCLIDEAN NATURE OF PAIRWISE FGW DISTANCE MATRIX
Definition 1 (Euclidean Distance Matrix). A K x K distance matrix D is said to be Euclidean if there exists set of points e =fe { in some Euclidean space Rd such that
Ii vk, E [K], DRI = |lex- - -eili.
The space of all Euclidean distance matrices (EDM) is denoted & E.
Fact (Conditions for Euclidean Distance Matrix, see, e.g., Gower 1985)) A matrix is an EDM and only it satisfies the following three conditions:
(i) Non-negativity: Dwa 2 0 for all ks k.le E [K].
(i) Hollow diagonal: DRA- for all k [K].
(iii) Positive semidefiniteness: the associated double-centered matrix F= -CDC positive semidefinite (PSD), where C II 1x 1 k is the centering matrix, and 1 k denotes the K <d-dimensionl vector of ones.
Recall that the pairwise FGW distance matrix D for a collection of K distributions is defined entry- wise by D&t FGW, a.a (S(S)), ((S)) for all k, e [K]. as introduced Sectioon The following result establishes that this matrix does not correspond to Euclidean distance matrix:
Lemma 1 -Non-Euclddenn Nature of Pairwise FGW Distance Matrix). Consider the case where dy =
Then the FGW distance matrix D, whose entries are given by FGW
(...(G11.9pp == +ell(w .w2)
min
((1-a) ((1-a)M + a L(A,,,A,LL)) & nT,))
with a E [0, 1), does not define a Euclidean distance matrix.
clidean distance. Theretore, which is proved in Appendix J.4, the distance FGW p.o. is not a Eu- interested in quantifying how accurately Eun-uuclde distance matrices can be approximated by pairwise distances between learned embeddings. To this end, we analyze the lower and upper bound of the set S in Appendices J.2 and respectively.
As established in Lemma I
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J.2 LOWER BOUNDS ON EMBEDDING CON-EUCLI FGW DISTANCES
We would like to find the lower bound of S. We note that the original formulation is non-convex, making analytically intractable. Nonetheless, by reparameterizing the objective as function of the pairwise squared distances Dµ := ||e el and Dxi == ||e ll2 induced by the embedding, and by incorporating the necessary conditions to ensure that D corresponds to a valid Euclidean distance matrix, the reformulated problem becomes convex for St. Note that we can prove that S has lower bound at L', where L' is minimizer of S1, that
S* = min Si( + Sa(D)), S2(D) = Σ DEE
(DKi -KI)
(16)
LE|K]
Si min (D)), Si E (DN (Dw- DDI A.EE[K]
(17)
DEE
Indeed, given the previous reformulation of S. we can establish the following lower bound via Proposition Notably, to simplify the problem, in inProposiiionn considering &c. containing & by keeping only the PSD property from the EDM definition in Fact 1
we relax the EDM constraint by
We will reintroduce the missing constraints in & and use the solution for the simplified problem to construct upper bound in Appendix J.3
Proposition 1 (Error Lower Bound of (*). The lower bound of S is provided as follows:
S* min DEg
S (D) + S2( (i"I")+ + SZ(") 2 L1 + =:
(18)
min S Si( Σ = Lt,
(19)
DEEc
k: <0
S2
min S2( = L2.
(20)
DEEc Here &g contains & by keeping only the PSD property from the EDM definition in Fact
Proof of Proposition
1 Note that if S₁ is minimized at L=, 1that is,
S Si
min S (i)),, Si(D) = Σ
(Dwr -DiI-Dal DRi
(21)
DEE
k.EE(K]
We then can find the lower bound of S = min Des (D) + SDD) via the minimizer П.
Using the definition of Frobenius norm and Eg, we can obtain:
S₁ Si = min S₁ (D) 2 min Si()), Si( = = - -DI
DeE
DEEc
We then obtain the following decomposition:
ID= D
||]
||
BIi A CDC CDC,
B K
oòC+ 1 K
còo+ + K²
1 ODO ( ODC
1 CDO + 1
K² ODO):
K
where C= IK- O is the centering matrix and 0 1x 1 k is the all-ones matrix. Indeed, using
the definition of the centering matrix C Ix 0 , we have I C +
[ D 1 I DIK DIK 1 IA+ + B = | f + |B +TT((BB = || + BB
Here we used the fact that the matrix product is invariant under eyclic permutation:
0 = Tr ( oc D)C( -D) 0,
Tr(AB) Tr C(D (DD-DJCI DDC
D) K
K
and
1 1 OC = K
0)
1
IK
- 0 00 =.
K
K
K
K2
Under only the PSD constraint, the optimal solution ī. that minimizes S (D) can be decomposed as:
i. i L + L where D and L respectively minimize the terms
|| and || B| independently.
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In particular, using the definition of the centering matrix C given by:
IK k O, the entries of D
i =
ODC +
CDC +
ODO
K
K
K²
kl
1 (OD)
OD
ODO
= + DI
K
K
K2
where D denotes the mean of the k-th row (or column) of D, and D is the global mean of all elements in D. Therefore, the rows/columns mean of D equal those D itself, and hence min DEEc
D' arg min BB|
B 0.
DEfc
Therefore,
min S2 min Σ
[Ki Dk.) )² =0
DEE;
DEEE [E(K]
Here we used the fact that the matrix D is given by Du = FGW p.a (G(S)), for k, [K] and the empirical FGW barycenter is given by
K
GK E arg min
FGWP P.a
(9,
arg min (EP)
FGW p.a (9,
K
K
(EP,(1))
1=1
K
1
D == FGW (. G(S))) = min (EP,(S2)
K FGW p.a. (9, (=: column /-th means of D);
Σ
1=1
where (22) denotes the space of attributed graphs with finite p-th order FGW distance. To ap- proximate this barycenter in embedding space, we require
|
112 FGW (Gx, ((S)) D',, for all I [K],
where EK = K Eb
E/ is the mean embedding and e == (H))) ) is the learned representation.
Now would like to find local analytic solution L' minimizing ||] such that the global solution Ż. = E'A + Σj minimizes both terms || and B| simultaneously. That is,
min |||| min ||C(ÍA LB) DEEg
CDC
DEE:
||C( + - - CDC = -
Here we used the fact that by definition of Li holds that C. 0. Hence, the optimization becomes:
min 1 DEE
This is in fact the problem of computing the nearest PSD approximation CLA C to a symmetric matrix CDC. Using the result from Higham (1988), we find the analytic solution follows:
LA=-A
AEULUY.
22)
A:A.00
Here (Ax, ) k the eigenvalues and eigenvectors of F -CDC. Because CDC has rows/columns means O, the ones vector 1x is an eigenvector of CDC with eigenvalue 0. This leads to 1 g is also in the null space ÉA and:
i = C.AC,
1 R
Oo
1 K
(Oi')
= 0.
Therefore,
I D 2 L Ln DIE
Σ X2.
k:AA<U
Combining all together, Proposition is derived follows: S" 2 min ||||| Deec
+ min Deec
B + min S₂(
Σ
X +0+0=
Σ
X L.
DeEc
k=Ax<0
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J.3 UPPER BOUNDS ON EMBEDDING OF PAIRWISE EMPIRICAL FGW BARYCENTER DISTANCES
As discussed in Appendix J.2 the lower bound stated in Proposition | is derived by simplifying the problem and relaxing the EDM constraint. Specifically, this relaxation involves considering the set Ec, which contains but retains only the PSD requirement from the EDM characterization given in Fact 1 In Proposition 2 we reintroduce the missing constraints excluded in &g and leverage the closed-form solution obtained from the relaxed problem to construct an upper bound under the original EDM constraint set E.
Proposition 2 (Error Upper Bound of S*). There exists matrix Ú &ucch that the following upper bounds hold:
S = min
(i + S2 - S,( + +2O su + U2 = U.
(23)
DEE
S,(Ū)) = min S S U1 == Σ X
Σ k
(Apk + Api)
DEg
k <0
1
Aph =
Σ
A pig Ak E [K]
(24)
2
1:A:00
S²Ú min S2 ( S Σ (Api) 2 =: U22
(25)
Beg where the aggregated error term is defined as:
AP:
Σ
d . Ugg.
2K
A= EAACCO
We aim to exploit the information derived from the truncation of the negative eigenspace of the matrix CDC, specifically the matrix introduced in Eq (2), defined as:
L' =- Σ ARAHUR T
A:A..0
where (Ax, Dk [K]Kk] denote the eigenvalues and corresponding eigenvectors of the matrix F -CDC.
Recall that the entries of Dn given by:
1 OOD
I R (OD) K2 ODO
DB
D Dr D.
K
As consequence, the sum i + L may not be strictly hollow or PSD when D is not an EDM. To address this, we seek construct symmetric matrix P be added to L'A resulting in the matrix Ú- := L + P. which is both hollow and PSD. This adjustment is designed to avoid any additional penalty the term || though it may introduce some approximation errors in |B|] and in the quantity S2. These induced errors contribute to the upper bound // for the optimal score S*.
We begin with the requirement that the matrix P does not contribute any additional penalty the term || This can be ensured by imposing the constraint CPC = 0. Under this condition, the
matrix U remains minimizer of || [ as demonstrated below: min DEEc
||
min |[C(ÍA + LB) - CDC
DEEc
||C( + P C - CCDC
|C CDC 2 where the final equality holds due to the constraint CPC =.
This leads to the condition (CP)C C(PC) 0. implying that CP lies in the left null space of C. and PC lies in its right null space. As a result, all rows of PC must be constant, and this expression can be written as:
1 T PC -P Ik
1 10) K
P lgc + +p 0
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where is column vector to be defined subsequently. Here, we have used the fact that C' is the centering matrix defined by C = IK= O..
Multiplying both sides on the left by K O yields:
OP Olxe 1 K K
1
(x) = lxc + K2
OPO.
K
This leads to
1
c K
1PP
OPOO
K
Indeed, via the definition
0 1x K. we can verify this as follows:
1
K KIPP
I
1
1 OPO
1xc²
OPO 1
+
K2
K2
K
[KIpP
K
K2
+
K
OPO
1
0000 +
OPO K
K
1
OP.
K
Hence,
P lx
1
I
1
TOPO)
P -
K
K
K
Ix(1IP)) K
1 + (Plx) K
1
1 OPO
K2
Since Plk k is a column vector, to satisfy this constraint, P must be of the form:
P P-lk
1 K K
-
P
where p R.K. is vector of free parameters, and p denotes its average. This construction implies that has only K degrees of freedom. However, to ensure that L'A + has zero diagonal (i.e., the resulting matrix is hollow), the diagonal entries of P must satisfy the following K linear constraints:
- [[AARRR
Px- 2 P
Vk E [K].
Solving this linear system yields:
pk
Σ
d
UN
K
2
K
1
p =
Pk
Σ
A
K
K
k=1
k=
where we have used the fact that L' = Endiso \aaaa and hence its diagonal entries are given
by [Ea = ELapo > vzg.
Consequently, the resulting matrix P can be expressed element-
Pa. [Ka + A
2 2 0.
where the inequality follows from the fact that i negative semi-definite.
In summary, the matrix 0- = Σ + P satisfies all three constraints specified in Definition
Although Ò preserves the value of || it differs from i' and introduces approximation errors in the [B term and the S, term. Note that the sum of the untruncated version of CDC and the matrix
1 LODC + CDO + K2
ODO
K
K
equal to D and remains hollow. Recall the decomposition:
|
D
||
|
A= CD= CDC, -CCC
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B
oDC+ 1 c K
1
0D0
K
K2
(H - CD0+ 1
- ODO).
K
K²
where C Ik
K
i is the centering matrix and 1x1 is the all-ones matrix.
The matrix
ODC CDO + K²
ODO
K
K
can be written similarly to P by including the contributions from the negative eigenvalues, resulting in the matrix P. parameterized by:
Pk
2
Σ
A
Ukl
K
1
1 K
Σ
Σ
Σ
K Ph
k=1
Define the correction due to negative eigenvalues
Apk
Σ
A
Vk E [K].
2
The resulting error in the |
2 term is given by:
B 2 P
PI
Σ (Apx + Api)²
Furthermore, the contribution S2 is bounded as:
S min S2(
Σ
D -KI)DK.)) DK.)² $ Σ (Ap,)² ) U22
DES
LE|K]
where the aggregated error term is defined
Ap:
E d -
2K
k=1EACC
J.4 PROOF LEMMA I
The proof is proved via leveraging Proposition 8.2 from Peyré et al. (2019), applied to the specific case a = 0, and relies on the relationships among FGW, Wasserstein (W), and Gromov-Wasserstein (GW) distances.
The FGW cost FGW, p.o (9,, G2) is defined via two components: the node feature cost matrix M ji = d, (H (i) H Hi and the structural discrepancy tensor L(A1, i,)
[A [ - Ag m.
Let Gi (H1, wI) and G2 (H2, A2, w2) be two attributed graphs with Nj and N2 nodes, respectively. Their associated probability measures are
#1 Σ (Ind(en)aaa)) #2 Σ (2)))
We define the marginals μr, Ex wrdn; and MAi En wrdas (and analogously for μH, and (A as projections of and μ2 onto the feature and structural spaces, respectively.
Using these definitions, introduce the following notation:
JA A2, = E Ligia (1, A2) PTi Thi
Thi
(26)
85kl
(Wp(H.,.P "H.)"" REIl(wo),w)
Jy(A1, A2, TT) min
(27)
Hz(M, Σ d (Zik: T2i) '
(28)
kl
Wy(HA,
min
Hp (M, TT)
(29)
(w
33

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Published as a conference paper at ICLR 2026
Let [11 w2) be any admissible coupling. If both μ4 and A2 are defined common
metric space (St, A./ μt) then the FGW distance is given by:
FGW (91, 92) == =E (
min ((1-a) aLL, A2) (T,
(30)
We now derive the following fundamental identity:
Ep (M, A1, A2, T
Σ t5k1
[[L - (adddd(((aaa + (i,
=
(M,, tJ π
(31)
Moreover, let To denote the optimal coupling that minimizes the FGW objective Ey,c (M, AI, A) Then the FGW distance admits the following decomposition:
FGWP (H1, p2)
min Bp.c (M, A1, A2, TT) Ep.a (M, A1, A2, To))
*eIl( m22)
(1 a) Hp M, n.) + aJ (A,, TT)
2 (1 0 WT (1 + a GW GW (1HH
(32)
This inequality follows from the optimality of the W and GW distances with respect to the cost functions Hp and Jp. respectively, and highlights the interpolation nature of the FGW distance between these two metries governed by the parameter
The generalized FGW cost Ep.o (M, , Ai. A2, T) admits the following explicit formulation: Ep.a (M, A1, A2, π ((1 a) M a
(aL(A11 A2) π. π.)
Σ [[L ( (2)) + A k) A2(i,0)² T Tki F.RR.
Based on the formulation above, we obtain the following upper bound on the FGW distance: FGW p.o (G1, G2) ((l-a M aL(A), @ π., π
d1 x21) + 2P- a A Alk, !] Thl
(33)
k.
where the second inequality follows from the convexity of the function I + x" for 2 1 and an application of Minkowski-type bounds on the structural term. Importantly, inequality in Eq. (33) holds for any admissible coupling π E (2))),w). and in particular, it remains valid when = the optimal coupling associated with the Wasserstein distance Wp(H1, (22) over the product metric space (22, d). Here, the effective distance d between structured nodes (I1. a1) and (x2,, a2) is defined
d((x1, a1), (x2, az)) (1 a) d x22) + 2n- A(a1, a2).
Combining this with the Wasserstein formulation in Eq. 29 we observe the following inequality: FGW FGWp.a (9r G2) S Wp(HA,.PA and FGW p.a (9,, G2) Wp(MA, (A,) when a
(34)
K E(3) INVARIANT PROPERTY
We utilize a 2D-MPNN, where node embeddings are iteratively refined across layers as follows:
h UPD (n AGG' ((M' (b ,,.,, u N(w))))
(35)
with M denoting the message function, AGG representing aggregation by summation, and UPD implemented either the identity function or multilayer perceptron. The final atom-level repre- sentation is obtained by integrating three modalities: the 2D molecular graph embeddings H.D the 3D conformational features HAD, and the geometry-based structural descriptors HŒr- This fusion performed using trainable linear projections:
Heomb W2D H2D WaD HaD + WeT HGT:
(36)
where W 2D, WAD, and WGT are trainable parameter matrices. Assuming that Hap and H&T are composed of K repeated copies of their respective feature vectors, compute the fused represen- lation as:
H Homb HaD + WaD WoTHeT,
(37)
34

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Published as a conference paper at ICLR 2026
where y is a hyperparameter controlling the influence of the barycenter features. This fusion scheme allows balanced contributions from all modalities, which is empirically beneficial.
To predict the molecular property, we perform
pooling over the K conformations and apply
a linear transformation:
K H
ÿ = ww
Homb [k]
+ +
(38)
K
where W and b are the weight matrix and bias vector used for the final prediction.
We demonstrate that the function specified in Eq. (35) through reemmmissss invariant under both the action of the (3) and permutations of the input conformers.
Theorem 2 (E(3) Invariant Property). Let 9 denote the 2D molecular graph, and let (Si...., ,gk) be collection of K conformers, where each Sy Tk.n: Zu 1 for k E [K]. Consider the function ý [o (S,,..., (K)) as defined by Eq.13
Eq. 38.
Then, for any transformations
91. 9K E3), the following holds:
Solg, ....,...... (KSKK Sa (S1...., Sk)).
Furthermore, for any permutation и E Sym [K]), have:
fo(g, (S.(1)) -.. sx(K))) fG (S.....,k]])
Proof of Theorem 2 We establish the result in several steps. First, we consider the invariance prop- erties of the geometric representation HGT- By construction, the geometry-aware embedding ag- gregation used to obtain H E (T.)) is invariant under permutation of conformers. Additionally, because E(3) transformations preserve Euclidean distances and given that the up- stream 3D MPNN assumed to be F(3)-invariant, the generated features H, are likewise invariant under such transformations.
Next, consider the aggregated representation defined in Eq.(37}:
Heomb W H2D HaD W WGTHGT.
From the prior step, we know that H&T is invariant under both E3) actions and conformer permu- tations. Additionally, HAD inherits E3) invariance from the 3D MPNN and is permutation equiv- ariant, i.e., permuting the conformer inputs permutes the columns of H3D accordingly. However, because the final prediction in Eq.(38) based on an average over the conformer-wise features:
K
ÿ w
H Heamb(M)
+ b
k=1
which is invariant column permutations of the matrix HAD. leading to the final ÿ invariant to E(3) group and permutation of 3D conformers.
35


Reviews:
[{"review_id": "tmEbTTMXnO", "reviewer": "Reviewer_NGyk", "summary": "This paper presents a well-executed study on a scalable framework for integrating conformational information into molecular property prediction. The authors propose a learned embedding of the Fused Gromov-Wasserstein (FGW) distance to efficiently aggregate 3D conformers, enabling the model to scale to large datasets. Additionally, fragment-level structural priors are incorporated to enhance representation learning. The proposed method achieves state-of-the-art performance across multiple benchmarks, demonstrating its effectiveness.", "questions": "Given that the proposed method approximates real FGW calculations using a learned embedding, is the performance theoretically upper-bounded by the true FGW metric? If so, under what conditions would the approximation match or exceed FGW-based methods in practice?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- **Scalability**: The use of learned FGW embeddings enables efficient conformer aggregation, making the method suitable for large-scale datasets.\n- **Structural Integration**: The combination of 3D conformers with 2D fragment-level information provides a rich, multi-scale representation of molecular structure.\n- **Empirical Performance**: The method consistently outperforms strong baselines across diverse tasks, offering compelling evidence of its advantage.\n- **Architectural Flexibility**: The framework is compatible with multiple backbone architectures and tasks, highlighting its versatility.", "weaknesses": "- **Split Strategy**: All experiments rely on random splits. Evaluating performance under scaffold splits would better assess the model’s ability to generalize across distinct chemical scaffolds—a key criterion for real-world applicability.\n- **Role of 2D Fragments**: The necessity of 2D graphs and fragment-level features is not fully justified, given that 3D conformers already encode topological and spatial information.\n- **Limited Ablation Scope**: Ablation studies are restricted to MoleculeNet tasks. Extending these analyses to other datasets (e.g., MARCEL) would strengthen the claims and demonstrate robustness across domains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mPA91E0xPd", "reviewer": "Reviewer_1ftv", "summary": "This paper introduces FACET, a hybrid model integrating a Message Passing Neural Network (MPNN) with a Graph Transformer (GT) to accurately predict molecular properties from 2D molecular graphs and 3D conformations. Existing methods for aligning graph and conformational features often suffer from high computational costs. FACET addresses this challenge through two key innovations: (1) It leverages a GT to approximate the computation of the Fused Gromov-Wasserstein (FGW) distance, enabling efficient feature alignment with a provable theoretical error bound; (2) It embeds fragment-level structural priors from the molecular graph and sampled conformational features into the MPNN and GT, effectively capturing multi-scale relationships between the two molecular representations.\nExperimental results demonstrate that FACET outperforms several baseline models, including CONAN-FGW, on the MoleculeNet and MARCEL datasets. Furthermore, FACET exhibits significantly faster training and inference speeds, validating its computational efficiency.", "questions": "1. Revise the abstract to more prominently and explicitly state the core methodological innovations.\n2. Redesign Figure 1 to provide a clearer, more detailed visual representation of the MPNN and GT components.\n3. Integrate the key points from the appendix's \"Limitations of FACET\" section into the main \"Conclusion\" section to offer a more critical summary and a concrete outlook for future work.\n4. Conduct a thorough review of all tables, especially Table 1, to ensure data consistency and accuracy across all entries.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The use of a GT to approximate the FGW distance is a notable innovation, combining theoretical rigor with algorithmic efficiency for feature alignment.\n2. The model design is sound, integrating fragment-level structural priors and conformational features through a multi-scale message-passing architecture.\n3. The model achieves a favorable balance between predictive performance and computational speed, underscoring its practical utility.\n4. Comprehensive evaluations on multiple benchmark datasets provide strong, empirical support for the model's effectiveness and efficiency.", "weaknesses": "1. The abstract inadequately highlights the core contributions, particularly the novel use of GT for efficient FGW approximation.\n2. The model diagram (Figure 1) lacks clarity. The MPNN for processing 3D conformations is represented only by the symbol \"Φ\", and the GT is depicted generically as an \"attention mechanism,\" which hinders a straightforward understanding of the model's architecture.\n3. The conclusion is underdeveloped, merely restating the main contributions and results without critical analysis or forward-looking perspectives.\n4. Table 1 contains errors. The column header \"Model\" is incorrect and should be \"Dataset\" based on the content. Furthermore, the sum of the values under the \"Train,\" \"Valid,\" and \"Test\" rows for the \"Lipo\" dataset does not match the value in the \"Total\" row.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "D7PiVCeNQu", "reviewer": "Reviewer_Csta", "summary": "This paper presents a novel architecture for molecular property prediction that integrates fragment-level chemical structure with 3D conformer ensembles. FACET introduces a Fragment-Conformer Attention Module (FCAM) that hierarchically aggregates information within fragments, across conformers, and at the whole-molecule level, ensuring rotational, translational, and permutational invariance. This fragment-aware design captures both local and global geometric interactions more effectively than atom-level models. FACET achieves state-of-the-art performance on multiple 3D molecular benchmarks while using fewer conformers and offering improved interpretability and computational efficiency", "questions": "1. **Fragment Definition and Generalization:** How are fragments defined and standardized across molecules? Is the fragmentation algorithm differentiable or fixed? Could learned or adaptive fragmentations improve performance on datasets with different chemical domains?\n2. **Conformer Ensemble Sampling Strategy:** The method relies on conformer ensembles from datasets such as GEOM. How sensitive is FACET to the number of conformers used during training and inference? Could the model degrade when applied to datasets lacking reliable conformer ensembles (e.g., generated on-the-fly via RDKit)?\n3. **Theoretical Guarantees on Invariance:** The paper asserts rotational, translational, and permutational invariance of FACET (Theorem 1), but the proof sketch in Appendix C assumes strict alignment across conformers. How does FACET ensure invariance when conformers differ significantly in spatial orientation or fragment positioning?\n4. **Fragment-Level vs. Atom-Level Trade-offs:** Have the authors evaluated the granularity trade-off - i.e., how predictive accuracy changes when using atom-level embeddings versus different fragment granularity levels?\n5. **Conformer Diversity Representation:** Does FACET perform any diversity regularization or weighting among conformers? Could the model collapse to a single representative conformer if conformer embeddings are highly correlated?\n6. **Data Efficiency and Scalability:** While FACET achieves good performance, training time and parameter counts are not discussed in detail. How does FACET scale with molecular size and number of conformers compared to GNN baselines?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. FACET introduces a novel multi-scale attention mechanism combining fragment, conformer, and molecule-level reasoning - an elegant fusion of chemistry-driven inductive bias with deep transformer architectures. It extends prior conformer-based GNNs by explicitly modeling fragment ensembles, bridging a gap between quantum-chemical interpretability and machine-learned representation power.\n2. The paper's architectural design is technically sound and well-motivated, supported by ablation studies showing the benefit of fragment-aware aggregation.\n3. The writing is clear and well-organized, moving systematically from intuition to formalism and experiments.\n4. FACET represents a meaningful advance for 3D molecular ML, offering a scalable, interpretable, and chemically grounded alternative to purely atom-level GNNs.", "weaknesses": "1. **Dependence on Predefined Fragmentation:** The model’s reliance on fixed fragment decompositions could limit adaptability to novel or unusual chemistries.\n2. **Theoretical Limitations:** The invariance proof (Theorem 1) assumes perfect fragment-conformer alignment and ignores potential numerical instabilities in coordinate normalization. The theorem's validity would benefit from a more rigorous group-theoretic formalization akin to SE(3)-equivariant models.\n3. **Ablation on Conformer Ensemble Size:** While the authors claim reduced conformer dependence, quantitative experiments varying ensemble size (1, 5, 10, 20) are limited. Such results would strengthen the argument for FACET's data efficiency.\n4. **Computational Overhead and Resource Reporting:** FACET adds multiple attention layers per fragment and conformer, but runtime and memory usage are not reported.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZwY0TxnL1F", "reviewer": "Reviewer_Jbmu", "summary": "This work focuses on molecular property prediction by integrating 2D graph information and 3D conformer ensemble information. The main contributions include the introduction of fragment-level structural priors, supported by comprehensive ablation studies, and a fused Gromov-Wasserstein (FGW) alignment pre-training stage that employs a Graph Transformer with an FGW distance loss. This replaces the computationally expensive FGW alignment process and leads to substantial efficiency improvements.", "questions": "1. Regarding Weakness [3], if a relevant discussion already exists, please indicate where it appears. Otherwise, additional discussion and experimental results on this point would be valuable.\n2. In the implementation, is the graph fragmentation step handled during data pre-processing (i.e., fragmented and stored in advance) or performed dynamically during each forward pass (i.e., fragmented in real time)? If it is the latter, what proportion of the total forward-pass time (from raw data input to loss computation) does it account for, and does it introduce noticeable computational overhead?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The paper is clearly written, and the proposed method is described in a detailed and logically consistent manner. The experimental setup is well documented, enhancing reproducibility.\n2. The overall pipeline design is reasonable. The fragmentation strategy for generating fragmented graphs, along with the subsequent fusion of atom node features and fragmentation features, is well-motivated. The use of 3D-MPNNs (VisNet/SchNet) for extracting 3D conformer features ensures E(3)-invariance, making the overall framework theoretically consistent.\n3. To address the high computational cost of FGW alignment, the work proposes replacing it with a Graph Transformer trained using a supervised FGW distance L2 loss. Section 4 provides theoretical analysis supporting this design, and experimental results (Fig. 2) show positive correlations. This approach significantly improves efficiency in both training and inference (Figs. 7 and 3).\n4. The method achieves strong performance on multiple regression benchmarks, and the ablation study provides clear evidence for the effectiveness of the fragment-level structural priors.", "weaknesses": "1. Since the pipeline consists of three training stages, the pipeline illustration (Fig. 1) could be reorganized to clearly delineate these stages and indicate the corresponding losses for each stage. This would make the process easier to follow.\n2. There is no comparison of model parameters and inference time across all baselines. While it may not be necessary to include every method, most recent approaches should be considered, rather than comparing only with CONAN-FGW.\n3. There is no ablation or discussion on whether Training Stage 2 (i.e., training the Graph Transformer to approximate the FGW distance) is necessary. Specifically, what if Stage 1 is retained, but the Graph Transformer is randomly initialized and trained directly without Stage 2 pre-training (i.e., without the approximate FGW distance loss $\\mathcal{L}_{enc}$)? Would this configuration still achieve good results, and how would it compare with the proposed approach? If the performance difference is small, it could indicate that Stage 2—and the overall FGW-based motivation—may not be critical to the method’s success.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)