FragFM: Hierarchical Framework for Efficient Molecule Generation via Fragment-Level Discrete Flow Matching
Authors: Joongwon Lee
Seonghwan Kim
Seokhyun Moon
Hyunwoo Kim
Woo Youn Kim
Venue: ICLR 2026 Poster (2026)
Keywords: Molecular Graph Generation
Discrete Flow Matching
Fragment-Based Drug Discovery (FBDD)
Natural Products
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We introduce FragFM, a novel hierarchical framework employing fragment‐level discrete flow matching for efficient molecular graph .generation, along with a new molecular generative benchmark focused on natural products.

Abstract: We introduce FragFM, a novel hierarchical framework via fragment-level discrete flow matching for efficient molecular graph generation. FragFM generates molecules at the fragment level, leveraging a coarse-to-fine autoencoder to reconstruct details at the atom level. Together with a stochastic fragment bag strategy to effectively handle a large fragment space, our framework enables more efficient, scalable molecular generation. We demonstrate that our fragment-based approach achieves better property control than the atom-based method and additional flexibility through conditioning the fragment bag. We also propose a Natural Product Generation benchmark (NPGen) to evaluate the ability of modern molecular graph generative models to generate natural product-like molecules. Since natural products are biologically prevalidated and differ from typical drug-like molecules, our benchmark provides a more challenging yet meaningful evaluation relevant to drug discovery. We conduct a comparative study of FragFM against various models on diverse molecular generation benchmarks, including NPGen, demonstrating superior performance. The results highlight the potential of fragment-based generative modeling for large-scale, property-aware molecular design, paving the way for more efficient exploration of chemical space.

Published as a conference paper at ICLR 2026
FRAGFM: HIERARCHICAL FRAMEWORK FOR EFFI- CIENT MOLECULE GENERATION VIA FRAGMENT- LEVEL DISCRETE FLOW MATCHING
Joongwon Leel.*, Seonghwan Kim- Seokhyun Moon".", Hyunwoo Kim*,,, Woo Youn Kim 1,4,5,1
DDepartment of Chemistry, KAIST, 'InnoCORE AI-CRED Institute, KAIST, CCollege of Pharmacy, Dongguk University, "Department of Data Science KAIST, SHITS
(lee jwon 942, dmdtkk00, mshmipp @kaist. st. ac.krr kr, wooyoun& brryyounkkiisee kaist.ac.k.
edu,
ABSTRACT
We introduce FragFM, a novel hierarchical framework via fragment-level dis- crete flow matching for efficient molecular graph generation. FragFM generates molecules at the fragment level. leveraging a coarse-to- fine autoencoder to recon- struct details at the atom level. Together with a stochastic fragment bag strategy effectively handle a large fragment space, our framework enables more efficient, scalable molecular generation. We demonstrate that our fragment-basee approach achieves better property control than the atom-based method and additional flexi- bility through conditioning the fragment bag. We also propose a Natural Product Generation benchmark (NPGen) to evaluate the ability of modern molecular graph generative models to generate natural product-like molecules. Since natural prod- uets are biologically prevalidated and differ from typical drug-like molecules, our benchmark provides a more challenging yet meaningful evaluation relevant to drug discovery. We conduct comparative study of FragFM against various models on diverse molecular generation benchmarks, including NPGen, demonstrating supe- rior performance. The results highlight the potential of fragment- based generative modeling for large-scale, property-awe -ware molecular design, paving the way for more efficient exploration of chemical space.
INTRODUCTION
Deep generative models are achieving remarkable success in modeling complex, structured data, with graph generation being a prominent application area (Jang al. 2023 Jo et al. 2022). Among various applications, de novo molecular graph generation, which has the potential to accelerate drug and material discovery, is particularly important. Recently, diffusion- and flow-based graph generative models have demonstrated the ability to generate complex molecular graphs (Vignac et al. 2023 Qin
al. 2024 Siraudin al. 2024 Eijkelboom et al. 2024).
However, these models, which are built on atom-based representations, face significant scalability challenges. particularly when generating large and complex molecules (Qin et al. [202).. The quadratic growth of edges with increasing graph size leads to computational inefficiencies. At the same time, the inherent sparsity chemical bonds makes accurate edge prediction more complex, often leading to unrealistic molecular structures or invalid connectivity constraints (Qin et al.) [023] [Cen et al.) 2023). Moreover, graph neural networks struggle to capture topological features like rings, leading to deviations from chemically valid structures. Although various methods incorporate auxiliary features (e.g., spectral, ring, and valency information) to mitigate these issues, they do not fully resolve the sparsity and scalability bottlenecks (Vignac al. 2023
"These authors contributed equally.
"Corresponding authors. Code and data available at httptttpp/ggthhhh.cmmmm coweo/ocomeeeeeeeeeeeeee j

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Fragment-based strategies, rooted in long-standing success in traditional drug discovery, offer an alternative (Hajduk & Greer 2007
et al. 2014 Kirsch et al. 2019). By assembling
molecules from chemically meaningful substructures, these approaches enable more efficient
exploration chemical space, preserve global structural coherence, and provide finer control over molecular properties than atom-based methods (Tin et al., 2018 SSo al., 2023 Hetzel al.) 2023 Jin et al., 2020a). Diffusion models also adopted the fragment-based approach, showing their potential in improving scalability and property control (Levy Rector- Brooks, 2023; Chen et al.)
2024). However, existing methods rely on a small, fixed- fragmeetgment vocabulary or employ automated fragmentation procedures, which severely limit coverage of chemical space and hinder integration of domain knowledge.
Here, we introduce FragFM, a novel hierarchical framework for molecular graph generation to address these challenges. FragFM first generates fragment-level graph using discrete flow matching and then reconstructs into an atom-level graph without information loss. To this end, we develop a novel stochastic fragment bag strategy that circumvents reliance fixed fragment libraries, along with a coarse-to- autoencoder that ensures direct atom-level reconstruction from the generated fragment-level graph. Consequently, FragFM can efficiently explore the molecular space, avoiding the generation of chemically implausible molecules with an extensive fragment space at moderate computational cost.
Our main contributions are summarized as follows:
We propose FragFM, novel hierarchical framework that combines fragment-Iewe discrete flow matching with a coarse-to-fine autoencoder, designed to operate effectively on large fragment libraries.
We introduce NPGen, a new benchmark for natural product generation, designed to evaluate larger and more complex molecules.
Extensive experiments show that FragFM not only outperforms prior molecular generative models, but also achieves substantially stronger performance on large, natural product-like molecules. Moreover, remains robust under fewer denoising steps.
FragFM enables more effective and flexible property-guides molecular generation through both fragment bag control and conventional guidance strategies.
RELATED WORK
2.1 MOLECULAR GRAPH GENERATIVE MODELS
Modern molecular graph generative models can be classified into autoregressive and one-shot generation models. Autoregressive models generate graphs sequentially based on their node, generally an atom or fragment, and edge representations (Lim et al. 22200) Mercado et al. 202T] [hi et al.. 2020 Jin et al., 2018). Despite their performance, these models have intrinsic issue in learning the permutation of nodes the graph, which must be invariant for a given graph, often making them highly inefficient. Among one-shot models, there exists model that directly generates molecular graphs (Kwon et al. 2020). Also, denoising models have recently become fundamental for generating molecular graphs by iteratively refining noisy graphs into structured molecular representations. Diffusion methods, which have been successful in various domains, have been extended to graph structure data (Jo et al. (2022) Niu et al., 2020)) demonstratigg the advantages of applying diffusion in graph generation. This approach was further extended by incorporating discrete stochastic processes (Austin et al. (02T), addressing the inherently discrete nature of molecular graphs (Vignac et al. 2023). The discrete diffusion modeling is reformulated using the continuous-time Markov chain (CTMC) (Xu et al. 2024, Siraudin et al., 2024; Kim et al., 2024), allowing for more flexible and adaptive generative processes. More recently. flow-based models have been explored for generating molecular graphs. Continuous flow matching (Lipman et al. 2022) has been applied to structured data (Eijkelboom et al., 2024), while discrete flow models (Campbell al. 2024: Gat et al.) (024) have been extended to categorical data generation, with recent methods showing that they can also model molecular distributions as diffusion models (Qin al. 2024 Hou et al. 2024).
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2.2 FRAGMENT-BASEE MOLECULE GENERATION
Fragment-based molecular generative models construct new molecules by assembling existing molec- ular substructures, known as fragments. This strategy enhances chemical validity and synthesizability, facilitating the efficient exploration of novel molecular structures compared to the atom-basee bane- ap- proaches. Several works have employed fragment- based approaches within variational autoencoders (VAEs) by learning to assemble
chemically meaningful way (Jin et al. 2020b] Kong et al. 2022) Maziarz al. 2021: Podda et al., 2020). Also, Jin et al. (2018) adopts stepwise generation approach, constructing a coarse fragment- levell graph before refining it into an atom-level molecule through substructure completion. The other strategies construct molecules sequentially assembling fragments, enabling better control over molecular properties during generation (Seo et al., 2023] Jin et al. 2020b). In contrast these assembly-basee based methods, Noutahi et al. (2024) proposed SAFE-GPT, a GPT-2-based transformer model that generates SAFE (Sequential ettachment-base Fragment Embedding) strings, a novel representation that expresses molecules as sequences of fragment-level tokens rather than through explicit fragment attachment.
Fragment-based approaches have also been explored in diffusion-based molecular graph generation. Levy & fector-Brooks (2023) proposed a method that utilizes a fixed set of frequently occurring fragments to generate drug-like molecules, ensuring chemical validity but limiting exploration beyond predefined structures. Since enumerating all possible fragment types is infeasible, the method operates solely within a fixed fragment vocabulary. In contrast, Chen et al. (2024) proposed an alternative, dataset-dependent fragmentation strategy based on byte-pair encoding, which provides more flexible molecular representation. However, this approach does not yet integrate chemically meaningful fragmentation methods (Degen et al. 2008; Liu al., 2017), which inspired by chemical synthesis and functionality, limiting its ability to leverage domain-specific chemical priors.
2.3 HIERARCHICAL MOLECULAR GENER ATIVE MODELS
Beyond these fragment-based approaches, several hierarchical generative models explicitly leverage multi-scale graph structure for molecule generation. MolGrow (Kuznetsov Polykovskiy, (2021) and MolHF (Zhu et al. 2023) introduce hierarchical normalizing flows that generate molecular graphs in a coarse-to-fine fine fashion by recursively refining coarsened graph representations, enabling the generation of larger and more complex molecules but operating non-chemical subgraphs rather than chemically defined fragments. In 3D molecule generation, Qiang et al. (2023) proposes a hicrarchical diffusion model that sequentially samples coarse-grained fragments, fine-grained fragments, and atom- level conformations, improving local structural validity without relying on explicit synthesis-driver fragmentation schemes. Compared to these serrarch--on-graphe methods, which induce fragment-like units implicitly from data and graph coarsening, fragment-based models with chemically meaningful fragment vocabularies provide more direct control over substructural composition and synthesizability, and our work follows this latter line by treating informed fragments as the fundamental generation units while still benefiting from modern generative architectures.
FRAGFM FRAMEWORK
We propose FragFM, a novel hierarchical framework that utilizes discrete flow matching at the fragment-level graph. As illustrated in fig. [ our framework introduces two key strategies: (i) a coarse-to-fine fine autoencoder. and (ii) a stochastic fragment bag strategy for coarse-graph -rapeggrpphh generation. The autoencoder compresses atom-level graphs G into fragment-level graphs G, while preserving atomistic connectivity in latent variable 2. This design enables the use of discrete flow matching (DFM) at the fragment level. The stochastic fragment bag strategy ensures the model handles comprehensive fragment libraries at a manageable computational cost. To realize this strategy, we adopt a graph neural network module for fragment embedding, which enables generalization to unseen fragments. In this section, first describe the conversion between fragment- and atom-level graphs in section 3.1 and then present the flow motchingg procedure for the coarse graph, including both training (section 3.2) and generation (section 3.3 at the fragment level.
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(a)
f1
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Full fragment beg
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Coarse Graph Encoder
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Coarse Graph Decodar
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Figure 1: Overview of FragEM. (a) FragFM utilizes hierarchical framework of coarse-to-fine autoencoder (section 3.1) and fragment-level graph flow matching (section 3.2). An input atom- level graph (G) is initially decomposed via the fragmentation rule. This is then processed by coarse-to-fine encoder, which compresses it into a joint representation X = (G, z) comprising a fragment-level graph G and latent vector designed capture fine-grained atomistic connectivity information not explicitly present G. During generation (section 3.3), neural network Se selects the most probable fragment from fragment bag B, which is a stochastically sampled subset of the full fragment bag F. FragFM then employs two fow-matching processes: (i) a discrete flow generates the target fragment-level graph 91 from an initial Go (mask and uniform prior for node and edge, respectively), operating with fragments from B; (ii) continuous flow generates the target latent vector 2 from Gaussian prior (0, 1) (from an initial z0). (b) Finally, given (91, 21 21 the coarse-to-fine decoder reconstructs the atom-level molecular graph by first predicting the probabilities of all possible atom-to-atom edges, and then applying the Blossom algorithm to select the edge set that maximizes the likelihood of the true graph. Further details and hyperparameters are described in section and fig.
3.1 MOLECULAR GRAPH COMPRESSION BY COARSE-TO-FING AUTOENCODER
While a fragment-level graph G offers higher-level abstraction of molecular structures, it also introduces ambiguity in reconstructing atomic connections. Specifically, single fragment-level connectivity E can map to multiple distinct, valid atom-level configurations. To achieve accurate end-to-end molecular generation, it is therefore crucial to preserve atom-level connectivity when forming the fragment-level representation. Drawing on hierarchical generative framework (Razavi
al. 2019 Rombach et al. 2022 Qiang al. 2023), we employ a coarse-to-fin autoencoder.
The encoder compresses atom-level graph G into its fragment-level counterpart G and, for each input molecule, outputs a single continuous latent vector z that encodes the committed connectivity details. Specifically, G is first converted into G using predefined fragmentation rule (e.g. BRICS (Degen et al. 2008 )) after which a neural network encodes (G, 9) into 2. Given 9 and z, the decoder reconstructs atom-level edges between adjacent fragments in G. This process combines neural network that outputs continuous edge scores with the Blossom algorithm (Edmonds) 1965). which discretizes these scores into valid atom-level connectivity. The marse-to-fine autoencoder is simply
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constructed as:
Encoder:
G
Rule
g,
(G,9)
denee
2,
Decoder:
g Odec
score
Blosssor
E.
We verified that the autoencoder can faithfully reconstruct atom-level graphs, achieving over 99% bond-level accuracy standard benchmarks (see section C.3 for details). Additional implementation details are provided in section A.1
3.2 FLOW MATCHING FOR COARSE GRAPH
We aim model the joint distribution the fragment-level graph and its latent representation, X = (G, z). through the flow-matching after continuous-time generative paradigm. Flow matching begins at known prior at f=0 and follows a Iearned vector field that continuously transforms this prior into the target data distribution at t=1.
In our coarse graph G. both nodes { and edges {e} are discrete categorical variables, for which we adopt DFM realized by continuous time Markov chain (CTMC) (Campbell et al.) (2024). In this section, we focus on the generation of fragment types. For latent vector and edges, we follow the standard flow- now-matching for continuous (Lipman et al. 2022) and discrete (Campbell et al. 2024) features, further described in section A.2
DFM for fragment types and Info-NCE Loss. Because realistic chemical spaces involve an extremely large vocabulary of fragment types [F], directly modeling CTMC over the entire space is computationally prohibitive. To address this, we adopt a masked version of DFM, which a node remains fixed once is de-masked, and further introduce stochastic fragment bag strategy to efficiently handle the large fragment vocabulary. Given noisy state X, (Ge, zt), we draw a subset C of size N from the full fragment vocabulary and then sample node 21 within this restricted subset. As a result, the model approximates the in-bag conditional posterior p(r1 X2, x1 E B) rather than the unconditional one (1) Xt).
To train the model, we follow the Info-NCE formulation (Oord et al. 2018), constructing fragment bags and parameterizing the density ratio (X.)/1 with a neural network fo. For each training step we build bag B that contains one positive fragment x (1|((11 (X) and N-1 negative fragments 2 sampled i..dd from the marginal fragments library distribution Pr (2)). Applying the Info-NCE formulation, we can write the in-bag posterior as
Pils (r X,, = 18(x) X.)/P((x)
(I)
Σ P1pl (,)/Pi
where 1g is an indicator function. We let the fo(X, x) approximate the unknown density ratio (t)) by optimizing with the standard Info-NCE loss:
So (x(
((0) = Es log EyeB fo(Xe.
E
which encourages the network to assign higher scores to the positive 2 within B while pushing down the negatives. Because the loss involves only E B, its cost scales with N rather than [F].
3.3 GENERATION PROCESS OF FRAGFM
During sampling, the model evolves nodes, edges, and the latent vector step by step from the prior distribution. This requires a discretized forward kernel expressed as: Pr+ (X++ X2))
PePe+sse X.) II (u (+ Xx) Pe+Aui(F44A Xt). Similar to Campbell et al. (202)). we modeled each transition of nodes and edges as independent. In this section, we focus on
(id)
the DFM process for nodes, while details of edges and latent vectors provided in section A.2
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In-bag transition kernel One-step transition kernel Pe+ be obtained by direct Euler integration of the rate matrix, which is computed by an expectation of 21 si-oondiiooned rate matrix over the full fragment set F. In standard DFM, this expectation is approximated by sampling 21 from the trained model P1 & (x1 X,). We instead define an in-bag transition kernel by restricting ø1 to a randomly selected subset B C F.
Following the conventional Info-NCE approaches, we construct B by drawing N i..d.. fragments from the marginal pt, assuming that B is independent of the current state. Consequently the B conditioned x -i-postrrior is
1s(z1)fe(Xx,,)) EyeB (o(
0 Pij P
B)
The bag conditioned forward kernel for node is then simply induced by:
pi (x == M ()),, (P
(3)
5 differs from the kernel without the bag. It nevertheless serves as a practical surrogate that converges to the exact one as the bag size N approaches the fragment-pon size |F] (Oord et al., (018)). When the Euler step size At is small, and the bag size N is moderately large, the discrepancy is negligible while the computational cost remains manageable.
Strictly speaking, eq.
Conditional generation While generating valid molecules is essential, steering them toward desired properties is crucial for practical use. Following Dhariwal & Nichol (2021); Vignac et al. (2023) we adopt classifier guidance, steering with an external property predictor.
Because framework employs a bag-conditioned transition kernel, conditioning introduces two key requirements. First, the selection of the fragment bag must be steered by the target property c, so that the candidate fragments align with the desired outcome. Second, the transition kernel must incorporate both the bag and the property, effectively forming a multi--onditiooed kernel. In practice, the first requirement is addressed by re-weighting fragment sampling probabilities when constructing B, guided by a property predictor Poipoo (c x) and a tunable fragment bag re-weighting parameter Ag. The second handled by guidance strength parameter Ax as in Vignac et al. (2023). Further details of the conditional transition kernel and the construction of are provided in section A.3
bags p(B
NPGEN: NATURAL PRODUCT GENERATION BENCHMARK
We introduce NPGen, new benchmark for molecular generative models focused on natural products (NPs), which are biologically synthesized compounds produced by organisms and characterized by distinctive structural features (Feher & Schmidt 2003 Stratton et al. 2015). NPs represent a biologically meaningful subset of chemical space, and serve a motivation for many approved drugs (Boutridi & Quinn 2018 Rosén et al.) 2009 Atanasov et al., 2021). However, existing benchmarks such MOSES and GuacaMol predominantly comprise small, structurally simple molecules, and their evaluation Ceatrics-Frécher ChemNet Distance, scaffold overlap, and KL divergence over simple properties-ant nearing saturation, limiting their ability capture NP-specific characteristics (Bechler-Speicheree et al. 2025). To address this limitation, we construct NPGen by selecting molecules from the COCONUT database (Sorokina et al. [2021] Chandrasekhar et al. (025), which occupies a distinct region of chemical space compared to existing benchmarks (lig. 2a) As result, NPGen contains 658.566 natural product molecules with an average heavy-atom count of 35.0 (larger than those in MOSES (21.7) and GuacaMol (27.9)) and with richer structural diversity characteristic of natural products (fig. 2b). NPGen's evaluation includes not only standard metrics (Validity, Uniqueness, Novelty) but also NP-oriented measures such as KL divergence of NP-likeness score distributions (Ertl et al. 2008) and NP's biosynthetic pathways and structure classes (Kim et al. (021), which capture molecular functionality and biological context related to its structure. We provide additional details on NPs (section B.I). dataset construction (section [B.2). baseline implementations (section B.3 evaluation metrics (section B.4). and dataset statisties (section B.5)

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MOSES
16
16
16
16-
12
12
12-
UMMP2
(a) UMAP plot of MOSES, GuacaMol, and NPGen datasets, each with 5,000 randomly sampled molecules.
OH
01
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Terporsids Iintaepennits
Tesperokts Tittarpencida /Cusurstaane
Shikinatts Phenstrropeniids /Hevonods
alikinatiss Phonolroopanids DDhyboflevonela
Flavosords
/Oleanans Interpenoids
Representative molecules from NPGen with NPClassifier pathwayssuperelasssclas annotations.
Figure 2: NPGen dataset overview. (a) UMAP visualization comparing MOSES, GuacaMol, and NPGen datasets. (b) Representative molecules from NPGen with annotations from NPClassifier (pathway, superclass, and class).
5 RESULTS
Here, we present the main results of FragFM on molecular generation benchmarks (sections and 3.2), conditional generation (sections C.8 and 5.3), sampling efficiency (sections C.9 and
5.
and an ablation on fragmentation rules (section 5.5
Extended analyses and ablation studies are provided in the Appendix, including an evaluation of retrosynthetic analysis (section C.2 an analysis of the coarse-to-fine autoencoder (section C.3) generalization to rare and novel fragment types (sections C.4 and C.5 further study of fragment
bag size (section C.6). experiments for novel molecule generation (section C.7) We also provide visualizations of generated molecules for both standard benchmarks and baseline models in NPGen (sections F.3 and F.4) as well as visualizations the latent space of fragment embeddings (section F2)
5.1 STANDARD MOLECULAR GENERATION BENCHMARRSS
We evaluate FragFM the MOSES (table I. GuacaMol (table 4p. and ZINC250K (table datasets,
which focus on small molecule generation. We compare against a range of baseline models spanning
different generation strategies and representation levels, with additional details provided in section D
For MOSES, we follow Vignac et al. (023) and report results on the scaffold-split test set. this benchmark, atomistic models typically underperform the fragment-based method on validity and FCD, Notably, FragFM achieves nearly 100% validity without any explicit validity constraints constraints-compardd comparable JT-VAE, which explicitly enforces molecular validity-and attains an FCD of 0.58, substantially outperforming all baselines. In addition, FragFM achieves state-of of-the-art (all FCDs, MOSES Filters, SNN, ZINC NSPDK) or near-best (GuacaMol KL Div.) results on property- and structure-based scores across benchmarks.
Despite strong results on most metries, performance on the MOSES Seaf and novelty metrics is relatively weaker. For novelty, we note that higher values do not always guarantee better molecular quality, as there exists a trade-off between fidelity and novelty (Mahmood et al., 2021 Geng et al.) 2023). In this regard, we demonstrate that FragFM can trade off fidelity for novelty through simple modification, which we describe in section C.7 For Scaf, this stems from the scaffold-spa -affold-split protocol of the MOSES benchmark: unseen scaffolds in the test set cannot be generated if they are absent from the fragment vocabulary, a limitation of fixed fragment vocabularies also observed in JT- VAE. Unlike prior fragment-basee based approaches, however, FragFM allows flexible replacement of the vocabulary due
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to the GNN-based fragment embedding. To verify this, we further demonstrate that it can generalize when equipped with a test-set fragment vocabulary, as detailed in section C.44
Table I:
Molecule generation on MOSES. We use 25,000 generated molecules for evaluation. The upper part comprises autoregressive methods, while the second part comprises one-shot methods, including caasioon-based and flow-based methods. Results for FragFM are averaged over three independent runs. The best performance is highlighted in bold, and the second -eett is underlined.
Model
Rep. Level
Valid
Unique
Novel f
Filters
FCD.
SNN 1
Scaf f
Training
100.0
100.0
100.0
0.48
0.59
0.0
GraphINWENT MMrcada et
2021
Atom (Graph)
96.4
99.8
95.0
1.22
0.54
12.7
VAE Jin et
2018
Frugment (Gruph)
100.0
100.0
99.9
97.8
1.00
0.53
10.0
SAFE-GPT C Noutahi e al 2024
Fragment (Sequeace)
98.1
100.0
90.9
98.2
0.71
0.54
9.8
MolHF Zhu d| 2023
Atom (Graph)
7100
100.0
100.0
17.8
35.4
0.23
1.5
MolGrom 100272500 Polykorskiy
2021
Fragment (Graph
100.0
100.0
99.8
82.7
6.39
7.7
DiGires Vienae et
2025
Atom (Graph)
85.7
100.0
95.0
97.1
1.19
0.52
14.8
DisCo NI
(Graph)
88.3
100.0
97.7
95.6
1,44
15.1
Cometh Strani
2024
Atom (Graph)
90.5
100.0
96.4
97.2
1.44
0,51
15.9
Cometthe PC Siratcn
2024
Atom (Graph)
90.5
99.9
92.6
99.1
1.27
0.54
16.0
DeFoG
Qin
Atom (Graph)
92.8
99.9
92.1
1.95
14.4
PMagFM (ours)
Pragment (Graph)
99.8
100.0
87.1
99.1
0.58
0.56
10.9
5.2 BENCHMARKING GENERATIVE MODELS ON NPGEN
We now present the comprehensive evaluation results for our proposed NPGen benchmark (section establishing baselines across diverse modalities including atom-fragmonn fragment--eeeeel and graph-sequence based models-alooe alongside our FragFM. As shown in table 2 while the sequence-based SAFE-GPT achieves high distributional aligmment compared to graph- based basclines, it exhibits limited novelty (73.5%). Among graph-based models, FragFM achieves superior performance on functionality-diin driven metrics, including NP-likeness and NP-Classifier scores, while maintaining high validity and novelty. Furthermore, surpasses atom-based discrete flow and diffusion models (DeFoG, DiGress) by substantial margin while offering significantly faster sampling (e.g., 5x faster than DiGress; fig. ( These results highlight the advantage of our fragment- based framework in efficiently capturing the complex structures of natural products. We provide visualizations of generated molecules for graph-based baselines in fig. and section
Table 2: Molecule generation results on NPGen. We use 30,000 generated molecules for evaluation. The upper part comprises autoregressive methods, while the second part comprises one-shot methods, including diffusion- based and Now-based methods. The results are averaged over three runs. The best performance highlighted in bold, and the second-best is underlined.
Model
Rep. Level
Val.1
Unique:
Novel
Score Div.
Pathway
Superclass
KL 1
Claes
FCD J
Triinin
100.0
100.0
0,0005
0.0002
0.0028
0.0094
0.01
GraphAE
Atom (Graph)
95.6
3.3907
6.6905
ITVAE di 2013
Fragment (Graph)
100.0
97.2
99.5
0.5437
0.1055
1.2895
2.5645
4.07
Hie m
Fragment (Graph)
100.0
$1.5
97.7
0,3021
0,4230
0.5771
1,4073
8.95
SAFE-GPT (Nattah
2024
Fragment (Sequence)
96.5
98.6
73.5
0,0024
0,0054
0.0414
0,1722
0.15
MolHF Zhu al. 2025
Atom (Graph)
71.0
59.6
97.6
0.883]
1 1.8022
9.1608
10.3760
31.26
DiGress Vigiaee
2026
Atom (Graph)
85.4
99.7
99.9
0.1957
0.0229
0.3370
1.0309
2.05
DeFoG
Qi
Atom (Grip))
85.9
98
99.2
0.1550
0.1252
0.4134
1,3997
4.46
FragFM (ouns)
Fragment (Graph)
98.0
99.0
95.4
0.0374
0.0196
0.1482
0.3570
1.34
5.3 ENHANCED CONTROLLABILITY VIA JOINT FRAGMENT AND CLASSIFIER GUIDANCE
key requirement for molecular generative models is controllability. iee., steering generated molecules toward desired properties while retaining validity and distributional fidelity. We evaluate the condi- tional generation ability of FragFM against the atom-based baseline DiGress, where both models employ classifier guidance scheme. For all conditional generation experiments, both FragFM and DiGress use property predictors trained on the full set of labeled molecules for each corresponding dataset.
To this end, we first vary the guidance strength (Ax) and plot the resulting trade-off curves (MAE-FCD and MAF-validity). When conditioning on simple molecular properties such QED
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ND
DiGress
HD
FragFMM
OH
Figure 3: Randomly selected molecules from DiGress (top) and FragFM (bottom) trained on NPGen. We randomly sample a moderate-sized molecule containing 31 to 40 heavy atoms. Chemically implausible moieties are highlighted in red. More examples are provided in section R.4
(figs. and 14). logP (fig. 15). and ring count (fig. [6). FragFM attains lower conditional MAE with lower FCD and higher validity, while the atom-based baseline often suffers sharp validity drops under strong guidance.
We further confirm that fragment-based generation provides additional flexibility through fragment- bag conditioning by λg. In the JAK2 docking score task (fig. 5). FragFM consistently outperforms the atom- atom-based baseline even without bag guidance (Ag O). Increasing Ag further shifts the generated distribution toward the target docking score while preserving nearly 100% validity, whereas DiGress not only suffers from substantial validity loss but also shows little shift the docking score distribution from the original data. Importantly, this conditioning is applied in particularly challenging region of ZIN2250k (top 0.08%), where such low docking scores are extremely rare, highlighting the robustness of Tragment-based design in realistic scenarios.
Finally, we analyze the joint effect of the two guidance terms, Ax and ÅB. As shown in the MAE-FCD curves (fig. 5), each curve corresponds to varying λx, while adjusting Äg consistently shifts these curves toward more favorable trade-off regions. This demonstrates that fragment- complements the standard classifier guidance, offering additional degree of controllability that is unique to fragment-basse the results of using Åσ only (red points in fig. indicate that employing conditional fragment bag alone can already achieve effective conditioning. Together, these findings align with the long-standing paradigms in medicinal chemistry (Sadybekov et al. 2022; Hajduk Greer (007)) and recent computational strategies (Lee et al. 2024). while emphasizing the importance of preparing property-oriee oriented fragment candidates.
reweighting
generation and unattainable in purely atomistic approaches. Moreover,
success of fragment-basee
Seap Lekee getwn
conncttin
7kc2554
DiGres
RoghH
000-13
TraghM U,
mayel Ua only
Grut
Denaity
0.70
6.1
0.00 ROTS 3.2 ouze kide a25 0200 a225 0229
6.0
3.5
2.2
35
3.0
2.5
0.50
Condition WAE
SMINA Docking Score (ocatimal)
Condition MAE
Figure 4: Property condi- Figure 5: Effect of À5 in conditioning. (left) A5 Ax 2.0: tioning results for QED. the DiGress guidance level is set as 2,000 for comparable FCD MAE-FCD curves for differ- ent target QED values. Each curve varies classifier guidance strength. Additional results are
values. (right) MAE-FOC curves ZINC250k with JAK2 docking score conditioning at -11.0 kcal/mol. Red markers indicate xx = 0.0. i.e. fragment-bag-oni guidance. Each curve shows results as the classifier guidance strength is varied. Additional results are shown in figs. to 20
shown in figs.
to 16

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Published as a conference paper at ICLR 2026
5.4 SAMPLING EFFICIENCY AND ROBUSTNESS OF FRAGEM
Iterative denoising in stochastic generative models involves a trade-off between the number of sampling steps and output quality. As shown in Figure 66 most diffusion- and flow-based models suffer declines in validity and FCD as the number of steps decreases, whereas FragFM remains robust, maintaining over 95% validity and FCD below 1.0 even with fewer steps. This efficiency arises from operating on fragment-level structures rather than individual atoms. Moreover, FragFM achieves substantially faster sampling time at the fragment level, as illustrated in Figure 7 Additional results and full tables are provided in section C.9
DiGress
DeFog
Cometh
FragFM
Thmee Ohoins)
NOARA (0 surgine
8.0
Validty
6.-
6
40
4
2
101
10a
10
10
10
Sampling Stops
Sampling Stops
DIG/mA
Detne
FrighM
TranIM
Figure 6: Analysis of sampling steps across multiple denoising models. FragFM maintains higher sampling quality than baseline atom-based denoising models as the number of sampling steps is reduced, exhibiting significantly less performance degradation. Ad- ditional results are provided section
Figure 7: Sampling time for MOSES and NPGen across different models. The num- ber in parentheses indicates the sampling steps.
5.5 ABLATION STUDY ON FRAGMENTATION RULES
We further ablate the fragmentation rule by replacing the default BRICS with RECAP (Lewell et al. 1998) and rBRICS (Zhang et al. (024). Although BRICS is used by default due its wide adoption and chemical interpretability, FragFM FM is not tied to a specific fragmentation scheme. As shown in Table FragFM maintains consistently strong performance under alternative fragmentation rules, while a finer fragmentation rule (rBRICS) slightly increases novelty. Overall, these results show that FragEM is robust to different Tragmentationniinuuce vocabularies when trained under each scheme.
Table 3: Molecule generation on MOSES and ZINC250K with different fragmentation rules. For IBRICS and RECAP, we used the same hyperparameter settings as FragFM (BRICS).
MOSES
ZINC250K
Model
Valid
Unique
Novel
Filters
FCD J
SNN
Scaf
Valid f
NSPDK J
FCD
Training set
100.0
100.0
100.0
0.59
100.0
0.0001
0.062
FragEM (BRICS)
99.8
100.0
87.1
99.1
0.58
0.56
10.9
99.81
0.0002
0.630
FM (RFCAP)
99.8
99.9
83.6
99.3
0.56
0.57
13.3
99.66
0.0003
0.580
FragUUM (rBRICS)
99.8
100.0
88.5
98.7
0.58
0.56
13.1
99.79
0.0003
0.563
CONCLUSION
In this paper, we have introduced FragFM, novel hierarchical framework with fragment-level discrete flow matching followed by lossless reconstruction of the atom-level graph, for efficient molecular graph generation. To this end, we proposed a stochastic fragment bag strategy with a coarse-to- fine autoencoder to circumvent dependency on limited fragment library cost-effectively. Standing on long-standing fragment-
strategies in chemistry, FragFM showed superior performance on the standard molecular generative benchmarks compared to the previous graph generative models. Additionally, applying classifier guidance at the fragment level and conditioning the fragment bag on the target property enables more precise control over diverse molecular properties. These significant improvements pave the way for a new frontier for fragment- taggeettbbased denoising approaches in molecular graph generation. Finally, to contribute the growth of the molecular graph-generatin domain, developed a new benchmark for evaluating models of natural products, which is also crucial in drug discovery.
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ACKNOWLEDGMENTS
We thank the anonymous reviewers for their constructive feedback and valuable comments, which helped improve this manuscript. We also thank Jun Hyeong Kim and Seonghwan Seo for helpful discussions and feedback that informed this work.
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF). funded by the Ministry of Science and ICT (Grant Nos. RS-2022- NR068758, 11-2223-NN0770H4 RS-2023-000 DS-2003-00255479, RS-2023-0021 1868). S.K. was additionally supported by Grant No. N10250153. It was also supported by a grant from the Korean ARPA-H Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health and Welfare, the Republic Korea (RS-2024-005124998)
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A ADDITIONAL METHOD DETAILS
A.1 COARSE-TO-FINE AUTORNCODER
To convert between atomistic and fragment-level representations, we introduce coarse-to-fin to-finc autoencoder. Since the coarse graph already captures the abstract molecular structure, the primary role of the autoencoder is restore atomistic connectivity information that is lost during the fragmentation process.
Coarse-graph encoder In the encoding phase, each molecule is first decomposed into fragments using the BRICS decomposition rules (Degen et al., 2008), yielding a fragment-level graph G. During this decomposition, connectivity information between atoms belonging different fragments is discarded; for example, the relative orientation of an antisymmetric fragment with respect to others cannot be determined solely from the fragment graph. Unlike conventional autoencoders, our encoder network is therefore only required to encode the missing atomistic connectivity into the latent variable 2, since the coarse graph already retains the overall molecular structure.
Fine-graph decoder In the decoding phase, the goal is to restore atom-level connections between fragments that are linked the coarse level. Each fragment defined by junction atoms that mark the cut sites introduced during fragmentation. As a result, we only need to consider connectivity between junction atoms across neighboring fragments, rather than all possible atom pairs. From the coarse graph, we extract candidate junction-atom pairs, and the neural network predicts their connectivity as a continuous score conditioned on the latent variable z. These scores are then discretized into final bond assignments using the blossom algorithm, the details of which are provided below.
Formally, the encoding and decoding process summarized as:
Encoder:
G Rule
G, (G, Pono
2,
Decoder:
(G, 2) denc, score
Blossom, E.
(4)
Training details We train the coarse-to-fing autoencoder by minimizing reconstruction losses over atomistic connectivitics. During training, each connectivity is treated as independent of the others, and the loss is formulated KL regularization term, as in VAEs, to the training loss on the latent variable in order to enforce a well-struetured and properly scaled latent space. The total loss is formulated as:
binary cross-entropy objective. In addition, add small
(AF(#)= = EG-pas
Σ (6.))E
LACK (eig, Big) + Dxt. (4s (z)()) p(2))
(5)
where eij and ěiy denote the ground truth connectivity and predicted score between atoms and j. and A is the set of candidate junction-ates -ttoo pairs derived from the coarse graph G. We set a low regularization coefficient of 3 0.0001 to maintain high-fidelity reconstruction.
Atom-level reconstruction by Blossom algorithm Although each connectivity is trained inde- pendently, reconstructing a valid graph from the predicted scores requires accounting for their dependencies. To this end, we employ the Blossom algorithm (Edmonds) 1965) to determine the opti- mal matching on the atom-level graph. More specifically, the Blossom algorithm returns a matching in which each atom is constrained to be paired with at most one partner, while maximizing the sum of connectivity scores (logit) predicted by the decoder. Within our framework, this procedure ensures accurate reconstruction of atom-level connectivity from fragment-level graphs, thereby yiclding chemically valid molecular structures.
The algorithm takes as input the matching nodes Vm, edges E,, and edge weights Wiy. Once the fragment-leve graph and the probabilities of atom-level edges from the coarse-to- torrse-to-fine to-fine autoencoder
computed, we define V.. as the set of junction atoms in fragment graphs, which are marked in fig. and E,, as the set of connections between junction atoms belonging to connected fragments.
Formally, edge est exists in Em if the corresponding atoms belong different fragments that connected the fragment-level graph, expressed as:
Ekl E E,,, if uk Vi, ey Vj, and Eij E E,
(6)
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where Eay denotes the coarse level edge between i-th and j-th fragments, and V; S Vm denotes junction atoms that included in the i-th fragments. The edge weights Waj correspond to the predicted logit of each connection obtained from the coarse-to-fin autoencoder. The Blossom algorithm is then applied to solve the maximum weighted matching problem, formulated as
M argmax
Σ
Way
degm (v) SIAEE
(.J)MM
Here, M represents the optimal set of fragment-level connections that best reconstructs atom-level connectivity, maximizing the joint probability of the autoencoder prediction. Although the algorithm has an O(N*) complexity for N fragment junctions, its computational cost remains negligible in our case, as the number of fragment junctions is relatively small compared the total number of atoms in molecule.
A.2 DETAILS OF FLOW MODELING
In this section, we describe the details of flow modeling in the FragFM framework. The generation process jointly produces the coarse graph G and the latent vector 2, where the generation of G can be interpreted as joint procedure over nodes and edges. total, three different types of variables are generated jointly: two discrete variables (node and edge states) and one continuous variable (the latent vector). Below, we detail the flow formulation and the corresponding loss functions for each type.
A.2.1 FLOW MODELING FOR NODE
Following the original DFM formulation (Campbell et al. 2024). we specify the distribution of nodes (fragment types) 1-1 pi (x) and define the x condditiooe time marginal by linear interpolation:
Pt (xze I1) = 16 .1) + (1-t) pa (2:4,
(8)
where Po is prior and -(- is the Kronecker delta. We adopt the masked version of DFM proposed by Campbell et al. (2024), , in which the prior distribution assigns probability only to the mask token. With this choice, the interpolation distribution in eq. (8) becomes particularly simple: it is nonzero only when F, equals the mask token or the data token IT
The corresponding CTMC transition rate is provided by the authors,
Re(ar, mn) (1 21) 0xPe xz))
Vo y,
S Pe (x x1)
with the number of states for which P= x1) > 0. brief algebraic manipulation of the Kolmogorov forward equation yields the 21 uncoondiiioo generator
Re(x, y) (x)=p))(((z)) [e(2,, y x1)]]-
(10)
from which we can sample trajectories { Realizing these trajectories requires the posterior distribution (2) Xt), where X, denotes the noisy version of the coarse-graph G, and latent vector z₁ As described in the main text, we approximate this posterior by modeling the bag-conditioned distribution (x) X,, B).
A.2.2 FLOW MODELING FOR EDGE
Let ed E & denote the absence (0) or presence (1) of an edge between the i-th and j-th fragments in the coarse graph. Because the edge state is binary, we adopt Bernoulli prior with probability density (mass) function (s for both states.
Following the discrete flow-matching (DFM) recipe, the -conditioned time marginal is Pr Prle E1)= = t t(E s1) + (1 Po(ee),
t E [0,
(11)
The CTMC rate that realizes this marginal is
ReLU 0 (
E1) - app. ((El
Re(e,e' |
VE FE
(12)
2p (=] E1)
Re(s,e') = Erro (1to es, s1)].
(13)
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The posterior pi ja(=)" X2) is parameterized by neural network 96 by minimizing the cross cross-entrope entropy loss:
elge (Xt)iss- We trained the model
Cdgge = Σ Exx,xx,1
[-e log 9a (Xt)ss - (1 - Log (1 (1- 98 (xi)
(14)
In the sampling phase, the neural network replaces the posterior in eq. expectation above is computed exactly no Monte-Carlo sampling is required. Thus, the forward kernel for j-th edge is as:
(13 Because [8] = 2, the
Pi X,) 8(E), at Ra(EEEEE At.
(15)
A.2.3 FLOW MODELING FOR LATENT VECTOR
Let z E Rd be the continuous latent vector attached to the fragment-level graph. We model its evolution with conditional flow matching (CFM; Lipman al. (022)), which views generation as integrating an ODE whose time-dependent velocity field (VF) is learned from data. Specifically, we linearly change the mean and standard deviation (1(x) tæ1 and (2) conditioned target VF is
The corresponding
Me(2)|z1)
21
(16)
Then, the trajectory for a prior sample 2g NO, I) and a data sample 2 ~ pi (2) under the target VF, ie., the solution di ue (1) with 20 given by:
21 (1 1)=0 + t21,
LE 10.
(17)
We fit neural vector field (X.) by minimizing the mean- -quared error
CCrm Ex..x.. [l)((X))
(18)
t
To generate a latent vector, we solve the ODE
da,
uo (Xx),
(19)
dt
forward from t 0 to t = with deterministic solver. The resulting & is then fed to the coarse-to- fine decoding network to obtain an atom-level graph.
CFM as a Limiting Case of VE Diffusion Bridge Unlike diffusion models, which first define reference process and then learn its drift, CFM directly prescribes the time-marging distribution and optimizes the corresponding velocity fields that "point" toward fixed data point. This raises question: How can we treat CFM with the transition kernel ( 2x) used in diffusion models?
A diffusion bridge is reference diffusion process conditioned to hit fixed endpoint. Its SDE is
dæ, f(z, t) + gt()) =., log Q(=r 21) 1=) dt + gff dw.,
(20)
where ((T | 2x) is the unconditioned transition kernel of the reference process, f its drift, and g its diffusion coefficients.
If the reference process is the variance-exploding (VH) diffusion dz, =(() dw,, Zhou et al. 2024
show that equation 20 reduces to
do 2 //tt o o?
dz
(2T 22) dt + g(t) dwz.
(21)
Setting T = 1 and of et (constant c) gives
dæ,
dt cdw,,
(22)
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Taking limit of 0 eliminates the stochastic term and leaves the deterministic drift
dæ Zz
(23)
dt 1 -
which is exactly the velocity field optimized by CFM, i.e., the VE diffusion bridge collapses CFM.
Given coupling (20, 21), we can form mixture bridge II by averaging the pinned-down trajectories over w. According to Proposition Shi et (2023), its Markov approximation M satisfies
21
2t d + edw,, with Mj = II, Vt.
dæ3- = E
(24)
When O, the drift term above coincides with the averaged velocity field learned by CFM eq. [18 confirming that M recovers the CFM dynamics in the zero-noise limit.
A.2.4 TRAINING DETAILS
Our objective is to learn a generative diffusion the coarse graph state (Recall X, = (G,, 2ς with
G: = ({a+), {= the node-type vector z/, binary edge matrix e and latent vector 2₁ E Rd). By combining the node-type Info-NCE loss, the edge binary-cross-entropp loss, and the latent CFM loss into a single training objective.
Sampling a training triple tt.
1. Data endpoint. Sample a atomistic graph G from the molecular dataset, and apply coarse- to-fine encoder to obtain X; = (91,*1). =1).
2. Time sampling. Sample a time / E [0, 1] from a uniform distribution.
3. Forward noise. Independently transform each component to its noised counterpart:
Node. For every node indexed with i, sample xt Papi( æf) with the masked prior.
Edge. For each pair (i.j), draw EX² (:pp E using eq. 111.
Latent. Sample 2n ~ (O, I) and set 2, = (1 =) + t2 as in eq. 17
4. Construct X,. Collect the three noised components into X, (G,, zt).
Joint loss.
fo x1)
Lroaee (0; B) - log
EyeB So (x,y))
Ledge(8) = E log 90
odge (Xt)is (1- )oogl 90 odge
(Xi)6)
14
Cissen (0) = -llulatent (Xt) - (21 20)1
18
We minimize the weighted sum
(0:tal(: ():B) Lnode (0; B3 + Ohedge Ledge (0) Oliteet Ltatent (0),
(25)
with Oredge; O'atten >> O. We fix Oedge 5.0 and O'laten = 1.0 for all of our experiments.
A.3 CONDITIONAL GENERATION WITH FRAGMENT BAG
While generating valid molecules is essential, steering them toward the desired property is crucial for the practical application molecular generative models. Direct conditioning via classifier-free guidance (Ho& & Salimans 2022) offers strong control but tightly binds the generative network to specific properties and often requires retraining when new targets are introduced. Instead, we adopt classifier guidance, steering generation sampling time with external property predictor. This decouples the generator from any single conditioning signal and allows the predictor to be trained or updated independently (Dhariwal & Nichol 2021 Vignae et al. 2023)
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To generate molecules conditioned external properties, require property transition kernel, denoted Pe+ |X,, c). The transition kernel X factorizes into kernels for nodes, edges, and the latent vector. Among these, node types require special treatment because they sampled from a candidate fragment bag B.
Analogous the unconditional forward kernel using the bag strategy as
described in the main text, we define the property-conditioned
Pe+Aef (Xe+ar Xt,c) s p(X++ X1,8%.c)
(26)
where B denotes the property-conditioned fragment bag. The original constructed by sampling one positive fragment from x p( X,) and N-1 negative fragments from x- p(z). Analogously, the c-conditioned bag 8 is constructed by sampling one positive fragment from xt p( Xx, c) and N-1 negative fragments from
p(> c). By Bayes' rule, the
conditioned in-bag transition kernel can be factorized into the c- c-unconditional kernel and a guidance ratio:
p(c XI+ B,, p(c Xx, B
p(X++ee Xx, = p(Xe+se B).
(27)
Guidance ratio
Guidance ratio modeling
The guidance ratio in eq. 27 p(c|Xe+sr, X2,
can be written by:
p(c]X++s1, B
((c)X,,B')
p(c|X;,, Bc)
p(c(X(+s))p( (B X(+4))/p(( ((-ai)
p(c(X,)p(BB1 x.)/p(BB|X,,
The ratio P(
(eX)). is intractable, yet it involves the difference between two consecutive states X₁ and X X(tA. Because an Euler step is very small, it can be assumed that the diffusion O(At). If the bag- sag-sampling distributions p(B* X) and p(B c, X) vary continuously with X, a first-order Taylor expansion yields
state evolves smoothly: X,+
p(B" X(4e))) = p( (x)))++ + O((t),
(28)
So the whole ratio is approximately 1, to be
p(c X(c|x XX, P(c| BC)
p(c|X++01)
(29)
p(c(X,)
Following Nisonoff et al. (2024): Vignac et al. (2023), we can estimate the ratio via noisy predictor o(X.) with 1tt order Taylo lor expansion, yielding
log p(c|X(+te) (V x, logp(c|XX,), -
p(c(X,)
logp(c(X,))
1 (2d + Σ
o0 log logp(cX))..
8)
V logp(c|X.), #4 2x) + C.
In practice we estimate X,) by Gaussian modeling with a time-conditioned noisy classifier parameterized by v. N. (Xe, v), )2. Thus, the guidance term is written as:
p(c)X++
x eexp(-1 -xx
(xx, x.2
(1)
(30)
p(c(,,))
x exp(-Xx -xx
[X,, - -c1,, c(())
x expl -xpl-Axxio (V | (xx - cll?, - =z)),
where Ax controls the strength of the guidance.
The smoothness assumption adopt is exactly the adopted by earlier discrete- gurirnne methods (Vignac et al. 2023 Nisonoff et al. 2024), our derivation remains consistent with the foundations laid out in those works.
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Conditional bag sampling To sample B first recall the unconditional case. When property specified, bag B = {r1..... ........ of size N drawn without replacement from the fragments vocabulary F with probability
II ((x)
P(B)
ZEB
(31)
Σ
p(y)
SCF
WES
|6|=N
i.e., bags that contain high-frequency fragments under the marginal distribution p(z) are sampled more often. To steer this toward a desired property we replace each p(x) by its conditional counterpart p(zlc). Viewing a fragment as part of molecule the condition can be written as the expected property value over all molecules that contain that fragment: Ex p(r X, c)p(X c).
The resulting bag distribution becomes
II ZEB=
p( le)
P
(c)
(32)
ply
cc)
DES
|S|=N
Applying Bayes' rule and dropping the constant factor p(c) gives
p(xle) x ((z)p(((/))
(33)
In practice, we estimate p(c|z) as Gaussian distribution with a light neural regressor parameterized by vi,
Po(c x) = (c; pB(z; 2²). p c) x p(z) exp(-AB || [B( e) cll²),
(34)
where (3, v)) is the predicted mean, o² is fixed variance, and A controls the strength of the property-guided bag selection.
To parameterize the guidance ratio in eq. (30), we train time-conditioned noisy property regressor μx (Xe, t) that predicts the target property from the corrupted molecule state Xx:
(x(r) Ex 10) (x,= px (X,, [|]]]
(35)
We also train fragment-level regressor 1us (x) that estimates the global property from each fragment, as required for the conditional fragment guidance in eq. (34)
(b()) = E(X,O
Σ (ET(X)
(uB(;; 1b) cl12
(36)
Both objectives are optimized jointly during training with shared parameters. Architectural details are provided in section E.2 and the corresponding hyperparameters are summarized in table 17
A.4 DETAILED BALANCE
The space of valid rate matrices extends beyond the original formulation of eq. (9; thus, alternative constructions can still satisfy the Kolmogorov equation. Campbell et al. 2204 show that if matrix RPB fulfils the sttailed-balanee identity:
(:j 11) RPB
(xt. y x1) (eji(1 x1) R (,, x1).
(37)
then,
R = R + RRP3 IR remains a valid CTMC generator. A larger n injects extra stochasticity, opening additional state- transition pathways.
(38)
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Although several designs are possible, follow Campbell et al. (2024). The only non-zero rates for fragment-type nodes are the transitions between a concrete type œ1 and the mask state M:
RPB (1, = 21 0(z M 21),
(39)
where M denotes the masked type.
For edges, whose states are binary ed {0, 1}. consider flip rate nedge and matching backward rate that satisfies eq. (37) which leads to:
+ (E,E' Ei) = 5(e, E1) + I -- ((5,,e)).
(40)
We set (Trode Medge) = 20.0) for the MOSES and ZINC250K datasets, and (2.0, 20.0) for the GuacaMol and NPGen datasets.
B DETAILS OF NPGEN BENCHMARK
Here, we provide additional details on (1) the dataset construction process from the COCONUT database, including the filtering steps, (2) the NP-specific evaluation metrics and how they are computed, (3) the statistics and distributional properties of the NPGen benchmark, and (4) the baseline models and setups used in our evaluation. These details complement the high-level description in the main paper and are intended facilitate reproducibility and further use of NPGen by the community.
B.1 BACKGROUND ON NATURAL PRODUCTS
Natural products (NPs) are chemical compounds biologically synthesized by organisms, e.g., plants, fungi, and bacteria. They have long served as a valuable resource in drug discovery, with their unique structural features compared to typical synthetic compounds- more complex ring architectures, higher heteroatom density, and abundant oxygen-basee functional groups that contribute to polarity, stereochemical diversity, and bioactivity (Feher & Schmidt [0033] Stratton et al., 2013). In essence, NPs occupy biologically meaningful subset of chemical space, often resembling endogenous metabolites, which makes them particularly effective as templates direct sources for drug design (Boufridi & Quinn, 2018: Rosén et al.) 2009; Attanaovv et al.) 2021p. Moreover, unlike many syn- thetic compounds, NPs frequently resemble endogenous metabolites, increasing their likelihood of interacting with biological targets in meaningful ways.
substantial portion of clinically approved small-molecule drugs are inspired by or derived from NPs, mimicking their structures or functionalities (Newman & Cragg] [2020). Multiple classes of drugs. such as antibiotics (penicillin, derived from Penicillium fungi; erythromycin, from Saccharopolyspora erythruea bacteria), anticancer agents (paclitaxel, originally isolated from the Taxus brevifolia (Pacific yew tree); doxorubicin, from Streptomyces peucetius bacteria; vincristine, isolated from Catharanthus roseus (periwinkle)), and immunosuppressants (cyclosporin A, from the soil fungus Tolypocladium inflatum) demonstrate the long-stann standing role of NPs in drug discovery. Consistent with the above, analyses of FDA approvals indicate that nearly one-third of all small-molecule drugs introduced since the 1980s are prouuaa-prouuu-deeriiee therapeutic relevance (Atanasov al. (02II. In this regard, generative models capable of reflecting NP-specific structural and biological characteristics represent an important frontier for computational drug discovery.
or natural- natural-product-ing produuct-inspired. underscoring their sustained
For the machine learning community, natural products (NPs) present a particularly valuable and chal- lenging benchmark: their structural diversity, functional complexity, and biological relevance extend far beyond the synthetic-like molecules that dominate current datasets. While modern generative models have achieved strong performance on widely used benchmarks such as MOSES (Polykovskiy et al. (020)) and GuacaMol (Brown et al.) 2019). these datasets are largely composed of small, structurally simple molecules, and their evaluation metrics-FFrčhet ChemNet Distance, scaffold overlap, and KL divergence basic descriptors are approaching saturation. As result, they cannot adequately assess models aiming to capture the complexity and biological meaningfulness of NPs. A benchmark centered NPs, therefore, not only reflects real-world NP inspired drug discovery needs but also provides a domain-speciie stress test for molecular generative models, probing their ability to extrapolate to richer regions of chemical space (Bechler-Speicher et al. 2025
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B.2 DATASET CONSTRUCTION
To construct the NPGen dataset, we utilized the 2024/12/31 version of the COCONUT database (Sorokina al. 2021; Chandrasekhar et al. 2025), which comprises 695, 120 natural product-like molecules. Given that the original database contains compounds with transition metals-Sseciie that are rarely encountered in typical organic natural products-w, applied a filtering procedure to retain only molecules composed exclusively of non-metal atoms: 'B', 'C', 'N', 'O', 'F', 'Si', 'P', s 'Cl', "As', Se', 'Br', 'I' Additionally, to exclude arbitrarily large macromolecules, we retained only those molecules whose heavy-atom counts fell between 2 and 99.
Furthermore, we consider only neutral molecules, excluding salts and molecules containing "." in their SMILES representations. After filtering, 658,566 molecules were retained. The resulting dataset was randomly partitioned into training, validation, and test subsets using an 80:5:15 split under the assumption of i.idd sampling, yielding 526,852, 32,928, and 98. 786 molecules, respectively.
B.3 IMPLEMENTATION DETAILS FOR BASELINES
As baselines, we selected set of molecular graph generative models with two aspects, i.e., generation strategy (autoregressive and one-shot) and representation level (atom and fragment). We provide more details on the baseline models.
GraphAF (Shi al. 2020) is flow-based autoregressive model for molecular graph generation that constructs molecules sequentially by adding atoms and their corresponding bonds. We used the authors' official implementation from /https://glt titbbbbtbb/bt/// com
CraphAF) with its default settings, extending only the preprocessing and generation steps to include atom types that the original implementation does not support 'B', 'As', 'Si', 'Se'. During generation, the official implementation terminates sampling once 40 atoms have been generated per molecule; increased this limit 99 to match the NPGen benchmark's maximum heavy-atom count.
JT-VAE (Jin et al. 2018) is fragment- based autoregressive variational autoencoder that generates molecules by building a junction tree of chemically meaningful substructures and then assembling the corresponding level graph (Tin et al., 2018). We used the authors' official implementa- tion (https: //tpbbb. com/ wengong- jin he//mmmlll 1- tnnn) with the acceleration module (fast_mad www.molvvvvao lva).. Because the codebase relies on Python 2 and is incompatible with newer GPU drivers, we performed training and sampling an NVIDIA GeForce RTX 2080 Ti. We performed random hyperparameter search over h fridde__iii and batch_s zi, and report the best results.
HierVAE (Jin et al. 2020a) builds on JT- VAE by introducing hierarchical latent space and scaffold- aware passsageppasiing scheme to boost structural diversity and sampling fidelity. We used the authors' official implementation (https: / /ig thub. com wwt.w//w/wwww.ee. www.ong jin http/ggaahhggaphl//m extending only the preprocessing step to include the "As' atom type. By default, VAE employs greedy motif-sampling strategy, which prioritizes the top-scoring fragments and may bias the output distribution. We observed that this led to artifacts, only generating single carbon chains on the NPGen benchmark. To provide fair comparison, we report the results of the alternative stochastic-sampling mode (enabled via a single option flag in the official implementation), without modifying the core codebase.
SAFE-GPT (Noutahi et al. 2024) is an autoregressive sequence generative model that produces SAFE (Sequential Attachment-base Fragment Embedding) strings-a novel representation proposed in the paper, where molecules are expressed as sequences of fragments. The model is built on a GPT-2-like transformer architecture. While the original work reported only basic results on validity, uniqueness, and diversity, we trained the SAFE-GPT-20M model using its official implementation to obtain comprehensive benchmark results. We trained the model using AdamW (learning rate 5c-4) with linear warmup schedule over the first 10k steps. The tokenizer was constructed for the NPGen benchmark, and training was conducted for 40k steps with a maximum sequence length of 275. For every step in data processing, training, and sampling, we used the original code from https::/githun com/ amo 1o/ safe.
MolHF (Zhu et al. 2023 is hierarchical normalizing flow model designed for molecular graph generation. It employs a coarse-to to-fine strategy, generating a fragment connectivity graph and sub- sequently refining it the atom level via conditional flows. We trained MolHF with its default
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setting provided in its httttt://///ttit wtt.://u.oooooilooeetttt/soo///e
The model uses condi- tional latent variables, Ixl convolution, rotation of node features, and a learned prior. Since the NPGen dataset has larger molecules than the previous benchmarks, we had to decrease the batch size, resulting in the training with Adam for 200 epochs (batch size 64, learning rate 2e - 4).
DiGress VVgnac et al. 2023) is atom-based generative model that employs discrete diffusion. We run the authors' official implementation (bttps: /tte::///ttttbb com/ cv
with all default hyperparameters, adding the atom types 'B', 'As' and their corresponding charges.
DeFoG (Qin et al. 2024) is an atom-based generative model that employs discrete flow matching. We run the authors' official implementation (https://github) wwt.cccmmmmmmmm mannue m maddee ra/ DeFoG) with all default hyperparameters, adding the atom types 'B', 'As' and their corresponding charges.
B.4 METRICS
As mentioned in the main text, utilize two methods for distributional metrics: NP-likeness score (Ertl et al.] 2008) and NP Classifier (Kim et al. 2021). Both strategies are developed by domain experts to effectively analyze the molecule through the lens of natural product.
NP-likeness score is developed to quantify the similarity of a given molecule to the structural space typically occupied by natural products. Since one of the major differences between NPs and synthetic molecules is structural features such as the number of aromatic rings, stereocenters, and distribution of nitrogen and oxygen atoms, the NP-likeness of a molecule is calculated as the of the contributions of its constituent fragments, where each fragment's contribution is based on its frequency in natural product versus synthetic molecule databases. We show the distributions of NP-likeness scores for existing benchmarks and NPGen in fig. 2 Since the source database of NPGen, COCONUT, aggregates compounds from diverse sources, it naturally contains molecules across a wide range of NP-likeness values. Importantly, the goal of generative model is to faithfully reproduce the dataset's 's distribution rather than simply maximizing NP-likeness. To this end, we measure the Lulbbr (KL.) divergence between the distributions of NP-likeness scores for generated and reference molecules.
Since NP-likeness scores are continuous, we adopt non-parametric estimation procedure from the prior benchmarks, such as MOSES and GuacaMol. Specifically, we calculate NP-likeness scores for all molecules in both sets, then estimate their probability density functions (PDFs) using Gaussian kernel density estimation (KDE). We evaluate both PDFs on a common range discretized into 1,000 points spanning the observed minimum and maximum scores, adding small constant (10-10) for numerical stability. The resulting discrete distributions are compared via KL divergence using the scipy- stats. ent ropy function, providing robust measure of how closely the generated distribution aligns with that of natural products in the reference set.
NPClassifier is deep learning-based tool specifically designed classify NPs. categorizes molecules at three hierarchical devels-Pathway (7 categories; e.g., Polyketides, Terpenoids), Superclass (70 categories, e.g., Macrolides, Diterpenoids), and Class (672 categories; e.g., Ery- (aspemooiddd-fflecciiiii the biosynthetic origins, broader chemical and chemo-
thromycins, Kaurane taxonomic properties, and specific structural families recognized by the NP research community. This multi-level system, built on an NP-specific ontology and trained on over 73,000 NPs using counted Morgan fingerprints, provides classification based on knowledge of natural products, including their biosynthetic relationships and structural diversity.
We employed Kullback-Leibler (KL) divergence as a metric for both methods. We compute the KL divergence for the NP-likeness score and the NPClassifier differently, they are continuous and discrete values, respectively. is worth noting that NPCIassifier often predicts *Unclassi aalllssssieed fied', which indicates molecule is not included in any classes, along with multiple class results (e.g., PPett ide alkaloids, Tetramate a alkaloido in Class). We treat all prediction results another unique class, since molecules can have multiple structural features.
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B.5 DATASET STATISTICS
We analyzed the distributions of several molecular properties to highlight NPGen's distinctions from standard molecular generative benchmarks (MOSES and GuacaMol). These properties fell into two categories: (1) simple molecular descriptors, such as the number of atoms, molecular weight, and number of hydrogen bond acceptors and donors (fig. 8 and (2) functionality-relate properties. including NP-likeness scores and NPClassifier prediction results (fig.P) Consistent with the nature of NPs, which are generally larger and more complex than typical synthetic drug- -ike molecules, NPGen molecules are, on average, larger in terms of the number of atoms and molecular weight compared to those in MOSES and GuacaMol (figs. Sa and Sb). Furthermore, molecules in NPGen exhibit higher numbers of hydrogen bond acceptors and donors (see figs. 8c and 8d), reflecting another characteristic of NPs.
The difference between benchmarks becomes significant when examining functionality-relatte properties. NPClassifier predictions for Pathway (fig. (aa indicate that NPGen molecules span diverse range of NP categories. In contrast, molecules from MOSES and GuacaMol mostly fall into 'Alkaloide" which are non-peptidic nitrogenous organic compounds, or remain unclassified. Focusing four selected Superclass categories for which NPClassifier had demonstrated high predictive performance (FI score higher than 0.95 for categories with more than 500 compounds (Kim al., 202T)) NPGen shows higher proportions of molecules in these specific categories. Conversely. molecules from the other benchmarks mostly fall into *Unclass" fied', implying that they are dissimilar NPs. The NP-likeness score further emphasizes this divergence (fig. (9c). In particular, NPGen's ' distribution is largely shifted toward higher scores (average: 1. 1.14) compared with MOSES (average: -1677) and GuacaMol (average: -0.90), where higher scores indicate greater similarity to NPs.
Additionally, we visualize the chemical space of existing benchmarks (MOSES, GuacaMol) and NPGen using UMAP (MeInnes et al. 2018) in Figure 2a While MOSES and GuacaMol largely overlap, NPGen extends into distinct regions, indicating coverage of different chemical subspaces.
These statistical analyses demonstrate that NPGen has distinct features compared with existing molecular generative benchmarks, establishing its suitability as unique molecular graph generative benchmark targeting NP-like chemical space.
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0.34
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1000
1250
1500
1750
Number of Atoms
Molecularr Weight
(a) Number of atoms distributions.
(b) Molecular weight distributions.
0.25
0.4
Dersity
0.3
0.30
0.06
0.00
0.0g
10
Numbar Hydrogen Accoptors
Numbor Hydragon Donors
(c) Number of hydrogen bond acceptors distributions.
(d) Number of hydrogen bond donors distributions.
Figure 8: Comparison of simple molecular property distributions among three benchmarks: MOSES, GuacaMol, and NPGen. The number of molecules in each dataset is 1,936,962, 1,591.378, and 658,566, respectively.
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80%
01%
50%
30%
30%
30%
22%
20%
17%
13% 13%
10%
0%
Alkololds
Sh Shiidmates Phenylpropenoids
4%
0%
Fatty acids
Tarpenoids
3%
Carbohydrates Amina acids Peplides
0%
1%
35
3%
3%
Poly eotiiess
2%
Unclassifled
Pathways
(a) Proportion of all Pathway, along with "Onelassified" Note that we excluded predicted results with mul- tiple categories, for clarity. Specifically, number of predictions decreased as 1,936.962 to 1,873,287, 1,591,378 185 and 658,566 628,840 for MOSES, GuacaMol, and NPGen, respectively.
0.7
60
20%
209
106
4.hh 9.7%
0.6%
5.0% 6.2%
8mh am
Superclasses
(b) Proportion of four Superclass, along with "Unclassifiee fied'. Note that we excluded pre- clarity.
0.2
0.1
dieted results with multiple categories, Specifically, number predictions decreased as 1,936.962 to 1,933,854, 1,591,378 to 1,588,486, and 658,566 to 650,013 for MOSES, GuacaMol, and NPGen, respectively.
0.0
NP-likoness score
(c) NP-likeness score distribution.
Figure 9: Comparison of NP-likeness score and NPClassifier prediction results among three benchmarks: MOSES, GuacaMol, and NPGen. The number of molecules in each dataset is 1,936,962, 1,591,378, and 658,566, respectively. Note that we also report the ratio of unclassified entities in the datasets in figs. 9a and 9b Class prediction statistics are not included since there 687 classes and the ratio of each class is too small compared to that of the MUnclassi i fied' class.
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C ADDITIONAL RESULTS AND ANALY
C.I ADDITIONAL BENCHMARKS
In the GuacaMol (table and ZINC250K benchmarks, similar MOSES, FragFM achieved the best performance among diffusion- and flow-based baselines in terms of Validity and V.U. metrics, obtained a state-of-the-ar FCD score, and ranked as a close second in the KL Div. score GuacaMol. On ZINC250K, FragFM achieved the best performance across all reported metrics, surpassing the strongest atom-based baseline with five-fold improvement in NSPDK and a two-fold reduction in FCD. These results underscore the effectiveneess of the fragment-basee based approach of FragFM in generating valid and chemically meaningful molecules. Visualization results for GuacaMol are provided in section
Table 4: Molecule generation results the GuacaMol benchmark. We total of 10,000 generated molecules for evaluation. All baselines except MCTS this table are one-shot methods. The results for FragFM are averaged over three independent runs. The best performance is shown in bold, and the second-best is underlined.
Model
Rep. Level
Val. f
V.U.
V.U.N. 1
KL. Div.
FCD 1
Training set
100.0
100.0
99.9
92.8
MCTS (Jensen 2019
Atom
100.0
100.0
95.4
82.2
1.5
NAGVAE (wonn et
Atom
92.9
88.7
88.7
38.4
0.9
DiGress Vignac
Atom
85.2
85.2
85.1
92.9
68.0
DisCo Xu Xu ml 2124
Atom
86.6
86.6
86.5
92.6
59.7
Cometh Siraudin al
2024
Atom
94.4
94.4
93.5
94.1
67.4
Cometh-
Atom
98.9
98.9
97.6
96.7
72.7
DellCCi
QQin et al. 2024
Atom
99.0
99.0
97.9
97.7
73.8
FragEM (ours)
Fragment
99.7
99.3
95.0
97.4
85.8
Table 5: Molecule generation on ZINC250K benchmark. We use 25,000 generated molecules for evaluation. The upper part comprises autoregressive methods, while the second part comprises one-shot methods, including diffusion- based and flow-based methods. The best performance highlighted in bold, and the second-best is underlined.
Model
Rep. Level
Valid +
NSPDK
FCD 1
Training
0.0001
0.062
GraphAF Shi al.
2020
Atom (Graph)
67.92
0.0432
16.128
GraphDF Ldo ct
2021
Atom (Graph)
89.72
0.1737
33.899
MolHF
Zhu al.
2023
Atom (Graph)
94.75
0.0709
22.230
GDSS
oo a 20
Atom (Graph)
97.12
0.0192
14,032
GSDM
Luo al 2023
Atom (Graph)
92.57
0.0168
12.435
GruM et 2025
Atom (Graph)
98.32
0.0023
2.235
SwinG.N
Yan
2023
Atom (Graph)
86.16
0.0047
4,398
DiGress Vignae
Atom (Graph)
94.98
0.0021
3.482
GGFlow Hou at
2024
Atom (Graph)
99.63
0.0010
14555
FMagFM (ours)
Fragment (Graph)
99.81
0.009
0.630
We also report the mean and standard deviation of the newly proposed NPGen benchmark across three runs for all baselines and FragFM in table
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Table 6: Molecule generation results on NPGen with error ranges. We use a total of 30,000 molecules for evaluation. The upper part comprises autoregressive methods, while the second part comprises one-shot methods, including diffusion- basee and flow-based methods. The results are averaged over three runs. The numbers with + indicates the standard deviation against each runs.
Model
Val. 1
Unique.
Novel 1
KL. Div.
NP Score
Paihway
Superclass
Class
FCD
Training
100.0
100.0
0,0006
0.0002
0,0028
0.0094
0.01
GriphAH
2020
79.1 gn1
63.6
95. 6:100
0.8546
0.911 3anorss
3.3007.0000
00000
25.11
IE-AE
me 2013
100.01i0
97.2,101
99.3 99.5.10
0.5437
0.1055 Lo000
1.289511412
2.5645 pp
4.07..0
Hit
100.0ane
81,5a14
97. 97,7500
0.3021
=0
0.4230sn0051
=50721
1.4073
8.95
SAPE-PP
73.5g03
0.00124.0
0,0054.0001
0,4114200m
0.1722.00er6
MolHIE
Zau et al. 2023
71.0zas
39.6400
97.6==00
0.883 =0.009
18072
9.1608
10.3760=:owers
31
DiGress Vipini et 2023
$5.4,0d
99.7.00
99.9
0.1957 00
0.02291nom0
0.3370
1. 1,0099 jam12
2.05 2,05
DeFoG
Vi
85.9amus
98.4
99.2
04134au
i.3597
4.46:90
FragHM (ouns)
Dzn0
99.0 99.0a
95.4201
0:0
0,01962n0098
0.1482:20006
0.3570=00006
1.34
C.2 RETROSYNTHETIC ANALYSIS
To assess whether FMagFM can generate structures that are not only chemically valid but also synthetically accessible, performed retrosynthetic analysis using AlZynthFinder (Genheden et al. (2020)), a widely used template- based retrosynthetic planning tool. We evaluated baseline models and FragFM in the MOSES benchmark by generating 25,000 molecules and recording (i) the fraction of molecules for which AiZynthFinder succeeded in identifying a synthetic route ( ("Solved') and (ii) the number of reaction steps required for successfully solved routes. As shown in fig. [0 FragFM achieves the lowest unsolved ratio among all learned generative models, and furthermore attains the highest proportion of solved routes requiring only single step. While the MOSES dataset itself contains 80% of solved molecules under the same setup, FragEM's solved rate remains close (77%), indicating that the model successfully learns fragment compositions that map to synthetically feasible structures. These results suggest that BRICS- based fragment- generation introduces chemically meaningful inductive biases that steer the model toward generating molecules that are more often experimentally accessible, even though no explicit synthetic constraints were used during training.
MOSES (Test)
JT-VE
0.7
DiGress
DeFog
0.6
SAFE-GPT
FragFM
0.5
Rabo
0.4
0.24
0.1
0.0
Solved
Step
Steps
Steps
Steps
Figure 10: AlzynthFinder synthesis-step distributions for the MOSES dataset and baseline models trained on MOSES.
C.3 COARSE-TO-FINE AUTOENCODER
Similar to the latent diffusion model (Rombach et al. 2022). we first train the coarse-to-fine autoen- coder, which is then kept frozen during flow model training. While the main experimental results imply that the autoencoder works effectively in conjunction with the flow model, we further assess whether the autoencoder alone can accurately reconstruct the original molecule from its coarse graph and latent representation. We measure reconstruction accuracy at both the bond and whole-graph
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levels. As reported in table bond-level accuracy exceeds 99% MOSES and GuacaMol, indi- cating nearly perfect recovery of individual chemical bonds. Graph-level accuracy is similarly high, confirming that overall connectivity patterns are faithfully preserved. Even on the structurally diverse and larger NPGen dataset, the autoencoder maintains strong performance with only a slight drop in accuracy, underscoring its robustness in handling complex molecular topologies.
We next examine the role the latent representation in enabling accurate reconstruction. While the coarse fragment graph captures higher-level structural motifs, alone is insufficient recover atom- level connectivity. This limitation underscores the importance of our coarse-to-fine design, where the latent representation complements the fragment graph by encoding fine- tineggrained connectivity details. To make this explicit, we conduct an ablation in which reconstruction is attempted with random latent vector (O, 1) instead of the encoded 2. As reported in table g reconstruction accuracy collapses in this case, particularly on the larger and more complex NPGen benchmark. In contrast, decoding with the encoded consistently yields near near-perfect reconstruction, empirically demonstrating that encodes nearly complete atom-level connectivity and highlighting the novelty of integrating fragment- level and atom-level information through our coarse-to-fine framework.
Table 7: Coarse-to-fine autoencoder accuracy. "Bond" denotes the accuracy of individual atom- to-atom bonds, while "Graph" denotes the percentage of graphs in which all bonds are predicted correctly.
Train set reconstruction (%)
Test set reconstruction (%)
Dataset Bond
Graph
Bond
Graph
MOSES
99.99
99.96
99.99
99.93
GuacaMol
99.99
99.43
99.98
99.42
ZINC250k
100.0
99.96
99.64
98.71
NPGen
99.98
97.62
99.71
97.43
Table 8: Coarse-to-fine autoencoder accuracy under ablation For each test dataset, recon- struction was attempted by decoding the fragment-level graph with a randomly sampled z ~ (0, T)-
Graph-level accuracy
(%)
MOSES
GuacaMol
NPGen
Random
55.2
46.7
34.4
Encoded
99.9
99.4
97.4
FragFM uses pre- pre-trained decoder from the coarse-to-fine autoencoder. Along the latent variable which carries connectivity information across multiple fragments, the decoder predicts the probability of junctions among fragments based the coarse graph and recovering the intended molecule. To isolate the role of this pre-trained decoder, we conduct an ablation in which its parameters are randomly re-initialized during inference. As shown in table 9 basic molecular properties such as validity, uniqueness, and novelty remain mostly unchanged, while distributional and structural metrics substantially degrade. We attribute this degradation to the incorrect generation of molecules from the coarse graph, which pushes the distribution away from the original dataset. This underscores the practical importance of our decoder and hierarchical design.
Table 9: Molecule generation on MOSES with ablation on coarse-to-fine autoencoder. We use 25,000 generated molecules for evaluation. Results for FragFM are averaged over three independent runs.
Model
Valid
Unique
Novel
Filters
FCD +
SNN
Scal 1
FragFM
99.8
100.0
87.1
99.1
0.58
0.56
10.9
FragFM (random decoder)
99.8
99.9
87.1
98.5
0.9
0.52
9.8
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C.4 FRAGMENT BAG GENERALIZATION
Tables and [1 report FragEM's performance when sampling with fragment bags derived from the test-set molecules both MOSES and GuacaMol. Recall that MOSES uses scaffold-split evaluation--est scaffolds are deliberately excluded from training-so sampling with only training-set fragments limits the model's ability recover those unseen scaffolds, resulting in depressed Scaf score. We re-ran the generation using fragment bags drawn from the test split to validate this. In MOSES, the training split contains 39, 39,247 fragments, while the test split contains 9,789 fragments, of which 2,588 are unseen during training. In GuacaMol, the training split contains 192,751 fragments, while the test split contains 63,608 fragments, with 22,962 unseen. Using only test-set fragments for generation, FragEMM's s Scaf score increases dramatically, while all other metrics on MOSES and GuacaMol remain unchanged. This demonstrates that, leveraging its fragment embedding module, FragFM generalize to novel fragments without compromising validity, uniqueness, other quality measures.
Table 10: Molecule generation with unseen fragment bag on MOSES dataset. We use total of 25,000 generated molecules for evaluation. The results averaged over three independent runs.
Model
Rep, Level
Valid
Unique
Novel
Filters
FCD J
SNN
Scaf
Training set
100.0
100.0
100.0
0.48
0.59
0.0
GrNT (Mereado al.
2021
Atom
96.4
99.8
95.0
1.22
0.54
12.7
JEVAE Jin et al. 2018
Fragment
100.0
100.0
99.9
97.8
1.00
0.53
10.0
DiGress Vignac al 2023
Atom
85.7
100.0
95.0
97.1
1.19
0.52
14.8
DisCo e
Atom
Cometh Staadin
2024
Atom
90.5
99.9
92.6
99.1
1.27
0.54
16.0
DeFoG QQi ct a. 2024
Atom
92.8
99.9
92.1
98.9
1.95
0.55
14.4
FragFMM (train fragments) FragFM (test fragments)
Fragment
99.8
100.0
87.1
99.1
0.58
0.56
10.9
Fragment
99.8
100.0
88.2
98.9
0.44
0.57
24.5
Table 1I: Molecule generation with unseen fragment bag on GuacaMol dataset. We use total of 10,000 generated molecules for evaluation. The results are averaged over three independent runs.
Model
Rep. Level
Val. t
V.U. 1
V.U.N. f
KL Div.
FCD 1
Training set
100.0
100.0
99.9
92.8
MCTS (Jensen 2019
Atom
100.0
100.0
95.4
82.2
1.5
NAGVAE (Kwon al.
2020
Atom
92.9
88.7
88.7
38.4
0.9
DiGress (Vignac el al.
2023
Atom
85.2
85.2
85.1
92.9
68.0
DisCo Xu
2024
Atom
86.6
86.6
86.5
92.6
59.7
Cometh (Siaaudin et
2024
Atom
98.9
98.9
97.6
96.7
72.7
DeFoG QQn et al. 2024
Atom
99.0
99.0
97.9
97.7
73.8
FragFM (train fragments)
Fragment Fragment
99.7
99.4
95.0
97.4
85.7
FMagEM (test fragments)
99.8
99.4
97.4
97.6
85.7
C.5 LONG TAIL FRAGMENT RECOVERY OF FRAGEM
Molecular fragment distributions in real-world datasets are highly imbalanced, with small number of frequent fragments and long tail of rare ones. When we generate molecules at fragment level, is important to consider whether generative models can effectively recover such rare fragments during sampling, rather than being biased to the frequent head of the distribution. This challenge closely aligns with the long-tail problem in language models, where models tend to underpredict or ignore infrequent categories despite their importance (Zhao et al. 2021 Kang & Choil 2023).
To this end, we group fragments in the training set according to their occurrence count k. and refer to them as k-rare fragments. We then compare the occurrence ratios across groups and quantify how often these fragments reappear in generated molecules, thereby assessing the model's ' ability to recover the long tail.
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Figure 11 shows the results. Across all datasets, the recovery ratios of k-rare fragments in generated molecules closely follow the distributions observed in the training sets. Even for fragments that occur fewer than five times, FragFM is able regenerate them at comparable frequencies. Unlike language models, which often underestimate rare tokens, we suppose that our fragment-to-vector module helps generalize across fragments, contributing to this effect. These findings suggest that the fragment-basee based representation, together with our modeling strategy, provides effective coverage of the long-tail space.
Traiiing
1.5
0.00
Trigh'/ Cenersted
144
Frag/t Cenerttd
1.5
Frigftt Cenerated
0335
1.25
1.00
0.75
0.50
1.2
000
292802226633 24 023300
23
3.00
201220173633 14-3123112
E-fare Wagments
2-ror magmennts
k-aars Nagmanns
(a) MOSES
(b) GuacaMol
(c) NPGen
Figure 11: Long-tail fragment recovery results. Occurrence ratios of k-rare fragments in (a) MOSES, (b) GuacaMol, and (c) NPGen. Generated sets consist of 25,000 molecules for MOSES, 10,000 for GuacaMol, and 30,000 for NPGen.
C.6 EFFECT OF FRAGMENT BAG SIZE
The size of the fragment bag B in eqs. to controls the number of fragments sampled at each step during training and inference. The posterior in eq. corresponds to a self-normalized importance sampling (SNIS) estimator of the true posterior, whose bias and variance vanish as N increases.
During training (eqs. I and (2), the bag size Nurai determines how many negatives are included in the Info-NCE loss. Larger Nimin generally improves stability, although the optimality of the density- ratio estimator fo itself does not directly depend on Nurin- By contrast, at inference time the bag size Ninfsrence directly controls the variance of the estimator: larger Ninference yields better results, as more closely approximates the full fragment library F. In the default setting of FragFM, set both Nurin and Naterence to 384. To analyze the effect of fragment bag size, we ablate Nusin and Ninterence separately quantify the trade-off between computational efficiency and fidelity.
As shown in fig. [2(a), increasing the inference-time bag size Ninference consistently improves validity, filter scores, and FCD by reducing estimator variance. In contrast, varying the training-time time bag size Nurin produces only minor differences once is moderately large, with slight upward trend observed in filter scores, shown in fig. b).. We speculate that this stability may partly stem from training regularization techniques such as EMA, which help smooth optimization dynamics.
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Validity (%)
Filters (%)
FCD
100
100
99
2.0
98
97
96-
94
10a
10a
10
10a
10a
10a
Ninference
Nnferance
Nasaracce
(a) Inference-time bag size
Validity (%6
Filters (%)
FCD
99
2.0
98
96-
0.5
94-
10 Merain
10 Nyzin
10'
Nyan
(b) Training-tine bag size
Figure 12: Effect of fragment bag size. Comparison across (a) inference-time bag size (with Nusin = 384), (b) training-time bag size (with Ninterence = 384). To isolate the effect of bag size, the detailed-balane term was excluded during sampling. The horizontal axis (fragment bag size) is log-scaled in all subplots, and the default FragFM configuration is indicated by black dashed line.
C.7 STEERING FRAGFM TOWARDS NOVEL MOLECULES
While FragFM achieves state-of-the-an art fidelity on most distributional metries, its novelty on small- molecule benchmarks (tables D and is slightly lower than some baselines. We note that this difference modest and reflects the trade-off between novelty and fidelity in molecular generation (Mahmood et al. 2021 Geng et al. 2023). In fact, higher novelty scores sometimes arise from atoe- models that generate valid (in terms of atomic valency) yet chemically implausible motifs that cannot stably exist in the real world, which daes necessarily correspond to meaningful chemical space exploration. Nevertheless, users may sometimes prefer sampling molecules from broader and previously underexplored regions of chemical space. To this end, we found that is possible to modulate the sampling algorithm to generate more unseen molecules, fully utilizing the flexibility of the stochastic bag strategy.
Specifically, we consider two types of temperature scaling: (1) applying temperature factor Tpred to reweight fragment logits within the in-bag transition kernel (eq. (3), and (2) applying temperature factor Toug when reweighting fragments during bag construction at each Euler step (eq. (3I).
Empirically, as shown table 12 increasing either Tpred or Thug improves novelty with only slight trade-off in fidelity metrics such as validity and FCD. Notably, the combined setting (Tpred = 1.5, Thug = 1.5) achieves over 94% novelty while still maintaining high validity (99.2%) and filters (98.3%) metrics, indicating that the generated molecules remain well within the MOSES filter constraints. This demonstrates that temperature scaling provides a simple and effective mechanism to balance fidelity and exploration, offering users practical control knob for tuning generation behavior.
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Table 12: Effect of temperature scaling on MOSES. Benchmark results under different settings of
prediction temperature Tred and fragment-bag temperature Tong.
Tpred
Toag
Valid f
Unique f
Novel
Filters
FCD 4
SNN f
Scaf
1.0
1,0
99.8
100.0
87.1
99.1
0,58
0.56
10.9
1.0
1.5
99.2
100.0
94.2
98.3
0,90
0.52
13.5
1.5
1.0
99.7
100.0
88.6
98.8
0.88
0.54
11.0
1.5
1.5
99.2
100.0
94.5
98.3
0.91
0.51
13.1
C.8 MORE RESULTS ON CONDITIONAL GENERATION
For simple molecular properties (logP, QED, and number of rings), we perform conditional generation on the MOSES dataset using regressors trained on its training split. The detailed illustration of property distributions and corresponding targets is depicted in fig. 13 For protein-target conditioning, we perform conditioning on the ZINC250K dataset. The targets were selected from the DUD-E+ virtual screening benchmark for our docking score experiments, following Yang et al. (2021). The established
reliability of Smina (Koes et al. 2013), which is forked version of AutoDock Vina (Trott & Olson
2010), evidenced by its high AUROC for discriminating hits from decoys on DUD-E led us to use an oracle.
Figure 17 depicts the Smina docking score distributions for ZINC250K molecules against three targets (fa7, jak2. p)rp Since lower scores correspond stronger predicted binding, we selected the conditioning value for each protein at the extreme left tail of its distribution (FA7: -10.0 kcal/mol, 0.01% %; JAK2: -1.. kcal/mol, 0.08%: PARP1: -12.0 kcal/mol, 0.09%), indicated by the vertical dashed lines in fig. 17 to focus generation on the most tightly binding candidates.
With the perspective of chemistry, the high FCD and low validity with conditions of DiGress shown in figs. 14 and 15 highlights a critical challenge for atom-based approaches: satisfying targeted property constraints while ensuring chemical correctness, simultaneously. From chemical perspective, this distinction can be attributed to the nature of the search space; atom- based approaches explore vastly larger space, far exceeding the molecular space, where many cases can lead to chemically invalid structures, especially when generation is heavily biased by property objectives. Conversely, FragFM's fragment- based construction inherently operates within a more chemically sound and constrained subspace by assembling prevvalidated chemical motifs. These findings collectively emphasize the intrinsic advantages of employing fragments as semantically rich and structurally robust building blocks, particularly for reliable, property-focuser molecular generation.
Moreover, the importance of the fragment bag's composition, which is shown in the main text (fig. 5) is intuitive: defines the accessible chemical space and, consequently, the possible range of achievable molecular properties (e.g., generating acyclic molecules is impossible if the fragment bag exclusively contains ring-based structures, among other structural constraints). Based on Äg, FragFM automatically modulates fragment selection probabilities, inducing drift in the fragment space generate the chemically valid molecules satisfying the given objective. enables the model to construct molecules with desired properties even if the initial ceneral-purposs fragment bag is not perfectly tailored to specific task, making our strategy powerful and practically manageable tool for fine-grained control.
We also analyze how conditioning strength affects the diversity (see fig. 20p. Specifically, we observe consistent trend that stronger conditioning leads to lower diversity: when we increase Ax alone, the diversity score decreases monotonically; a similar monotonic decrease is observed when we increase Äg alone; and in the joint setting, where both Ax and Äg are varied, diversity again exhibits an overall decreasing trend as the combined guidance strength grows. Notably, the absolute reduction in diversity remains modest (from approximately 0.880 in the unconditional setting to approximately 0.855 under the strongest guidance), indicating that FragFM continues to produce structurally rich and varied molecules even under strong conditioning. Overall, fig. 20 illustataees this trade-off between conditioning strength and diversity and shows that conditional generation in FragFM introduces only a small diversity cost in exchange for improved alignment with fragment- -bg and property constraints.
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04
Tirget
Target
0.4
0.3
Deenitt
Density
0.2
0.2
0.1
0.6
logp
nrings
qed
logP
Ring count
(c) QED
Figure 13: Distribution of molecular properties (logP, ring count, and QED) for the MOSES
dataset. Colored vertical lines denote the conditioning scores applied for each target property value.
10
Target
QED-0.6
QED-0.
DiGress FrigFM
Validity
80
Target
Q1D-0.6
0ED-0.5
OED=1.0
Model
DiGress
FragFM 1b, 20.0)
0.050 0.075 0.100 0.125 0.150 0.75 0.200 0:225 0.250
0.050 0.075 0.100 0.125 0.150 0:175 0,200 0.225 0,250
Condition MAE
Condition MAE
(a) MAE-FCD curves for QED conditioning on the MOSES dataset.
(b) MAE- Malivilid curves for QED conditioning on the MOSES dataset.
Figure 14: Conditioning results on QED. MAE-FCD and MAE-validity curves for FragFM and DiGress under QED conditioning on the MOSES dataset. Different conditioning values are color- coded.
rareet value
100
lopFw1.0
logP-3.0
logiiw5 D
5-
Model
90
Validity
FragFM My - 0.5)
85
Tanget value
legh-16
logP=3.0
75-
logP=5.0
Model
DiGress
FrooFM
65
1.0
1.5
2.5
Condition MAE
Condition MAE
(a) MAE-FCD curves for logP conditioning the MOSES dataset.
(b) MAE-validity curves for logP conditioning on the MOSES dataset.
Figure 15: Conditioning results on logP. MAE-FCD and MAE-validity curves for FragFM and DiGress under logP conditioning on the MOSES dataset. Different conditioning values are color- coded.
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Target
12
Ring-1
Ringm5
10
Model
Validity
85
FCD
FragFM (A,- 0.5)
80
Ring=1
Ningm3
Ringus
2-
65
FragFM (A 0.5)
DiGresss
1.0
1.5
2.5
Condition
Condition MAE
(a) MAF-FCD curves for the number of rings con- ditioning on the MOSES dataset.
(h) MAF-valldity curves for the number of rings conditioning the MOSES dataset.
Figure 16: Conditioning results on number of rings. MAE-FCD and MAE-validity curves for FragFM and DiGress under a number of rings conditioning on the MOSES dataset. Different conditioning values are color-coded.
0.5
Target
fa7
0.4
jak2
parpl
Density
0.3
0.2
0.0
-12
-10
-8
-6
-4
-2
SMINA Docking Score (kcal/mol)
Figure 17: Distribution of SMINA docking scores for the ZINC250K dataset across different target proteins. Vertical lines denote the conditioning scores applied for each target protein.
DiGress
FragFM (A,
FragFM MA 0.0)
FragFM
95
FragFM (Ae 0.2)
90
Validity
FCD
85
80
70
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Condition MAE
Condition MAE
(a) MAE-FCD for FA7 docking condition- ing.
(b) MAE-Validin tioning.
for FA7 docking condi-
Figure 18: Conditioning results on FA7 (target: -1000.. Each point represents 10,000 generated molecules. Results are shown for DiGress FragFM with Äg 0.0 and 0.2.
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FragFM (Ag 0.0)
100
DiGress
FragFM (A, 0.0)
FragFM
FragFM (A, 0.2)
96
Validity
94
3
92
88
2.5
4.0 Condition MAE
4.0
Condition MAE
(a) MAE-FCD curves for PARPI docking score condi- (b) MAE- Val-diyyy curves for PARPI docking score con- ditioning.
tioning.
Figure 19: Conditioning results on PARP1 (target: -12.0). Each point represents 10,000 generated molecules. Results shown for DiGress and FragFM with Äg 0.0 and 0.2.
0.880
0.50
Internal Diversity
0.875
0.870
Leve
0.865
0.20
0.10
0.855
FragFM
1.5
2.0
2.5
3.0
3.5
Condition MAE
Figure 20: Effect of guidance strength on diversity for JAK2-conditioned generation.
C.9 SAMPLING EFFICIENCY: SAMPLING STEPS AND TIME
Diffusion- and flow-based models typically require multiple denoising iterations, which slows sampling. Table [3 shows the performance of MOSES benchmark metrics of FragFM and baseline denoising based models with different denoising steps. For small sampling steps, FragIIM outperforms the baseline models with minimal metric degradation, particularly at low step counts, by wide margin. With only 10 sampling steps, FragFM achieves higher validity and lower FCD than competing models that run 500 steps.
We also compare sampling time across different models in table 14 By operating both node- and predictions, edge computations scale quadratically with graph size, making them the fastest approach among the compared models. Coupled with its robust performance at far fewer steps, FragFM could be further optimized with substantial speedups over atom-level methods with high generative quality.
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Table 13: Performance of denoising- graph generative models on the MOSES dataset across different sampling step counts. All the models are one-shot models. Results for DeFoG and Cometh are taken from their original publications; DiGress (excluding the 500-step setting) were obtained by retraining the model with the differing sampling steps from the official implementation. The best performance is shown bold for each sampling step.
Sampling steps
Model
Rep. Level
Val. f
V.U.
V.U.N.1
Filters f
FCD
SNN
Scaf f
Training
100.0
100.0
100.0
0.48
0.59
0.0
DiGress
Atom
6.3
6.3
6.3
66.4
9.40
0.38
7.4
Cometh
Atom
26.1
26.1
26.0
59.9
7.88
0.36
8.9
10
DeFoG
Atom
FragFM
Fragment
96.8
96.6
89.4
96.8
0.92
0.52
15.3
DiGress
75.3
75.3
72.3
94.0
1.35
0.51
16.1
Cometh
Atom
82.9
82.9
80.5
94.6
1.54
0.49
18.4
50
DeFoG
Atom
83.9
83.8
81.2
96.5
1.87
0.59
14.4
FragFM
Fragment
99.5
89.1
98.5
0.65
0.54
11.2
DiGress
Atom
82.6
82.6
79.2
95.2
1.14
0.51
15.4
Cometh
Atom
85.8
85.7
82.9
96.5
1.43
0,50
17.2
100
DeFoG
Atom
FragFM
Fragment
99.7
99.7
88.5
98.8
0.62
0.55
11.6
DiGress
Atom
81.1
96.5
1.11
0.52
13.5
Cometh
Atom
86.9
86.9
83.8
97.1
1.44
0.51
17.8
300
DeFoG
Atom
FMagFM
Fragment
99.8
99.8
87.2
98.9
0.58
0.55
11.6
DiGress
Atom
85.7
85.7
81.4
97.1
1.19
0.52
14.8
Cometh
Atom
87.0
86.9
83.8
97.2
1.44
0.51
15.9
500
DeFoG
Atom
92.8
92.7
85.4
98.9
1.95
0.55
14.4
FragFM
Fragment
86.9
99.1
0.58
0.56
10.9
DiGress
Atom
85.5
85.5
82.6
95.0
1.33
0.50
15.3
Cometh
Atom
87.2
87.1
83.9
97.2
1.43
0.51
15.9
700
DeFoG
Atom
FragFM
Fragment
99.9
99.9
86.9
99.1
0.61
0.56
10.8
DiGress
Atom
84.7
81.3
96.1
1.31
0.51
Cometh
Atom
87.2
84.0
97.2
1.44
0.51
17.3
1000
DeFoG
Atom
FragEM
Fragment
99.8
99.8
86.6
99.1
0.62
0.56
12.9
Table 14: Comparison of sampling time across different datasets and methods. Experiments were conducted on a single NVIDIA GeForce RTX 3090 GPU and an Intel Xeon Gold 6234 CPU @ 3.30GHz. *Results for DeFoG are taken from the original paper, where experiments conducted on an NVIDIA A A100 GPU. Note that DeFoG shares most of its backbone architecture with DiGress, with only minor adjustments.
Sampling steps
MOSES
GuacaMol
NPGen
Min. nodes
8
2
Property
Max. nodes
27
88
99 30000
# Samples
25000
10000
DiGress
500
3.0
36.0
500
5.0
7.0
Sampling Time (hour)
FragFM
500
0.9
1.3
7.0
FragFM
50
0.2
0.2
0.9
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D EXPERIMENTAL SETUP AND DETAILS
D.1 DATASETS
We provide details of the datasets used in our experiments, including MOSES (Polykovskiy et al. 2020), GuacaMol (Brown et al. 2019), and ZINC250K (Irwin & Shoichei, 2005
MOSES. The MOSES benchmark is constructed from a subset of the ZINC Clean Leads dataset, containing approximately 1.9 million drug-like molecules. The molecules are curated and filtered for training and evaluation of molecular generative models, with a predefined data split of training, test, and scaffold split test sets, All molecules in the MOSES dataset meet drug- -ilgeiiseeess criteria, including molecular weight and logP ranges.
GuacaMol. GuacaMol is based on the ChEMBL database and provides a large-scale benchmark for both Iestribution-learnin and goal-directed molecular generation tasks. The training dataset comprises around 1.6 million molecules extracted from ChEMBL v24.
ZINC250K. The ZINC250K dataset comprises 249,456 molecules selected from the larger ZINC database.
D.2 FRAGMENTATION
We use the BRICS DDeenn et al. 2008) decomposition scheme to construct our fragment library, with all mnter-fragment connections restricted to single bonds. The fragment library is built by fragmenting every molecule in each dataset and collecting the resulting unique fragments into a fragment bag. The final fragment counts in the training, validation, and test splits are: MOSES (39,247 / 12,212 9,789), GuacaMol (192,751 / 30,418/ / 63,608), NPGen (117,998) / 18,063 / 40,214), and ZINC250K (28,235 4,231 2.683). In the test sets, the number of fragments unseen during training is 2,588 for MOSES, 22,962 for GuacaMol, 12,116 16 for NPGen, and 367 for ZINC250K.
D.3 METRICS
We provide details of common metrics in both MOSES (Polykovskiy al. 2020) and GuacaMol BBrwnn et al. 2019) benchmarks.
Common Metrics. These metrics are fundamental for assessing the basic performance molecular generative models. Note that V.U. and V.U.N. metrics are multiplied values of each metric, i.e., V.U.N. is computed by multiplying validity, uniqueness and novelty.
This metric measures the proportion of generated molecules that are chemically valid according to a set of rules, typically checked using tools like RDKit. can be successfully parsed and represents
Validity (Valid):
A SMILES string is considered valid chemically sensible molecule (e.g., correct atom valencies, no impossible structures).
Uniqueness (Unique): This indicates the percentage of unique molecules among valid generated molecules. A high uniqueness score suggests the model is generating diverse structures rather than repeatedly producing the same few molecules.
Novelty (Novel): This metric quantifies the fraction of unique and valid generated molccules that are not present in the training dataset. It assesses the model's ability to generate novel chemical molecules.
MOSES Metries.
The MOSES benchmark focuses on distribution learning. Key metrics beyond the foundational ones include:
This refers to the percentage of valid, unique, and novel molecules that pass a set of medicinal chemistry filters (PAINS, MCF) and custom rules defined by the MOSES benchmark (e.g., specific ring sizes, element types). which are used in curating dataset of MOSES. This evaluates the drug-lliknee or suitability of generated molecules according to predefined structural criteria.
Filters:
Fréchet ChemNet Distance (FCD); FCD (Preuer et al. 2018 measures the similarity between the distribution of generated molecules and reference (test) dataset based on the
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latent representation of molecules using pre-trained neural network (ChemNet). A lower FCD indicates that the generated distribution is closer to the reference distribution.
Similarity to Nearest Neighbor (SNN): This metric calculates the average Tanimoto simi- larity using Morgan fingerprints (Rogers & Hahn 2010) between each generated molecule and its nearest neighbor in the reference (test) dataset. A higher SNN suggests that the generated molecules are similar in structure to known molecules in the target chemical space.
Scaffold Similarity (Scaf):
This metric specifically assesses the diversity of molecular scaffolds. calculates a cosine similarity between the vectors of the occurrence of Bemis- Murcko scaffolds (Bemis & Murcko 1996) of the molecules in the reference (test) dataset and the generated ones. A higher score suggests that the generated scaffolds are similar to reference scaffolds.
GuacaMol Metries. GuacaMol provides benchmarks for distribution -eaiinnng and goal-directed generation. For its distribution-leanii benchmark, which is utilized for our main results, the primary aggregated metrics are:
Kullback-Leibler Divergence (KL Div.) Score: This metric computes the KL divergence between the distributions of several physicochemical and topological properties of the generated molecules and the training set. These individual KL divergences (Dxr) then combined into a single score by averaging the negative exponentials of them (i.e., exp(-Dxx.) to reflect how well the model reproduces the overall property distributions. Due to the nature of the calculation method, score closer to I indicates better similarity.
Fréchet ChemNet Distance (FCD) Score: Similar to the MOSES FCD, GuacaMol also uses an FCD metric to compare the distributions of generated molecules and the training set. The only difference that GuacaMol, the raw FCD value (where lower is better) is transformed into score where higher is better.
ZINC250K Metries. ZINC250K is a collection 250k molecules from ZINC (Irwin & Shoichet 2005 Multiple generative models evaluate their performance with the following metrics:
NSPDK:
The Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) a graph kernel that measures structural similarity between molecular graphs by considering neighborhoods of atoms and their pairwise distances. For generative evaluation, the NSPDK distance is com- puted between the generated set and the reference dataset, capturing differences in both local and global graph structures. A lower NSPDK value indicates that the generated molecules are structurally closer to the reference distribution. We used the official implementation from Jo et al. (2022) to compute the NSPDK.
Fréchet ChemNet Distance (FCD): The FCD used in ZINC250K follows the same defini- tion as in the MOSES benchmark.
D.4 BASELINES
Next, we briefly introduce baseline strategies that we compared FragFM in the main results. We focused on molecular graph generative models, which are categorized by autoregressive and one-shot generation models. Each model uses either an atom- or a fragment-level representation.
GraphInvent (Mercado et al. 202T) employs graph neural network (GNN) approach for de novo molecular design. first computes the trajectory of graph decomposition based on atom-level representation, and then trains GNN to learn the action of atom and bond addition on given subgraph a molecule. During the inference stage, it builds molecules in an atom-ve atom-wise manner.
JT-VAE (Jin al. 2018) or Junction Tree Variational Autoencoder, generates molecular graphs in a two-step process. It first decodes a latent vector into a tree-structured scaffold representing molecular components (like rings and motifs) and then assembles these components into complete molecular graph, ensuring chemical validity. Since it iteratively decides whether to add a node during the sampling process, we consider it autoregressive model despite its use of VAE.
SAFE-GPT (Noutahi et al. 2024) is an autoregressive sequence generative model that produces SAFE (Sequential Attachment- based Fragment Embedding) strings- novel representation proposed
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in the paper, where molecules expressed sequences of fragments. The model is built GPT-2-like transformer architecture. While the original work reported only basic results on validity, uniqueness, and diversity, we retrained the SAFE-GPT-200 model using its official implementation to obtain comprehensive benchmark results.
MCTS (Jensen 2019) is non-deep-leariin-- based strategy that uses Monte Carlo Tree Search for molecular graph generation. Using atom insertion or addition as an action, it sequentially builds molecules from a starting molecule.
MolHF ZZuu et al. 2023 a hierarchical normalizing flow model designed for molecular graph generation. It employs coarse-to-fine strategy, generating fragment connectivity graph and subsequently refining it at the atom level via conditional flows. As MolHF refers to itself as one-shot flow-based model, we treat it as a one-shot model.
MolGrow (Kuznetsov & Polykovskiy 2021 is hierarchical normalizing flow model that constructs molecular graphs via recursive node-splitting process. Starting from a single node, iteratively divides nodes to generate complex structures, enabling multi-level control over molecular properties through its hierarchical latent space. Although MolGrow a hierarchical normalizing flow model, we follow (hu et al. 2023) and categorize it as one-shot model for consistency with prior taxonomy.
NAGVAE Kwon al., 2020). non-autoregressive Graph Variational Autoencoder, VAE-based one-shot graph generation model utilizing compressed graph representation. It reconstructs the molecular graphs from latent vectors, aiming for scalability and capturing global graph structures.
DiGress Vignac et al. 2023 is the first discrete diffusion model designed for graph generation. It operates by iteratively removing noise from both graph edges and node types, learning a reverse diffusion process construct whole graphs from a noise distribution.
DisCo (u et al., 2024 is graph generation model that defines a forward diffusion process with a continuous-ting Markov chain (CTMC). The model learns reverse generative process to denoise both the graph structure and its attributes simultaneously.
Cometh (Siraudin et al. 2024) is continuous-time discrete-state graph diffusion model. Similar to DisCo, it formulates graph generation as reversing a CTMC defined on graphs, in which the model learns the chain's 's transition rates generate new graph structures.
DeFoG (Qin et al. 2024) is a generative framework that applies the principles of flow matching directly to discrete graph structures. After training with flow-matching strategy, it uses CTMC for denoising to generate graphs.
E PARAMETERIZATION AND HYPERPARAMETERS
E COARSE-TO-FING AUTOENCODER
Our coarse-to-fin autoeneoder (eq. 4h) compresses the atom-level graph into single latent vector and then uses it, together with the fragment-level graph G, to reconstruct all atom-aton connections. The encoder, an MPNN (Gilmer et 2017), takes G and pools its node features into 2. The decoder conditions G and to predict a distribution over every possible atom-atom edge between them for each pair of linked fragments. Internally, it propagates messages along original intra-fragment bonds and across all candidate inter- fraemmf edges, enabling the recovery of the complete atomistic structure from the coarse abstraction.
E.2 FRAGMENT EMBEDDER, PREDICTION MODEL., AND PROPERTY DISCRIMINATOR
To parameterize the neural network fo(_ x) in eq. we jointly train two components: fragment embedder and a Graph Transformer (GT). Figure 21 provides an overview.
The fragment embedder, built to a dieed-dimessoonal embedding vector. Given a fragment-level graph Xt, we apply this shared embedder to every fragment node, producing set of local structure embeddings. Multiple GT layers, then process these embeddings to capture inter-fragment interactions and global context. The GT layers were designed with the sample architecture and hyperparameters prior atom-based
MPNN backbone Gilmer et al. (201)). maps each fragment
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diffusion- and flow-basee molecule generative models (Vignac et al. 2023 Qin et al. 2024 Siraudin et al. 2024 Xu et al. 2024). We directly predict the discrete fragment graph edges & and the continuous latent vector from the final GT output embeddings. To predict fragment types, compute the inner product between each candidate fragment type emhedding and its corresponding GT embedding infer the scores of different fragments. We reuse the flow model's s architecture for property discrimination: We aggregate both the fragment-level embeddings and the GT's global readout to produce fragment-and molecule-level property predictions.
We train our flow model with AdamW optimizer, using (31, (2) (0.9, 0.999), learning rate of 10-4, and gradient sradient-norm clipping 4.0. We employ the exponential moving average (EMA) scheme of Karras et al. (2024) to stabilize training. Training is performed on single NVIDIA A100 GPU for 96 on MOSES and GuacaMol and 144 on NPGen, and we select the checkpoint with the lowest validation loss.
E.3 AUXILIARY FEATURES
Although graph neural networks exhibit inherent expressivity limitations etu al. 2018)) augmenting them with auxiliary features has proven effective mitigating these shortcomings. For example, Vignac et al. (2023) augments each noisy graph with cycle counts, spectral descriptors, and basic molecular properties (e.g., molecular weight, atom valence). More recently, relative random walk probabilities (RRWP) have emerged as highly expressive yet efficient encoding (Ma et al. (023): by stacking the first K powers of the normalized adjacency matrix M D-1 A. RRWP constructs k-step transition probabilities that capture rich topological information. Accordingly, we integrate RDKit-derived molecular descriptors into the fragment embedder, and RRwP features into the graph transformer, enriching the model's 's ability to capture complex molecular semantics.
Coarse Graph Embedder
S
Graph
(Mask]
Embeddar
Continuous
Max
L
21
2 L+
Fragment Emitedding Layers
0.1
0.1
Discrete
Sahima
FM G4
Fragment Embedder
Graph Tranalomer
.HH
0.6
:
Figure 21: Overview of the parameterization of the fragment embedder and prediction model, Le., in fig. [1 (left) Each fragment in the fragment bag B is embedded by the fragment embedder, while each node the coarse graph is mapped to a fixed-size vector. We compute fo(,,, x) by taking the inner product of the two embeddings. (right) The coarse-graph embedder first maps every node an embedding, producing a coarse graph whose nodes are single vectors; the resulting graph is then passed through the graph-transformer layer.
E.4 NOISE SCHEDULE
Selecting appropriate noise schedule is crucial for the performance of diffusion- and flow-based
models (Vignac et al. 2023; Siraudin et al. 2024 Qin et al., 2024). Following QQn et al. (2024), we adopt the polynomially decreasing (polydec) time distortion, which skews the initially uniform time distribution so that more steps are allocated near the data manifold. Specifically, a uniformly sampled
(0, is warped by ((1) 21 - which preserves the endpoints (f( 0, f() 1)
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while increasing the density toward large t. Under Euler discretisation, this concentrates integration steps where fine-grained denoising is most critical.
E.5 HYPERPARAMETERS
For reproducibility, report the full hyperparameter settings of FragU', including the coarse-to-fine autoencoder (table [5]. flow matching module (table TT). and discriminator module for guidance (table[7]. Our primary contribution the development a fragment- based framework for molecular generation, rather than architectural novelty; however, since no prior models have adopted this design, we implemented the necessary modules accordingly. The hyperparameters were selected through preliminary exploratory experiments and kept fixed across all datasets and benchmarks for consistency. For the MPNN, we adopt the implementation from Gilmer et al. (2017): Battaglia et al. (2018), and for the graph transformer, the fragment- tommeenttt-veectee to-vector module already precedes it. For the property discriminator, we followed prior classifier- guidance studies, which typically used smaller networks than the corresponding diffusion classifier-uiidance
follow Vignac et al. (023) but reduce the number of layers from to 5, as
models, and set the number of parameters accordingly.
Table 15: Hyperparameters for the coarse-to-fine autoencoder.
Name
Value
Model architecture
Number of parameters
6, 648, 793 793
Backbone type
Sparse MPNN (Gimmer et al.
2017
Encoder/decoder layers
4
Node embedding dimension
256
Edge embedding dimension
128
Hidden dimension
256
Latent dimension
32
Activation function
SiLU
Layer normalization
Yes
Initialization
Xavier
Training setup
Batch size
256
Optimizer
AdamW (3) 0.9, 32 0.999)
Learning rate
1.0 x 10 10
Learning rate warm-up
Linear, 2000 iterations
Weight decay
1.0 x 10-12
Gradient clipping
1.0
KL divergence coefficient
1.0 10-4
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Table 16: Hyperparameters for the fragment-to-vector encoder and coarse graph propagation module.
Name
Value
Fragment-to-vector encoder
Number of parameters
4, 257, 290
Backbone type
Sparse MPNN Gilmer et al. 2017
Number of layers
5
Node embedding dimension
256
Edge embedding dimension
128
Hidden dimension
256
Activation function
SiLU
Dropout
0.1
Layer normalization
Yes
Initialization
Xavier
Coarse graph propagation module
Number of parameters
17, 418, 274
Backbone type
Graph Transformer (Vignac et al. 2023
Number of layers
5
Attention heads
8
Node embedding dimension
256
Edge embedding dimension
128
RRWP walk length
6
Dropout
0.1
Layer normalization
Yes
Initialization
Xavier
Training setup
Batch size
256
Fragment bag size
384
Optimizer
Adam (3 0.9, 12 0.999)
Learning rate Learning rate warm-up
5.0 x 10
Linear, 10,000 iterations
Weight decay
0.0
Gradient clipping
4.0
Exponential moving average (EMA)
Yes (0.999)
Loss coefficients
Fragment type (1.0), Fragment edge (5.0), Latent (1.0)
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Table 17: Hyperparameters for the property discriminator module.
Name
Value
Fragment-to-vector encoder
Number of parameters
923, 402
Backbone type
Sparse MPNN (Gilmer et al. 2017
Number of layers
4
Node embedding dimension
128
Edge embedding dimension
64
Hidden dimension
128
Number of property readout layers
Activation function
SiLU
Dropout
0.1
Layer normalization
Yes
Initialization
Xavier
Coarse graph propagation module
Number of parameters
5, 646, 572
Backbone type
Graph Transformer (Vignac et al.
2023
Number of layers
4
Attention heads
Node embedding dimension
128
Edge embedding dimension
64
Number of property readout layers
Dropout
0.1
Layer normalization
Yes
Initialization
Xavier
Training setup
Batch size
256
Optimizer
AdamW W (8) = 0.9, 82 = 0.999)
Learning rate
5.0 x 10-4
Learning rate warm-up
Linear, 10,000 iterations 0.0
Weight decay
Gradient clipping
4.0
Exponential moving average (EMA)
Yes (0.999)
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F VISUALIZATION
F.1 VISUALIZATION OF FRAGMENTS
We visualize the top-50 frequent fragments from each dataset (MOSES, GuacaMol, and NPGen).
Figure 22: Top 50 most common fragments extracted from the MOSES dataset. More frequently occurring fragments are positioned toward the top left.
Figure 23: Top 50 most common fragments extracted from the GuacaMol dataset. More fre- quently occurring fragments positioned toward the top left.
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P
Y
D
X
R
Figure 24: Top 50 most common fragments extracted from the NPGen dataset. More frequently occurring fragments are positioned toward the top left.
F.2 VISUALIZATION OF FRAGMENT EMBEDDINGS
To examine the structure of the fragment embedding space learned by the fusion of the fragment- to-vector encoder module, we projected the fragment-level embeddings into two dimensions using UMAP and visualized them with respect to several physicochemica properties that can defined at the fragment level. As shown in fig. 25 fragments with similar structures tend to occupy close regions the latent space. We additionally visualized property-specific projections in figs, 26 and 27 including LogP, TPSA, MolMR, the number of rings, and the number of hydrogen-bome donors and acceptors. These results demonstrate that fragment-level properties align smoothly with the geometry of the learned embedding space, indicating that the model captures chemically meaningful organization at the fragment level.
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30
25
20
15
UMAP
10
-5
-10
NI
-15-
UMAP
Figure 25: UMAP visualization of MOSES fragments using fragment embeddings, color-coded by molecular weight with example molecules. Structurally similar fragments are positioned close to each other in the embedding space.
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-15 -12
-15
-50
(a) LogP
(h) TPSA
(c) MolMR
17
12
0N073
USMPP
U0AF1
(d) Number of Rings
(c) Number of Hydrogen Bond
(f) Number of Hydrogen Bond Ac-
Donors
ceptors
Figure 26: UMAP visualization of MOSES fragments using fragment embeddings, color-coded by fragment level properties.
1940
LogP
(b) QED
(c) TPSA
Figure 27: UMAP visualization of MOSES fragments using fragment embeddings, color-coded by molecule-level properties.
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F.3 VISUALIZATION OF GENERATED MOLECULES FROM MOSES AND GUACAMOL
We visualize samples generated by FragPM on the MOSES and GuacaMol datasets figs. and
Foree rape Xooro 1030 tros
codd
Figure 28: Molecules generated by FragEM on the MOSES benchmark. Molecules were randomly selected for visualization.
eafpe ofoup goinato
Supo
Figure 29: Molecules generated by FragFM on the GuacaMol benchmark. We randomly select molecules for visualization.
F.4 VISUALIZATION AND ANALYSIS GENERATED MOLECULES ON NPGEN
For the NPGen task, we show generated molecules from FragFM alongside baseline models (GraphAF, VAE, HierVAE, and Digress) in figs. 30 to 34 Although all visualized molecules are formally valid in terms of valency (which rdkit can process), atom-based generative models often introduce chemically implausible motifs, such as aziridine or epoxide rings fused directly to aromatic systems, inducing severe angle strain (Sweeney (2002); anti-aromatic rings with 4n r-electrons (violating Hückel's 's rule), resulting in high electronic instability Carpenter 1983); and bonds between nonadjacent atoms in ring system, causing extreme geometric distortion Nishiyama al. (1900). Fragment- based autoregressive models largely avoid these issues, yet they, too, exhibit limitations: JT-VAE tends to generate only small, homogeneous ring systems, while HierVAE is strongly biased toward long aliphatic chains and simple linear moieties. Consequently, these approaches show distinct distribution of molecules from the trained dataset, matching the benchmark results table
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Figure 30: Valid molecules generated by FragFM on NPGen. The top two rows show molecules
with up 30 heavy atoms, while the bottom two rows show molecules with 31-60 heavy atoms. Molecules were randomly selected for visualization.
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Yomr
Figure 31: Valid molecules generated by GraphAF on NPGen. The top two rows show molecules with up 30 heavy atoms, while the bottom two rows show molecules with 31-60 heavy atoms. Molecules were randomly selected for visualization.
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Figure 32: Valid molecules generated by JT-VAE on NPGen. The top two rows show molecules with up 30 heavy atoms, while the bottom two rows show molecules with 31-60 heavy atoms. Molecules were randomly selected for visualization.
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Figure 33: Valid molecules generated by HierVAE on NPGen. The top two rows show molecules with up 30 heavy atoms, while the bottom two rows show molecules with 31-60 heavy atoms. Molecules were randomly selected for visualization.
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Figure 34: Valid molecules generated by DiGress on NPGen. The top two rows show molecules with up 30 heavy atoms, while the bottom two rows show molecules with 31-60 heavy atoms. Molecules were randomly selected for visualization.
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Reviews:
[{"review_id": "kkNOn55QzT", "reviewer": "Reviewer_PJ7B", "summary": "The paper proposes a fragment-based approach for molecular generation, with a particular focus on scalability to larger molecules (natural products; NPs).\n\nThe focus on NPs makes it difficult for me to judge the impact (of both the method and the dataset); I suspect it's somehow niche compared to the general small molecule generation, but at the same time it offers a comprehensive contribution (with both a benchmark and a new method).", "questions": "- The authors provide a figure for sampling times; how does the model compare in terms of training time?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is clearly written and easy to follow.\n- The approach focuses on scaling to the generation of large molecules (namely natural products), which is an interesting application area.\n- The approach seems novel, and has domain-specific contributions (e.g. coarse encoding)\n- A new benchmark dataset is introduced, which is an additional contribution.\n- Besides the comment below, the evaluation is in my opinion sound and fairly typical for 3D generation papers; it supports the claims made.", "weaknesses": "- The authors rely on fairly old baselines in experiments past unconditional generation. This raises questions about relative performance of the method compared to SotA. It’s also not clear why only an atom-based baseline is used in conditional generation.", "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": "M47fMIRhDl", "reviewer": "Reviewer_RJhm", "summary": "This paper introduces FragFM, a hierarchical framework for molecular graph generation that combines fragment-level discrete flow matching with a coarse-to-fine autoencoder for efficient and scalable drug-like molecule generation. The authors also propose NPGen, a new benchmark designed to evaluate the generation of complex, natural product–like molecules.", "questions": "1. How sensitive is FragFM to the choice of fragment decomposition rules (e.g., BRICS vs. other methods)? \n2. Coarse graphs are generated by fixed chemical tools, can this process be learned by another graph autoencoder instead?\n3. How is the fragment property predictor trained?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Interesting ideas that construct flows on latent variables and coarse graphs. \n2. The “in-bag” InfoNCE formulation is computationally efficient, and provides a surrogate for different computation budget, with the bag size increasing, $x_1$ posterior becomes unconditional.\n3. Strong performance on both standard benchmarks and NPGen.", "weaknesses": "1. The paper only evaluates 2D molecular graphs. It would strengthen the contribution to include 3D molecular generation. \n2. While fragment-level generation is inherently interpretable, the paper does not analyze fragment–property correlations or latent space structure.", "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": "RL9CJTSfos", "reviewer": "Reviewer_hbdo", "summary": "This paper studies the problem of molecular graph generation. The authors propose FragFM, a novel hierarchical framework via fragment-level discrete flow matching. The authors also propose the Natural Product Generation benchmark for comprehensive evaluation. FragFM enables effective distribution learning and property-guided molecular generation.", "questions": "See weaknesses", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- FragFM includes a coarse-to-fine autoencoder to deal with the ambiguity in reconstructing atomic connections.\n- FragFM introduces a dynamic fragment vocabulary.", "weaknesses": "- In related work, hierarchical generation models such as MolGrow[1], MolHF [2], and Coarse-to-fine [3] should be included. And their comparison with fragment-based methods should be discussed.\n- The coarse-to-fine autoencoder is not fully data-driven, which depends on predefined fragmentation rule and Blossom algorithm.\n- Several important baselines are missing in the proposed benchmark NPGen. For example, SAFE-GPT, a fragment-based method that performs well in MOSES.\n- Lacking ablation. The effectiveness of the autoencoder and the dynamic fragment vocabulary should be discussed.\n\n[1] MolGrow: A graph normalizing flow for hierarchical molecular generation\n\n[2] Molhf: A hierarchical normalizing flow for molecular graph generation\n\n[3] Coarse-to-fine: a hierarchical diffusion model for molecule generation in 3d", "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": "21ViTilusC", "reviewer": "Reviewer_FxE9", "summary": "The paper introduces FragFM, a hierarchical framework for molecular graph generation that operates at the fragment level by combining a discrete and continuous flow matching approach with a coarse-to-fine autoencoder. The design enables efficient and scalable molecular generation by modeling chemically meaningful fragments, rather than individual atoms, while ensuring all-atom validity during molecular reconstruction. The authors also propose NPGen, a new benchmark focusing on natural products, which challenges existing approaches to generate larger, more complex, and more diverse biologically relevant compounds. Experimentally, FragFM outperforms atom- and fragment-based models on both standard molecule benchmarks and NPGen, and also demonstrates the ability to conditionally generate molecules based on desirable properties while maintaining validity.", "questions": "1. It is somewhat unclear how the regressor, which decides conditional fragment bag sampling weights, is trained. Does the model predict the property values of each fragment directly? If not, is it instead trained to predict the average property value of all molecules in the training dataset containing the fragment? \n2. Does the NPGen dataset contain macrocycles? \n3. Does conditional generation with higher $\\lambda_B$ result in lower diversity? It would be interesting to see the trade-off between the strength of conditioning signals like $\\lambda_B$ and $\\lambda_X$ and the diversity of the results.\n4. Is there any merit to changing the relative schedules of the latent vector and molecular graph during sampling? For instance, would it be possible to construct an analog design experiment by conditioning on a fully denoised latent vector of a difficult-to-synthesize molecule, then allowing the discrete graph flow matching module to run and generate analogs with a chosen fragment library?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The fixed-length graph embedding for fragments and fragment bag strategy allows scalable training for large fragment libraries without sacrificing chemical expressivity. By introducing the conditional fragment bag sampling term $\\lambda_B$, the fragment bag restriction does not hinder conditional generation. \n2. For both conditional and unconditional molecule generation, FragFM maintains high RDKit validity, demonstrating that the hierarchical approach and Blossom algorithm are effective methods to ensure chemical reasonableness.\n3. FragFM outperforms atom-based and fragment-based methods in most metrics while also demonstrating a high sampling efficiency, even compared to an existing flow matching model DeFoG.", "weaknesses": "1. Despite the inclusion of chemical validity as a benchmark, the authors omit retrosynthetic analyses (i.e. Syntheseus) for generated molecules. This is especially important considering the reliance of the model on connecting building blocks via junction atoms, which does not necessarily guarantee a feasible experimental synthesis route.\n2. Although many relevant natural products are macrocycles, the ground-truth graphs are derived directly from the results of BRICS decomposition, which results in acyclic graphs. As the Blossom algorithm employed only considers pairs of junction atoms with edges decided by this fragmentation, the model is likely unable to generate macrocycles.\n3. While the hierarchical combination of fragment-level flow matching and coarse-to-fine reconstruction is well-executed, the scope of the work is somewhat incremental. Flow matching and autoencoder-based 2D molecular generation at the fragment level are individually present in the literature, as is the architecture used during training. \n\nMinor:\n1. The work omits training details for the conditional property regressor.\n2. Though it may exceed the scope of the work, the model does not explicitly consider information relevant to some conditional generation tasks, such as the local geometry of the protein pocket for binding affinity.", "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)