GraphNOSE: A Graph Transformer in Olfaction

# GraphNOSE: A Graph Transformer in Olfaction
Source: [https://arxiv.org/html/2609.05694](https://arxiv.org/html/2609.05694)
###### Abstract

Predicting olfactory qualities from molecular structure is an open problem in chemo\-informatics\. Although linear models can link molecular features to odor descriptors, they often fail when extrapolating to novel chemical scaffolds, extreme molecular weights, or complex odor mixtures\. To address this, we introduceGraphNOSE, an open\-source graph transformer framework that predicts multi\-label odor descriptors from simplified molecular\-input line\-entry system \(SMILES\) strings for single molecules and binary mixtures\. By integrating positional and structural encodings within a transformer\-based graph architecture,GraphNOSEachieves strong performance with six times fewer parameters than standard graph neural network \(GNN\) baseline while consistently outperforming linear models, molecular language model embeddings, molecular fingerprints, and baseline GNNs by an average area under the ROC curve \(AUROC\) margin of 4\.52% \(p < 0\.01\)\.GraphNOSEachieves an AUROC of 84% on out\-of\-distribution compounds \(OODs\)\. This exceeds the current state\-of\-the\-art GNN for OOD in olfaction \(OpenPOM: 81%, p < 0\.001\), and identifies conditions under which linear models empirically fail\. Finally, we apply XAI \(explainable AI\) methods to identify which substructures and molecular features drive odor predictions, yielding insights consistent with chemical intuition and grounded in the model’s learned representations\. Together, these results establishGraphNOSEas a scalable and interpretable architecture for olfactory prediction that generalizes to structurally distinct compounds underrepresented in current perceptual databases\.

###### keywords

American Chemical Society,LaTeX

††affiliation:These authors contributed equally to this work\.††affiliation:CSIR\- Central Scientific Instruments Organisation, Sector 30\-C, Chandigarh\-160030, India††affiliation:Academy of Scientific and Innovative Research \(AcSIR\), Ghaziabad\-201002, India††affiliation:Department of Higher Education,Himachal Pradesh,Shimla \-171001,India††email:mrityunjay\.csio24j@acsir\.res\.in††affiliation:These authors contributed equally to this work\.††affiliation:Indian Institute of Science Education and Research Bhopal\(IISERB\), Madhya Pradesh\-462066,India††affiliation:Monell Chemical Senses Center, Philadelphia, Pennsylvania, USA 19104††affiliation:Department of Otolaryngology, Head and Neck Surgery, University of Pennsylvania, Philadelphia, Pennsylvania, USA 19104††email:riteshkr\.csio@csir\.res\.in††affiliation:CSIR\- Central Scientific Instruments Organisation, Sector 30\-C, Chandigarh\-160030, India††affiliation:Academy of Scientific and Innovative Research \(AcSIR\), Ghaziabad\-201002, India††abbreviations:IR,NMR,UVAbbreviations:1\-Weisfeiler\-Lehman \(1\-WL\) isomorphism test, atoms in molecule localization and delocalization matrices \(AIMLDM\), area under the receiver operating characteristic curve \(AUROC\), Chemistry Bidirectional Encoder Representations from Transformers \(ChemBERTa\), Chemical European Molecular Biology Laborator \(ChEMBL\), physico\-chemical deviance score \(Dphys\\text\{D\}\_\{\\text\{phys\}\}\), Geometric Ensemble of Molecules \(GEOM\), graph convolutional network \(GCN\), graph isomorphism Network \(GIN\), graph isomorphism network with edge features \(GINE\), graph neural network \(GNN\), general, powerful and scalable \(GPS\), graph positional and structural encoder \(GPSE\), GraphNOSE model utilizing linear transformer \(GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\), GraphNOSE model utilizing standard self\-attention with quadratic complexity \(GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\), curated GoodScents\-Leffingwell dataset \(GS\-LF\), graph transformer \(GT\), kernel density estimation \(KDE\), K\-nearest neighbor \(KNN\), Laplacian positional encodings \(LapPE\), limited\-memory Broyden\-Fletcher\-Goldfarb\-Shanno \(LBFGS\), electron localization\-delocalization matrix \(LDM\),DMPNN\+LDM\\text\{DMPNN\}\+\\text\{LDM\}\(LDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}\),GCN\+LDM\\text\{GCN\}\+\\text\{LDM\}\(LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}\) , Molecular ACCess System \(MACCS\), median absolute deviation for featureii\(MADi\\text\{MAD\}\_\{i\}\), Multi\-Layer Perceptron \(MLP\), Molecular Language Transformer \(MolFormer\), Message Passing Neural Network \(MPNN\), out\-of\-distribution \(OOD\), open\-source counterpart of POM \(OpenPOM\), Principal Odor Map \(POM\), Positional and structural encoding \(PSE\), Receiver Operating Characteristic curve \(ROC\), Random\-walk structural encodings \(RWSE\), simplified molecular\-input line\-entry system \(SMILES\), topological polar surface area \(TPSA\), XAI \(explainable AI\)

## 1Introduction

Predicting human odor perception from molecular structure remains an open and challenging problem in cheminformatics\. Olfaction plays a crucial role in human survival and thriving, shaping physiological regulation[Tarumi et al\. \(2026\)](https://arxiv.org/html/2609.05694#bib.bib29);[Boesveldt and Parma \(2021\)](https://arxiv.org/html/2609.05694#bib.bib4), feeding behavior[Boone et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib67), and social mating choices[Blazing and Franks \(2020\)](https://arxiv.org/html/2609.05694#bib.bib68), and it underpins the multibillion\-dollar food, flavor, and fragrance industries[Chen et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib5)\. Translating physico\-chemical properties into perceptual space is particularly complex in olfaction, as structurally similar molecules may evoke qualitatively different odors, revealing a disconnect between molecular structure and perceived odor[Sell \(2006\)](https://arxiv.org/html/2609.05694#bib.bib3)\(see Supplementary Material, Figure S1\)\.

Linear machine learning approaches can nevertheless link molecular features to odor descriptors[Schicker et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib57);[Keller et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib39)and show strong predictive power for odorous molecules dominated by isolated functional groups \(e\.g\., aldehydes or thiols\)[Sanchez\-Lengeling et al\. \(2019\)](https://arxiv.org/html/2609.05694#bib.bib56)\. Although these models reach a reasonable performance ceiling \(≈80%\\approx 80\\%AUROC\)[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42), they fail to predict molecules with physico\-chemical properties different from those included in the training set\. For example, linear classifiers show a marked performance degradation when evaluating structurally novel molecules[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42);[Schicker et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib57), particularly those with extreme molecular weights\. In addition, these models fail to generalize to mixtures unseen during training[Ravia et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib55), limiting their utility for ecological olfactory stimuli[Arctander \(2017\)](https://arxiv.org/html/2609.05694#bib.bib28);[Kumar et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib47);[Garg et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib27);[Ruddigkeit et al\. \(2014\)](https://arxiv.org/html/2609.05694#bib.bib26);[Hastings et al\. \(2016\)](https://arxiv.org/html/2609.05694#bib.bib24);[Sharma et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib25)\.

A compounding bottleneck in overcoming these predictive limitations is the fragmentation and lack of standardization in publicly available olfactory databases[Parma et al\. \(2026\)](https://arxiv.org/html/2609.05694#bib.bib49)\. Odor datasets that combine chemical features and odor descriptors are typically small, often on the order of10210^\{2\}to10310^\{3\}molecules[Hamel et al\. \(2024\)](https://arxiv.org/html/2609.05694#bib.bib48), compared to the millions of samples commonly used in other machine learning domains[Deng et al\. \(2009\)](https://arxiv.org/html/2609.05694#bib.bib22)\. Even when aggregation efforts exist, such as Pyrfume library[Hamel et al\. \(2024\)](https://arxiv.org/html/2609.05694#bib.bib48), which integrates olfaction data from multiple sources[Arctander \(2017\)](https://arxiv.org/html/2609.05694#bib.bib28);[Kumar et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib47);[Garg et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib27);[Ruddigkeit et al\. \(2014\)](https://arxiv.org/html/2609.05694#bib.bib26);[Hastings et al\. \(2016\)](https://arxiv.org/html/2609.05694#bib.bib24);[Sharma et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib25), annotations remain sparse and inconsistent, particularly with respect to human perceptual descriptors and functional chemical group labels\. Despite these limitations, Pyrfume has catalyzed the development of predictive models evaluated on shared benchmarks through initiatives such as the DREAM Olfaction Challenges[Keller et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib39);[Satarifard et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib40)\. These efforts enabled the first models to systematically map chemical structure to perceptual space by leveraging predefined, high\-dimensional representations of molecular features \(i\.e\., chemical fingerprints[Moriwaki et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib43);[Morgan \(1965\)](https://arxiv.org/html/2609.05694#bib.bib44)\), thereby facilitating supervised learning of odor descriptors\.

However, while these aggregated repositories significantly increase the number of odorant\-label pairs available for model training, they still lack critical dimensions needed to capture the ecological validity of odor perception, most notably stimulus intensity\. Odor quality is not a static property of molecular structure but varies with concentration, with some molecules exhibiting qualitative perceptual shifts across intensity ranges\. For instance, furanol is described as strawberry\-like at low concentrations but as caramel\-like at higher concentrations[Zabetakis et al\. \(1999\)](https://arxiv.org/html/2609.05694#bib.bib2)\. As a result, descriptors as fixed labels independent of concentration impose a fundamental limitation on current predictive models, which fail to account for the dynamic and context\-dependent nature of olfactory perception\.[Keller et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib39);[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42);[Hamel et al\. \(2024\)](https://arxiv.org/html/2609.05694#bib.bib48)\.

To overcome the limitations of fingerprint\-based odor prediction, researchers initially augmented structural descriptors with mass spectra data[Nozaki and Nakamoto \(2016\)](https://arxiv.org/html/2609.05694#bib.bib31), and subsequently the field shifted from predefined chemical features toward learned representations, in which deep neural networks discover structure\-odor mapping directly from the data[Zhou et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib32)\. Deep learning models have demonstrated strong predictive capability in the field of molecular property prediction by employing graph based approaches, such as message passing neural networks \(MPNNs\), which represent molecules as atoms \(nodes\) and bonds \(edges\)[Gilmer et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib6)\. Through iterative message passing, atoms exchange information with their neighbors, integrating learned molecular representations to capture structural features and non\-linear relationships, enabling more accurate prediction of such properties as drug efficacy, toxicity, and receptor binding[Li et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib50), across domains including quantum chemistry[Gilmer et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib6)and drug discovery[Wang et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib30)\. This transition is substantial: while predefined fingerprints record the presence of isolated chemical substructures, graph\-based approaches preserve exact molecular topology\. By allowing atoms to iteratively exchange information with their neighbors, the model dynamically learns specific geometric arrangements while explicitly incorporating localized node and edge features, such as formal charge, valence, and structural degree\.

In computational olfaction, the most prominent application of these graph\-based models is the Principal Odor Map \(POM\)[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42)and its open\-source counterpart, OpenPOM[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)\. These models successfully map 2D molecular graphs of single molecules directly to latent perceptual dimensions\. Although recent frameworks like PharmaGNN[Liu et al\. \(2026\)](https://arxiv.org/html/2609.05694#bib.bib64)incorporate static pharmacophore features to capture 3D spatial alignments, all these monomolecular approaches remain critically insufficient for real\-world applications, as they do not extend to multi\-component odor mixtures\. While MPNNs excel at capturing local chemical environments \(e\.g\., functional groups and bonding patterns that impact reactivity\), they are constrained by a limited receptive field: information propagates only between neighboring atoms at each step, so many layers are required to propagate information across distant parts of the molecule[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)\.

Deep learning architectures with global receptive fields directly address this aggregation bottleneck by allowing every element to attend to every other in a single step\. Such models have proven powerful at capturing long\-range dependencies, namely, relationships between elements that are far apart in sequence or space, such as a pronoun and the noun it refers to earlier in a sentence[Vaswani et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib54)or two residues that are distant in a protein’s sequence yet adjacent once it folds[Jumper et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib19)\. Global architectures, particularly graph transformers \(GTs\)[Dwivedi and Bresson \(2020\)](https://arxiv.org/html/2609.05694#bib.bib60);[Ying et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib65), have not yet been applied to structure–odor prediction, even though odor descriptors depend on the molecule’s overall geometry and topology rather than on isolated functional groups alone[Amoore et al\. \(1964\)](https://arxiv.org/html/2609.05694#bib.bib58)\. Even small structural variations \(e\.g\., the exact distance between functional groups or the specific branching of a carbon chain\) can dictate receptor binding and evoke different perceptual outcomes[Uchida et al\. \(2000\)](https://arxiv.org/html/2609.05694#bib.bib20)\. The position of an ester’s carbonyl group, for instance, more strongly governs which olfactory receptors it activates than does the identity of the functional group itself[Poivet et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib21)\. Similarly, while molecules can differ only in the position of a single substituent on the ring, such as carvacrol and thymol, they may trigger the distinctly different odors of oregano and thyme, respectively\. This specificity underscores that odor prediction depends on spatial distribution of features across the entire molecule, not on their presence alone\. By combining local atomic features into a single global structure, GTs directly mimic how the biological olfactory system integrates receptor signals into the brain\.[Blazing and Franks \(2020\)](https://arxiv.org/html/2609.05694#bib.bib68)\. This spatial awareness is necessary to predict labels for both single molecules and mixture odors\.

Traditional GTs allow every atom in a molecule to attend to every other atom at once, but they typically treat the molecule as a disconnected “bag of atoms,” ignoring its geometric structure\. In other words, they are geometry\-blind, failing to capture molecular topology or spatial organization\. Positional and structural encodings \(PSEs\) bridge this gap by providing geometric context: where an atom sits and how its neighborhood is arranged\. By injecting these geometric rules directly into the attention mechanism, PSEs force the model to evaluate the molecule as a 3D architecture with explicit topological and geometric information rather than as a mere collection of atoms\. Despite several types of PSEs \[e\.g\., random\-walk structural encodings \(RWSEs\) and Laplacian positional encodings \(LapPEs\)[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)\] to select from, no principled guideline optimizes PSE selection for a specific downstream task\. Naively concatenating a variety of different PSEs introduces computational overhead \(time spend managing rather than doing a task\) while often failing to yield proportional performance gains\. To bypass the manual selection of specific PSEs, Cantürket al\.[Cantürk et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib23)proposed the highly transferable Graph Positional and Structural Encoder \(GPSE\)\. Rather than relying on domain\-specific chemical attributes, GPSE derives highly transferable encodings from graph connectivity alone\. These representations can augment any GNN and generalize across graphs of varying sizes, connectivity patterns, and modalities, including both MPNNs and GTs\.

Here, we presentGraphNOSE, an open\-source GT\-based architecture that combines global attention with GPSE to predict multi\-label odor prediction across both single molecules and binary mixtures\. We trained and evaluated the model using the largest benchmark dataset available for olfaction \(4,983 molecules, 138 descriptors\)[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42);[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)\. We hypothesized that \(a\) GPSE will provide more informative positional and structural information than commonly used positional/structural encodings, yielding systematically higher predictive performance on odor\-relevant descriptors across attention architectures\. Furthermore, we hypothesize that \(b\)GraphNOSE, a GT architecture equipped with GPSE\-based spatial encodings, will outperform state\-of\-the\-art MPNNs for molecular odor prediction, as measured by the area under the ROC curve \(AUROC\) on held\-out test data, generalization to out\-of\-distribution \(OOD\) molecules, and robustness under reduced training set sizes\.

Our experiments were organized to directly test these two hypotheses and to answer four main questions:

- •Q1Which GPSE pretraining corpus provides the most effective inductive bias forGraphNOSE, as a model designed for molecular odor prediction?
- •Q2How doesGraphNOSEcompare to existing GNN\-based baselines, particularly under extreme physico\-chemical out\-of\-distribution \(OOD\) conditions and in the low\-data regime?
- •Q3Does the learned odor space preserve known hierarchical relationships across odor labels and reflect perceptual similarity beyond mere structural similarity?
- •Q4How effectively doesGraphNOSEgeneralize to binary odor mixtures?

Our experiments are organized to directly test our two hypotheses and to answer questions Q1–Q4 through four groups of analyses: \(i\) GPSE pretraining corpora and inductive bias \(Q1\); \(ii\) benchmark comparisons, OOD generalization, and data efficiency \(Q2\); \(iii\) structure of the learned odor space and its alignment with perceptual hierarchies \(Q3\); and \(iv\) generalization to binary odor mixtures \(Q4\)\.

## 2Methods

### 2\.1Dataset and Molecular Graphs

This work uses a molecular odor dataset derived from the publicly available version of POM[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42), namely the OpenPOM release by Barsainyanet al\.[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)\. The dataset contains 4,983 molecules and remains the benchmark for representation learning in olfaction, as each molecule is annotated with up to 138 expert\-labeled odor descriptors \(e\.g\., meaty, sweet; listed in Supplementary Material, Table S1\), allowing each odorant to be associated with multiple odor descriptors\. Importantly, this is the dataset curated from GoodScents[Luebke \(2019\)](https://arxiv.org/html/2609.05694#bib.bib36)and Leffingwell PMP \(2001\)[Leffingwell \(2005\)](https://arxiv.org/html/2609.05694#bib.bib37)datasets[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)\(hereafter jointly referred to as the GS\-LF dataset\), which were also used to train the original POM model[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42), ensuring direct comparability with prior work\. In the GS\-LF dataset odor descriptor annotations are binary rather than continuous, indicating the presence or absence of a given perceptual quality but not its intensity, and they exhibit a highly imbalanced label distribution \(see Supplementary Material, Figure S2a\), where highly frequent descriptors like fruity \(1,902 molecules\) and sweet \(1,429 molecules\) significantly outnumber rare perceptual qualities such as musk \(98 molecules\) and garlic \(120 molecules\), posing significant challenges for model generalization and minority class learning\.

![Refer to caption](https://arxiv.org/html/2609.05694v1/PostNeuripsPaper.png)Figure 1:Overview of theGraphNOSEframework architecture for multi\-label molecular odor prediction\. \(A\)Raw\- input SMILES strings are processed through a custom featurizer using RDKit[Landrum et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib13)and DeepChem[Ramsundar et al\. \(2019\)](https://arxiv.org/html/2609.05694#bib.bib15), constructing undirected molecular graphsG⁡\(V,E,Xn,Xe\)G\(V,E,X\_\{n\},X\_\{e\}\), where atoms and bonds represent nodes and edges, respectively\.\(B\)The structural graph and node features are augmented using a graph positional and structural encoder \(GPSE\) pre\-trained on diverse molecular and graph datasets \(e\.g\., ZINC, ChEMBL, GEOM, PCQM4Mv2, MolPCBA\) or standard baseline \(RWSE, LapPE\) to generate comprehensive spatial representations \(Xn′X^\{\\prime\}\_\{n\}\)\.\(C\)The enriched graph representations are fed into a hybrid GraphGPS[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)backbone that integrates local message\-passing neural networks \(MPNNs\) for near\-neighbor functional group interactions with global self\-attention mechanisms for long\-range molecular topology\.\(D\)Finally, the model predicts labels for the input molecule from the 138 odor descriptors \(e\.g\., fruity, sweet, musky, garlic\)\.To construct molecular graphs,we employed a custom featurizer on DeepChem[Ramsundar et al\. \(2019\)](https://arxiv.org/html/2609.05694#bib.bib15)\(version 2\.8\.0\), utilizing RDKit[Landrum et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib13)\(version 2025\.03\.6\) for SMILES parsing and feature extraction, converting SMILES strings into undirected graphs, where atoms are nodes and bonds are edges\. Because we employ the pre\-validated OpenPOM dataset[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53), all SMILES strings pass standard RDKit sanitization without any removals\. For each node we extract 92 atomic attributes, and for each edge we extract 7 bond attributes \(Table[1](https://arxiv.org/html/2609.05694#S2.T1)\)\. These features include standard chemoinformatics descriptors such as atomic number, valence, formal charge, hybridization state, aromaticity, and bond order\. As the OpenPOM dataset utilizes a strictly non\-isomeric SMILES, stereochemical information is not encoded in the model’s input features\. In other words, molecules that differ only in 3D configuration share identical input representations, and odor descriptors of enantiomers will not be distinguishable\.

The overallGraphNOSEpipeline is illustrated in Figure[1](https://arxiv.org/html/2609.05694#S2.F1): raw SMILES strings are converted into molecular graphs via the custom featurizer; node embeddings are augmented with GPSE\-based positional and structural encodings, and the resulting representations are processed by GT to yield multi\-label odor predictions\.

FeatureCategoriesAtomic number1 \(H\) to 54 \(I\), UNKValence0, 1, 2, 3, 4, 5, 6, UNKDegree0, 1, 2, 3, 4, 5, UNKFormal charge−2\-2,−1\-1, 0, 1, 2, UNKNumber of hydrogens0, 1, 2, 3, 4, 5, 6, 7, 8, UNKHybridizationsp, sp2, sp3, sp3d, sp3d2, UNKEdge featuresCategoriesBond typesingle, double, triple, aromatic, UNKIs conjugatedtrue, falseIn ringtrue, falseTable 1:Node \(atomic\) and edge \(bond\) features used in molecular graphs\. All features are one\-hot encoded and concatenated into a single bit vector\. UNK denotes unknown values and serves as a catch\-all category\.
### 2\.2Model

A fundamental limitation of many GT models is their restricted use of edge information, which can limit their ability to capture local chemical environments[Dwivedi and Bresson \(2020\)](https://arxiv.org/html/2609.05694#bib.bib60)\. In these architectures, bond features typically only indirectly influence how information is aggregated across the graph, rather than driving explicit, edge\-conditioned updates of node representations\. While some GT variants, such as SAN[Kreuzer et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib35)and Graphormer[Ying et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib65), inject edge features into the mechanism that determines how strongly nodes weight information from their neighbors, edge attributes still do not directly control the feature updates applied to individual atoms\. Rampášeket al[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)therefore proposed the General, Powerful and Scalable \(GPS\) architecture, which combines local message passing with global attention\. In GPS, edges play a central role in updating node representations via the message\-passing pathway, making the architecture well suited for olfactory modeling, where both local functional groups and global molecular geometry are crucial\.

GraphNOSEadopts the GPS architecture[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)to integrate local and global molecular information within each layer\. GPS processes node features through two parallel branches: a local MPNN and a global attention module\. The MPNN pathway captures localized functional groups and near\-neighbor interactions, which are primary drivers for odor prediction[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42), while the global self\-attention pathway models long\-range dependencies across the entire molecular graph\.

Formally, at each layerℓ\\ell, the node features are updated by aggregating the output of the local MPNN and global attention branches\. For a graph withNNnodes, adjacency matrixA∈ℝN×NA\\in\\mathbb\{R\}^\{N\\times N\}, edge feature matrixEE, and thedℓd\_\{\\ell\}\-dimensional node and edge featuresXℓ∈ℝN×dℓX^\{\\ell\}\\in\\mathbb\{R\}^\{N\\times d\_\{\\ell\}\}andE∈ℝE×7E\\in\\mathbb\{R\}^\{E\\times 7\}, the state transition for the\(ℓ\+1\)\(\\ell\+1\)\-th layer is defined as

Xℓ\+1=GPSℓ​\(Xℓ,E,A\),X^\{\\ell\+1\}=\\text\{GPS\}^\{\\ell\}\(X^\{\\ell\},E,A\),\(1\)
with the following decomposition:

XMℓ\+1\\displaystyle X^\{\\ell\+1\}\_\{M\}=MPNNeℓ​\(Xℓ,E,A\)\\displaystyle=\\text\{MPNN\}^\{\\ell\}\_\{e\}\(X^\{\\ell\},E,A\)\(2\)XTℓ\+1\\displaystyle X^\{\\ell\+1\}\_\{T\}=GlobalAttnℓ​\(Xℓ\)\\displaystyle=\\text\{GlobalAttn\}^\{\\ell\}\(X^\{\\ell\}\)\(3\)Xℓ\+1\\displaystyle X^\{\\ell\+1\}=MLPℓ​\(XMℓ\+1\+XTℓ\+1\)\\displaystyle=\\text\{MLP\}^\{\\ell\}\(X^\{\\ell\+1\}\_\{M\}\+X^\{\\ell\+1\}\_\{T\}\)\(4\)
Here,MPNNeℓ\\text\{MPNN\}^\{\\ell\}\_\{e\}denotes a message\-passing network that explicitly uses edge features to model local chemical environments, andGlobalAttnℓ\\text\{GlobalAttn\}^\{\\ell\}denotes a global attention mechanism operating on all node pairs\. The fused node embeddings are passed through a two\-layer multi\-layer perceptron \(MLPℓ\\text\{MLP\}^\{\\ell\}\)\. InGraphNOSE, edge features are updated and used exclusively within the MPNN component, while global attention operates on node embeddings\. This modular design allows the choice of specific MPNN and attention implementations to be tailored to the task\.

The core model consists of 4 GPS layers with a hidden dimensionality of 64, utilizing residual connections and layer normalization within each layer to stabilize training\. Node features are updated through two parallel branches: a graph isomorphism network with edge features \(GINE\) for local message passing, where edge attributes are exclusively updated, and single\-head attention for global dependencies\. Finally, node representations are aggregated via mean pooling and passed through a 512\-dimensional MLP classifier for graph\-level odor prediction\. For detailed model architecture and hyper\-parameter configurations, see Supplementary Material, Table S3\. We instantiate two architectural variants: GraphNOSElinuses a linear transformer[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34)for global attention, while GraphNOSEtransuses standard self\-attention[Vaswani et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib54)with quadratic complexity\. GraphNOSElinis primarily employed to avoid the heavy computational overhead associated with standard quadratic attention\.

### 2\.3Positional and Structural Encoding

Standard MPNNs are upper\-bounded in expressivity by the 1\-Weisfeiler\-Lehman \(1\-WL\) isomorphism test[Xu et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib33)\. To enhance their ability to capture complex molecular topology and to restore structural inductive biases that may be attenuated in global attention, we employ PSEs \(positional and structural encodings\)\. These encodings are concatenated with node features, allowing the model to incorporate global structural information into its learned representations\. Random\-walk structural encodings \(RWSE\) have proven effective for molecular prediction tasks, while Laplacian positional encodings \(LapPE\) are well suited to capturing long\-range dependencies[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59);[Dwivedi et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib70)\. Because no single PSE is universally optimal across tasks, we adopted the graph positional and structural encoder \(GPSE\)[Cantürk et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib23)as a pre\-trained structural backbone\. GPSE is a self\-supervised model trained to predict a suite of PSEs, which include RWSE, LapPE[Dwivedi et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib71), heat kernel diagonals, Laplacian eigenvalues, cycle counting, and electrostatic potential encodings\. Canturket al[Cantürk et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib23)showed that latent representations from GPSE can be used to augment both GNNs and GTs as rich positional and structural features\.

InGraphNOSE, we use the pre\-trained GPSE latent embeddings as additional node features and systematically evaluate the impact of different GPSE pre\-training corpora on odor prediction\. Specifically, we consider GPSE variants pre\-trained on high\-dimensional biological assays \(MolPCBA\)[Hu et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib7), large\-scale quantum properties \(PCQM4Mv2\)[Hu et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib10), conformational ensembles \(GEOM\-Drugs\)[Axelrod and Gomez\-Bombarelli \(2022\)](https://arxiv.org/html/2609.05694#bib.bib8), and drug\-like chemical space \(ZINC/ChEMBL\)[Gaulton et al\. \(2012\)](https://arxiv.org/html/2609.05694#bib.bib9);[Gómez\-Bombarelli et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib11)\. All positional and structural encodings are first projected to a fixed dimensionality via an MLP and then concatenated with node attributes during training\. The pre\-trained GPSE parameters are kept completely frozen during this process\.

### 2\.4Training

We partitioned the GS\-LF dataset using an 80:20 train/test split, following the protocol of Leeet al[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42)\. Within the 80% training portion, we reserved 20% as a validation set for hyperparameter optimization\. Due to the multi\-label nature of the dataset, we employed iterative stratified splitting to preserve the distribution of label combinations across training, validation, and test sets\. No data augmentation techniques \(e\.g\., SMILES enumeration or 3D conformer sampling\) are utilized\. The model parameters are selected via Bayesian optimization using the Optuna framework[Akiba et al\. \(2019\)](https://arxiv.org/html/2609.05694#bib.bib52)\(the final configuration is summarized in Supplementary Table S3\)\. Specifically, the network is trained using theA​d​a​mAdamoptimizer with an initial learning rate of6\.82×10−46\.82\\times 10^\{\-4\}and a batch size of1616for a maximum of150150epochs\. To address label imbalance, we optimize an imbalance\-weighted binary cross\-entropy loss[Tarekegn et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib69), where the contribution of each odor descriptor is weighted inversely to its frequency in the training set\. This ensures that rare odor labels are not overwhelmed by more frequent classes\. All reported performance metrics, including the area under the ROC curve \(AUROC\), are calculated as macro\-averages to provide an unbiased assessment across the diverse odor label space\. To rigorously assess performance differences, pairedtt\-tests were performed on seed\-matched runs, treating random seeds as paired observations rather than independent samples;pp\-values were reported uncorrected for multiple comparisons\.

### 2\.5Physico\-chemical Out\-of\-Distribution Splitting

Out\-of\-distribution \(OOD\) evaluation assesses whether a model can generalize to molecules that differ substantially from those seen during training, rather than merely interpolating within the observed chemical space\. To testGraphNOSEunder pronounced distribution shift, we constructed an OOD split of the GS\-LF dataset based on physico\-chemical properties\. We consider 11 physico\-chemical features: molecular weight, logP, hydrogen\-bond donors, hydrogen\-bond acceptors, topological polar surface area \(TPSA\), number of rotatable bonds, number of aromatic rings, number of heteroatoms, ring count, number of saturated rings, and number of aliphatic rings\. For each molecule, we compute a physico\-chemical deviance score \(Dphys\\text\{D\}\_\{\\text\{phys\}\}\) that quantifies its displacement from the dataset median across all properties:

Dphys=111​∑i=111\|xi−mediani\|MADi\\text\{D\}\_\{\\text\{phys\}\}=\\frac\{1\}\{11\}\\sum\_\{i=1\}^\{11\}\\frac\{\|x\_\{i\}\-\\text\{median\}\_\{i\}\|\}\{\\text\{MAD\}\_\{i\}\}\(5\)whereMADi\\text\{MAD\}\_\{i\}represents the median absolute deviation for featureii\. Molecules are ranked byDphys\\text\{D\}\_\{\\text\{phys\}\}, and we define a 65:35 train/test split by assigning the most physico\-chemically extreme 35% forming the test set\. This is unlike 80:20 iterative stratified split used for training, which randomly samples across the chemical space to preserve odor label balance in training and test sets\.

## 3Results and Discussion

Table 2:Different PSE augmentations\.These macro AUROC scores are averaged over five random seeds\. Positional and structural encodings \(PSEs\) are explicitly augmented to the node features to provide topological context to the global attention mechanism\.GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}uses a linear transformer[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34)for global attention, whileGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}uses standard self\-attention[Vaswani et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib54)with quadratic complexity\. Significance levels \(∗\\astdenote pairedtt\-test comparisons relative to theGPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.PSEGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}RWSE87\.356±0\.15∗∗∗87\.356\\pm\{\\scriptstyle 0\.15\}^\{\*\*\*\}84\.474±1\.04∗∗∗84\.474\\pm\{\\scriptstyle 1\.04\}^\{\*\*\*\}LapPE87\.796±0\.13∗∗∗87\.796\\pm\{\\scriptstyle 0\.13\}^\{\*\*\*\}88\.378±0\.06∗∗∗88\.378\\pm\{\\scriptstyle 0\.06\}^\{\*\*\*\}GPSEChEMBL\\text\{GPSE\}\_\{\\text\{ChEMBL\}\}88\.864±0\.0888\.864\\pm\{\\scriptstyle 0\.08\}89\.084±0\.0789\.084\\pm\{\\scriptstyle 0\.07\}GPSEGEOM\\text\{GPSE\}\_\{\\text\{GEOM\}\}88\.916±0\.0588\.916\\pm\{\\scriptstyle 0\.05\}89\.124±0\.1389\.124\\pm\{\\scriptstyle 0\.13\}GPSEPCQM4Mv2\\text\{GPSE\}\_\{\\text\{PCQM4Mv2\}\}88\.930±0\.1288\.930\\pm\{\\scriptstyle 0\.12\}88\.990±0\.0988\.990\\pm\{\\scriptstyle 0\.09\}GPSEZINC\\text\{GPSE\}\_\{\\text\{ZINC\}\}88\.958±0\.1888\.958\\pm\{\\scriptstyle 0\.18\}89\.026±0\.1089\.026\\pm\{\\scriptstyle 0\.10\}GPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}89\.024±0\.1689\.024\\pm\{\\scriptstyle\\boldsymbol\{0\.16\}\}89\.146±0\.0889\.146\\pm\{\\scriptstyle 0\.08\}### 3\.1GPSE pre\-training corpora and inductive bias \(Q1\)

We investigated the impact of augmenting PSEs such as RWSE[Dwivedi et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib71), LapPE, and learned representations from GPSE[Cantürk et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib23)to our GT model\. We evaluated these encodings with respect to two architectural variant of our model:GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\.GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}utilizes a linear transformer \(Performer[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34)\) as the global attention, whereas,GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}uses self\-attention[Vaswani et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib54)for the global attention mechanism, which has quadratic complexity\.

Table[2](https://arxiv.org/html/2609.05694#S3.T2)demonstrates that the GPSE trained on varied chemical datasets consistently outperforms explicitly constructed PSEs\. WhileGraphNOSEachieves its highest numerical performance with MolPCBA pre\-training, the framework exhibits robust performance across all evaluated pre\-training corpora, indicating thatGraphNOSEis not overly sensitive to the selection of the underlying training domain\. Although we hypothesize that this minor numerical advantage may stem from a favorable alignment between MolPCBA biological assay targets and olfactory receptor\-binding mechanisms, the differences across corpora remain largely statistically indistinguishable\.

From a theoretical perspective, linear transformers like Performer[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34)are characterized by lower expressivity compared to self\-attention mechanisms\. Across most positional and structural encodings,GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}perform equivalently, showing negligible performance gaps \(≤0\.12%\\leq 0\.12\\%\) with overlapping standard deviations\. An exception is observed with RWSE, whereGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}degrades markedly in performance relative toGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\(Table[2](https://arxiv.org/html/2609.05694#S3.T2), line 1\)\. The performance degradation may be attributed to the attention collapse in RWSE, where the model pays more attention to the local structural features, and this leads to an attention matrix, which is often sparse\. This does not persist inPerformers[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34), which handles the attention matrix by the low\-rank kernel approximation, or in LapPE, which constructs it from a graph Laplacian matrix that encodes global graph geometry\.

Table 3:Baselines\. Models are grouped into descriptor\-based methods, LLM embedding approaches, standard GNN architectures, other GNN variants, and the proposed method\. Linear models using the limited\-memory Broyden\-Fletcher\-Goldfarb\-Shanno \(LBFGS\) solver are deterministic and therefore do not report standard deviations, pairedpp\-values, or effect sizes\. Significance levels and effect sizes \(∗\\astdenote pairedtt\-test comparisons and Cohen’sddrelative to theGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}baseline\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\. LR, logistic regression; RF, random forest\.ClassModelROC \(%\)pp\-valueCohen’sddLinear \[logistic regression \(LR\)\]LR \(TopTorsion\)[Nilakantan et al\. \(1987\)](https://arxiv.org/html/2609.05694#bib.bib63)80\.8280\.82––LR \(RDKit\)[Landrum et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib13)82\.0582\.05––LR \(AtomPair\)[Carhart et al\. \(1985\)](https://arxiv.org/html/2609.05694#bib.bib61)82\.8382\.83––LR \(Morgan\_ECFP4\)[Rogers and Hahn \(2010\)](https://arxiv.org/html/2609.05694#bib.bib66)83\.5683\.56––LR \(MACCS\)[Durant et al\. \(2002\)](https://arxiv.org/html/2609.05694#bib.bib62)84\.7684\.76––LLM embeddingChemBERTa[Chithrananda et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib12)84\.78±0\.04∗∗∗84\.78\\pm\{\\scriptstyle 0\.04\}^\{\*\*\*\}4\.93×10−74\.93\\times 10^\{\-7\}26\.4026\.40MoLFormer[Wu et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib38)84\.97±0\.09∗∗∗84\.97\\pm\{\\scriptstyle 0\.09\}^\{\*\*\*\}3\.12×10−83\.12\\times 10^\{\-8\}52\.6652\.66FingerprintRF\-bFP[Rogers and Hahn \(2010\)](https://arxiv.org/html/2609.05694#bib.bib66)85\.00±0\.08∗∗∗85\.00\\pm\{\\scriptstyle 0\.08\}^\{\*\*\*\}9\.67×10−79\.67\\times 10^\{\-7\}22\.3122\.31RF\-cFP[Rogers and Hahn \(2010\)](https://arxiv.org/html/2609.05694#bib.bib66)85\.01±0\.09∗∗∗85\.01\\pm\{\\scriptstyle 0\.09\}^\{\*\*\*\}1\.81×10−61\.81\\times 10^\{\-6\}19\.0819\.08RF\-Mordred[Moriwaki et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib43)85\.81±0\.15∗∗∗85\.81\\pm\{\\scriptstyle 0\.15\}^\{\*\*\*\}2\.27×10−62\.27\\times 10^\{\-6\}18\.0118\.01GNNLDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}[Saha et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib73)86\.60±0\.25∗∗∗86\.60\\pm\{\\scriptstyle 0\.25\}^\{\*\*\*\}8\.34×10−58\.34\\times 10^\{\-5\}7\.287\.28LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}[Saha et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib73)86\.91±0\.15∗∗∗86\.91\\pm\{\\scriptstyle 0\.15\}^\{\*\*\*\}3\.73×10−53\.73\\times 10^\{\-5\}8\.928\.92OpenPOM[Barsainyan et al\. \(2023b\)](https://arxiv.org/html/2609.05694#bib.bib51)87\.14±0\.57∗⁣∗87\.14\\pm\{\\scriptstyle 0\.57\}^\{\*\*\}2\.77×10−32\.77\\times 10^\{\-3\}2\.942\.94Graph transformerGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}89\.02±0\.1689\.02\\pm\{\\scriptstyle 0\.16\}––GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}89\.15±0\.0889\.15\\pm\{\\scriptstyle 0\.08\}––

### 3\.2Benchmark comparisons, OOD generalization, and data efficiency \(Q2\)

To compare the predictive performance ofGraphNOSE, we benchmarked it against various state\-of\-the\-art GNNs, chemical language models, and classical chemoinformatics fingerprint\-based methods\.

For traditional machine learning baselines, we evaluated both linear and non\-linear architectures\. Linear baselines utilized logistic regression paired with traditional 2D molecular fingerprints \[Morgan \(Extended\-Connectivity Fingerprint, diameter 4\),[Morgan \(1965\)](https://arxiv.org/html/2609.05694#bib.bib44);[Rogers and Hahn \(2010\)](https://arxiv.org/html/2609.05694#bib.bib66)RDKit, MACCS[Durant et al\. \(2002\)](https://arxiv.org/html/2609.05694#bib.bib62), AtomPair[Carhart et al\. \(1985\)](https://arxiv.org/html/2609.05694#bib.bib61), and TopTorsion[Nilakantan et al\. \(1987\)](https://arxiv.org/html/2609.05694#bib.bib63)\] to establish the foundational performance boundary\. For our advanced non\-linear baselines, we utilized random forests, LLM embeddings augmented by multi\-layer perceptrons \(MLPs\), and inherently non\-linear graph neural networks \(GNNs\)\. To ensure that the performance comparison was fair, all baseline models were trained and evaluated using the exact same GS\-LF dataset splits as for training and the extreme physico\-chemical OOD split as described above \(see[2\.5Physico\-chemical Out\-of\-Distribution Splitting](https://arxiv.org/html/2609.05694#S2.SS5)in Methods, above\)\. All data splits and baseline evaluation scripts are publicly accessible from our[GitHub repository](https://github.com/CSIO-FPIL/GraphNOSE)\.

##### LLM embedding\-based methods

As representations derived from pre\-trained LLMs align closely with perceptual representation of odors[Taleb et al\. \(2024\)](https://arxiv.org/html/2609.05694#bib.bib46), we incorporated embeddings of chemistry\-informed LLM models ChemBERTa \(768 dimensions\)[Chithrananda et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib12)and MoLFormer \(768 dimensions\)[Ross et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib17)in the baselines\. We fed the embeddings into a robust MLP classifier optimized via Optuna[Akiba et al\. \(2019\)](https://arxiv.org/html/2609.05694#bib.bib52)and trained with the same weighted loss function to have fair comparision with our model\.

##### GNN\-based methods

We comparedGraphNOSEwith several inherently non\-linear, state\-of\-the\-art architectures optimized specifically for olfaction\. Primarily we compared it with OpenPOM,[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)the publicly available version of the model proposed by Leeet al[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42)\. OpenPOM uses a custom featurizer and utilizes MPNN to map chemical structures to perceptually meaningful latent space\. We compared the model with two variants of a GNN that utilizes the atoms in molecule localization and delocalization matrices \(AIMLDM\)[Sumar et al\. \(2015\)](https://arxiv.org/html/2609.05694#bib.bib72)approach for odor prediction and incorporates quantum\-informed electron density distributions into the graph’s adjacency matrix:[Saha et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib73)GCN\+LDM\\text\{GCN\}\+\\text\{LDM\}\(LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}\) andDMPNN\+LDM\\text\{DMPNN\}\+\\text\{LDM\}\(LDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}\)\.

Table 4:Parameter counts for single molecule odor prediction models, reported in thousands \(K\) and millions \(M\)\.ModelParametersOpenPOM2MLDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}674KLDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}671KGraphNOSE314KThe results for model performances on the held\-out test dataset are displayed in Table[3](https://arxiv.org/html/2609.05694#S3.T3)\. To establish a foundational performance boundary, linear models achieved a maximum AUROC of 84\.76% \(logistic regression with MACCS\)\.GraphNOSEsignificantly outperformed OpenPOM \(pairedtt\-test,p=0\.0028p=0\.0028,d=2\.94d=2\.94\),LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}\(p=3\.73×10−5p=3\.73\\times 10^\{\-5\},d=8\.92d=8\.92\), andLDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}\(p=8\.34×10−5p=8\.34\\times 10^\{\-5\},d=7\.28d=7\.28\), while using only 314K parameters, compared to OpenPOM \(2M parameters\) and LDM baselines \(≈670\\approx 670K parameters\) \(Table[4](https://arxiv.org/html/2609.05694#S3.T4)\)\. Though the performance of GNNs may appear marginal in absolute terms compared to fingerprint\-based methods, these gains are significant when considering the high\-dimensional sparsity and long\-tailed label distribution inherent in the GS\-LF dataset \(see Supplementary Figure S2a\)\. TheGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}results in Table[3](https://arxiv.org/html/2609.05694#S3.T3)are derived usingGPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}augmentation, as evaluation across pre\-trained corpora identified it as yielding the highest numerical performance \(Table[2](https://arxiv.org/html/2609.05694#S3.T2)\)\. The same was used in subsequent experiments\.

![Refer to caption](https://arxiv.org/html/2609.05694v1/performance_errorbars.png)Figure 2:Impact of training set size on model performance:Mean predictive performance on the held\-out test set as a function of the training data proportion\. Error bars represent the standard deviation across 5 random seeds\. While all architectures exhibit performance gains as the training set size increases, bothGraphNOSEvariants consistently outperform baseline GNNs \(OpenPOM and LDM variants\) across all sampling rates, demonstrating superior data efficiency even in low\-data regimes\.
##### GraphNOSEoutperforms GNNs in low\-data regimes

We evaluated the performance of our model on data\-constrained conditions\. This can be visualized by measuring performance on the held\-out test dataset \(Figure[2](https://arxiv.org/html/2609.05694#S3.F2)\)\. BothGraphNOSEvariants \(GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\) consistently outperformed other GNN baselines across the entire spectrum of data availability \(30–90%,p<0\.05p<0\.05top<0\.001p<0\.001; see Supplementary Tables S4–S6\)\. The performance margin is largest in the 30\-50% regime, where the baseline performance diminishes\. We hypothesize that the parameter efficiency of the proposed model, coupled with the strong structural priors provided by GPSE, facilitates this generalization\.

Table 5:Out\-of\-distribution \(OOD\) training results for molecule odor prediction across five random seeds\. As linear models using the LBFGS solver are deterministic, they do not report standard deviations, pairedpp\-values, or effect sizes\. The best\-performing score is highlighted in bold\. LR, logistic regression\. Significance levels \(∗\\astdenote pairedtt\-test comparisons relative to theGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}baseline\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.ModelAUROC \(%\)pp\-valueCohen’sddGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}84\.34±0\.18\\boldsymbol\{84\.34\\\>\\pm\\\>\}\{\\scriptstyle\\boldsymbol\{0\.18\}\}––GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}84\.26±0\.3384\.26\\pm\{\\scriptstyle 0\.33\}––OpenPOM80\.84±0\.35∗∗∗80\.84\\pm\{\\scriptstyle 0\.35\}^\{\*\*\*\}4\.39×10−74\.39\\times 10^\{\-7\}27\.1927\.19LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}79\.13±0\.44∗∗∗79\.13\\pm\{\\scriptstyle 0\.44\}^\{\*\*\*\}1\.52×10−51\.52\\times 10^\{\-5\}11\.1811\.18LDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}79\.03±0\.73∗∗∗79\.03\\pm\{\\scriptstyle 0\.73\}^\{\*\*\*\}5\.63×10−55\.63\\times 10^\{\-5\}8\.048\.04LR \(MACCS\)78\.46––LR \(Morgan\_ECFP4\)77\.31––LR \(TopTorsion\)75\.77––LR \(AtomPair\)75\.47––LR \(RDKit\)74\.52––
##### GraphNOSEoutperforms GNNs in extreme out\-of\-distribution \(OOD\) setting

To evaluate the generalization capacity ofGraphNOSE, performance was tested under a severe distribution shift using the physico\-chemical\-based split of the GS\-LF dataset \(see[2\.5Physico\-chemical Out\-of\-Distribution Splitting](https://arxiv.org/html/2609.05694#S2.SS5), above\)\. Rather than evaluating with molecules structurally similar to the training data, the models were tasked with predicting odor labels for the most physico\-chemically extreme compounds\. For reproducibility, we provide the exact OOD training and test splits in the[GitHub repository](https://github.com/CSIO-FPIL/GraphNOSE/tree/main/data)\. Despite this challenging setup, bothGraphNOSEvariants outperform the baseline GNNs by an average margin of4\.65%4\.65\\%\(p<0\.001p<0\.001, paired t\-test\)\. For context, the strongest linear baseline performance on this extreme split topped out at an AUROC of 78\.46% \(logistic regression with MACCS; see Table[5](https://arxiv.org/html/2609.05694#S3.T5)\)\.GraphNOSE’s superior performance can be hypothesized to stem from the global attention mechanism that captures the long\-range dependencies between distant functional groups that remain consistent even when the local scaffold changes \(see Table[5](https://arxiv.org/html/2609.05694#S3.T5)\)\. Alternative explanations for this performance might include the robust structural priors injected viaGPSEand the parameter\-efficient modeling \(∼\\sim314K parameters\)\.

![Refer to caption](https://arxiv.org/html/2609.05694v1/Example1_Nearest_neighbour.png)Figure 3:Nearest neighbor retrieval for a reference molecule\.a\)Reference moleculeb\)Top five nearest neighbors retrieved using cosine similarity onGraphNOSEcontinuous embeddings\.c\)Top five nearest neighbors retrieved using Tanimoto distance on bit\-based Morgan fingerprints \(serving as the established baseline in chemo\-informatics[Jasial et al\. \(2016\)](https://arxiv.org/html/2609.05694#bib.bib14)\)\.

### 3\.3Structure of the learned odor space and its alignment with perceptual hierarchies \(Q3\)

We examined the learned space ofGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\(chosen for its computational efficiency via linear attention while yielding performance comparable toGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}as detailed in Table[2](https://arxiv.org/html/2609.05694#S3.T2)\) by employing the activations from the penultimate layer of the model \(512 dimensions\)\. This analysis follows the experiments by Leeet al[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42)\.

##### Perceptually similar molecules cluster together

To evaluate the representational quality of the learned embedding space, we assessed whether local proximity also corresponds to perceptual similarity\. Five nearest neighbors were identified from the learned space ofGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}and compared with those neighbors identified via Tanimoto similarity on bit\-based Morgan fingerprints \(Figure[3](https://arxiv.org/html/2609.05694#S3.F3)\)\. To provide a descriptive diagnostic of the local neighborhood structure, a K\-nearest neighbor \(KNN\) classifier \(k=50k=50\) was trained on a single evaluation split of the penultimate\-layer embeddings ofGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}and compared against bit\-based Morgan fingerprints[Morgan \(1965\)](https://arxiv.org/html/2609.05694#bib.bib44)\. This descriptive evaluation yields an AUROC of 86\.91% forGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}versus 79\.73% for Morgan fingerprints, illustrating how the embedding space efficiently captures perceptual similarity in local neighborhoods\.

![Refer to caption](https://arxiv.org/html/2609.05694v1/FP_kde.png)\(a\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/OpenPOM_embedding_kde.png)\(b\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/embedding_kde.png)\(c\)

Figure 4:Analysis of odor space:Kernel density estimation \(KDE\) plots for Morgan fingerprint\(a\)and predicted labels by OpenPOM\(b\)and GraphNOSE\(c\)Shaded and contoured areas are kernel density estimates of the distribution of labeled data\. Colors denote odor categories, with shaded areas for broad classes and contours for specific descriptors \(e\.g\., lavender, roasted\)\.
##### Odor\-label semantics are preserved in embedding space

The presence of neighborhoods consisting of molecules exhibiting similar odor profiles is necessary but not sufficient for an optimal odor space\. The distribution of molecules in the embedding space must indicate the logical hierarchies between the odor labels\. The learned space should preserve logical hierarchies and semantic relationships between odor labels\. Figure[4\(a\)](https://arxiv.org/html/2609.05694#S3.F4.sf1)illustrates the kernel density estimation \(KDE\) plots derived from Morgan fingerprints, where the boundaries between different odor categories are largely indistinct\. This highlights the limitations of purely structural representations in capturing perceptual organization\. In contrast, the KDE of the true labels \(see Supplementary Material, Figure S3\), serves as a benchmark, demonstrating both the separation of dissimilar odors \(e\.g\., floral vs\. meaty\) and the hierarchical relationships within categories \(e\.g\., jasmine, lavender, and muguet within floral\)\. Figure[4\(c\)](https://arxiv.org/html/2609.05694#S3.F4.sf3)presents the embeddings learned by our model, where the global odor space exhibits clear separation between distinct odor families while also preserving their hierarchical structure\. In particular,GraphNOSEsuccessfully distinguishes between broad perceptual classes while simultaneously capturing fine\-grained relationships among subcategories\. These results suggest thatGraphNOSElearns a perceptually relevant latent representation that more closely reflects human odor perception than conventional fingerprint\-based embeddings\.

![Refer to caption](https://arxiv.org/html/2609.05694v1/alcoholic.png)\(a\)Alcohol
![Refer to caption](https://arxiv.org/html/2609.05694v1/fruity.png)\(b\)Fruity
![Refer to caption](https://arxiv.org/html/2609.05694v1/ketonic.png)\(c\)Ketonic
![Refer to caption](https://arxiv.org/html/2609.05694v1/musk.png)\(d\)Musk
![Refer to caption](https://arxiv.org/html/2609.05694v1/phenolic.png)\(e\)Phenolic
![Refer to caption](https://arxiv.org/html/2609.05694v1/sulfurous.png)\(f\)Sulfurous

Figure 5:Explainability through integrated gradients[Sundararajan et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib45)of different odor descriptors\.The visualizations highlight the molecular features most important for predicting each odor category\. Green highlights atom\-level contributions; red highlights bond\-level contributions\. Node and edge importance are visualized by the opacity and are proportional to the values\. For more examples in each category, see Supplementary Material, Figure S4\.

### 3\.4Explainability: Structure\-Odor Relationship

Integrated gradients[Sundararajan et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib45)analysis reveals that the GT model consistently focuses on canonical odor\-bearing functional groups for each class\. Figure[5\(a\)](https://arxiv.org/html/2609.05694#S3.F5.sf1)shows that alcoholic odors show high attribution on hydroxyl \(O–H and C–O\) groups, whereas Figure[5\(e\)](https://arxiv.org/html/2609.05694#S3.F5.sf5)shows that phenolic odors emphasize the phenolic OH and adjacent aromatic carbons\. The molecules for which the model prediction was sulfurous odors highlight sulfur atoms and adjacent carbon chains \(Figure[5\(f\)](https://arxiv.org/html/2609.05694#S3.F5.sf6)\)\. Ketonic odors emphasize carbonyl \(C=O\) groups andα\\alpha\-carbons in conjugated systems \(Figure[5\(c\)](https://arxiv.org/html/2609.05694#S3.F5.sf3)\)\. The musk odor label highlights the importance of constituent called muscone, a macrocyclic ketone \(Figure[5\(d\)](https://arxiv.org/html/2609.05694#S3.F5.sf4)\)\. Fruity odor labels revealed that the ester groups were key contributors \(Figure[5\(b\)](https://arxiv.org/html/2609.05694#S3.F5.sf2)\)\. These patterns are consistent with chemical intuition regarding the functional groups that influence olfaction\. This is consistent with the model learning chemically meaningful structure–odor relationships\. The insights gained increase confidence in model predictions\.

Table 6:Results for binary odor mixture prediction\.Scores are macro AUROC scores averaged over five random seeds\. Linear models \(LR, logistic regression\) are deterministic and therefore do not report standard deviations, pairedpp\-values, or effect sizes\. Significance levels \(∗\\astdenote pairedtt\-test comparisons relative toGPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.ModelPSE / FeaturesROC \(%\)pp\-valueCohen’sddGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}GPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}77\.96±0\.5877\.96\\pm\{\\scriptstyle 0\.58\}––GPSEChEMBL\\text\{GPSE\}\_\{\\text\{ChEMBL\}\}77\.84±0\.5477\.84\\pm\{\\scriptstyle 0\.54\}0\.7780\.7780\.140\.14GPSEpcqmv2\\text\{GPSE\}\_\{\\text\{pcqmv2\}\}77\.64±0\.9977\.64\\pm\{\\scriptstyle 0\.99\}0\.3490\.3490\.470\.47GPSEzinc\\text\{GPSE\}\_\{\\text\{zinc\}\}77\.58±0\.6777\.58\\pm\{\\scriptstyle 0\.67\}0\.3960\.3960\.420\.42LapPE77\.54±1\.2477\.54\\pm\{\\scriptstyle 1\.24\}0\.6010\.6010\.250\.25GPSEGeom\\text\{GPSE\}\_\{\\text\{Geom\}\}77\.24±0\.63∗77\.24\\pm\{\\scriptstyle 0\.63\}^\{\*\}0\.0380\.0381\.361\.36RWSE74\.66±1\.13∗⁣∗74\.66\\pm\{\\scriptstyle 1\.13\}^\{\*\*\}5\.00×10−35\.00\\times 10^\{\-3\}2\.502\.50MPNN\-GNN\-76\.64±0\.06∗⁣∗76\.64\\pm\{\\scriptstyle 0\.06\}^\{\*\*\}8\.31×10−38\.31\\times 10^\{\-3\}2\.172\.17LR \(Morgan\_ECFP4\)\-74\.05––LR \(TopTorsion\)\-73\.74––LR \(MACCS\)\-72\.91––LR \(AtomPair\)\-72\.87––LR \(RDKit\)\-71\.28––
### 3\.5Generalization to binary odor mixtures \(Q4\)

Odor mixing is intricate rather than being a simple additive process\. It is characterized by complex interactions where existing odors are often suppressed and entirely new olfactory qualities emerge[Sisson et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib41)\. We evaluated the capability ofGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}to generalize to binary mixtures\. To process these binary mixtures computationally, the SMILES strings of the two constituent molecules were concatenated using a standard dot \(\.\) separator\. This generates a single, disconnected molecular graph as input\. We adopted exactly the same "graph carving" train/test split \(50:50\) proposed by Sissonet al\.[Sisson et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib41), which partitions the dataset on the basis of unseen molecular structures and not unseen combinations of known molecules\. This strict separation \(44,000 training pairs and 40,000 test pairs\) prevents data leakage and learns representations for novel chemical structures with 74 odor labels\. The MPNN framework proposed by Sissonet al\.[Sisson et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib41)outperformed other baselines \(including GIN, support vector machine \(SVM\), and random forest\) on this specific split\. We benchmarked our model directly against this state\-of\-the\-art MPNN baseline and linear models \(Table[6](https://arxiv.org/html/2609.05694#S3.T6)\)\. We used the same optimal hyperparameters chosen for the single\-molecule model, increasing only the training epochs that significantly reduced the computational overhead\.

GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}utilizing MolPCBA pre\-training significantly outperforms the MPNN architecture[Sisson et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib41), the best\-performing model to date on this dataset \(pairedtt\-test,p=0\.0083p=0\.0083with only 281K parameters compared to the MPNN’s 2M parameters; other GPSE variants are numerically higher but untested \(Table[6](https://arxiv.org/html/2609.05694#S3.T6)\)\. Notably, linear models maintain highly competitive baseline scores for mixtures, with logistic regression utilizing Morgan fingerprint reaching an AUROC of 74\.05%\. Overall, these results demonstrate thatGraphNOSEgeneralizes effectively to binary mixtures under a structure\-disjoint split while remaining highly parameter efficient\.

In summary, in this work we systematically resolved four open questions in computational olfaction\. First, we demonstrated that augmenting graph transformers with learned GPSE representations significantly outperforms hand\-crafted structural encodings \(RWSE, LapPE\)\. Furthermore, while the choice of GPSE pre\-training corpus does not significantly alter predictive performance, MolPCBA was selected as the numerically optimal configuration and used throughout \(Q1\)\. Second,GraphNOSEachieves robust performance in low\-data regimes \(30%​–​90%30\\%\\text\{\-\-\}90\\%training data\) and strong generalization under extreme physico\-chemical OOD shifts, outperforming baseline GNNs despite operating with a parameter\-efficient footprint \(Q2\)\. Third, learned space neighborhood analysis and label distribution density estimations confirmed that the model’s continuous latent space preserves hierarchical label clustering and perceptual similarities \(Q3\)\. Finally, by processing concatenated molecular graph pairs,GraphNOSEsuccessfully generalizes to binary odor mixtures, under a structure\-disjoint split \(Q4\)\. Together, these empirical findings directly validate our initial hypotheses, confirming both that GPSE\-derived spatial encodings deliver a superior inductive bias relative to hand\-crafted encodings \(hypothesis a\) and thatGraphNOSEoutperforms state\-of\-the\-art MPNNs in structural generalization, low\-data sample efficiency, and multi\-label olfactory prediction \(hypothesis b\)\.

These findings substantially improve the accuracy–parameter trade\-off in computational olfaction, balancing computational overhead with structural generalization and ecological validity\. By integrating structural priors \(GPSEMolPCBA\\text\{GPSE\}\_\{\\text\{MolPCBA\}\}\) with global attention in a parameter\-efficient architecture,GraphNOSEproves that superior predictive performance does not require massive computational overhead or static high\-dimensional chemical fingerprints\. By integrating single\-molecule topology, binary mixture dynamics, and human perceptual space, this framework delivers a scalable and interpretable foundation for robust odor prediction\. Nevertheless, we observe sensitivity of the framework to variations in hyperparameter settings\. As our model considers only the 2D representation, it assigns identical predictions to chiral enantiomers \(mirror\-image structures\) that have distinct odors\. Future work could address this limitation by incorporating explicit stereochemical features into node or edge representations\. Furthermore, though we utilize exclusively the Performer based linear transformer, alternative sparse attention models such as BigBird, can be implemented in future work to assess potential gains in expressive capacity\. Additionally, while highly computational, inherently non\-linear deep learning models currently dominate predictive accuracy, there remains significant potential to improve classical linear models\. Future work could focus on engineering richer, more expressive chemical descriptors that might allow these computationally lightweight, deterministic models to bridge the performance gap and potentially rival deep learning frameworks\. Finally, the model was evaluated on curated GS\-LF dataset\. Future work is required to validate its scalability across larger, more diverse olfactory repositories to address open questions regarding robustness against label noise and the integration of concentration\-dependent perceptual shifts that cannot be resolved using static binary datasets\.

## 4Conclusions

In this work, we have introducedGraphNOSE, a parameter\-efficient graph transformer \(GT\) framework that bridges the complex non\-linear gap between molecular structure and odor perception\. We demonstrate that pre\-trained structural and positional embeddings provide significantly richer inductive biases than do traditional explicitly constructed encodings\. By combining chemical priors with a global attention mechanism,GraphNOSEestablishes a new benchmark by outperforming existing message passing architectures and quantum informed models while requiring significantly fewer parameters\. Crucially, this framework not only offers a scalable solution for single molecules through the use of linear attention mechanisms but also successfully generalizes to the challenging task of prediction of binary odor mixtures and extrapolating to physico\-chemically novel compounds\. Ultimately,GraphNOSEpaves the way for future research in computational olfaction, serving as a foundational architecture for highly generalizable predictive modeling\.

### 4\.1Data and Software Availability

### 4\.2Acknowledgments

The authors acknowledge and are thankful to Council of Scientific and Industrial Research–Central Scientific Instruments Organisation \(CSIR–CSIO\), Sector 30, Chandigarh, for providing access to CSIR\-4PI supercomputer\. No external funding was involved in this project\.

### 4\.3Notes

The authors declare no competing financial interests\.

Supplementary Material GraphNOSE: A Graph Transformer in Olfaction

## 5Contents

S1\. The Complex Relationship Between Molecular Structure and Odor

Figure S1: Structural similarity does not imply perceptual similarity

S2\. Dataset Analysis

Figure S2: Dataset insights

Table S1: Odor labels used in the curated dataset

S3\. Model Architecture

S4\. Loss Function

S5\. Baselines

S6\. Positional and Structural Encoding

S7\. Pre\-training Dataset

Table S2: Summary of molecular datasets used in pre\-training of GPSE

S8\. Hyper\-parameter Search

Table S3: Hyper\-parameters forGraphNOSE

S9\. Training data efficiency and comparative GNN performance

Table S4: OpenPOM comparison in low\-data regime

Table S5:LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}comparison in low\-data regime

Table S6:LDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}comparison in low\-data regime

S10\. KDE Plot for True Labels

Figure S3: Analysis of odor space: KDE plot for true label

S11\. Explainability Analysis of Compounds

Figure S4: Explainability through integrated gradients for different odor descriptors

## 6S1\. The Complex Relationship Between Molecular Structure and Odor

![Refer to caption](https://arxiv.org/html/2609.05694v1/FamousExample.drawio.png)Figure 6:Figure S1:Structural similarity does not imply perceptual similarity\.Perceptual odor similarity to a reference molecule \(here, m\-cresol,a\)can be observed among some structurally related molecules \(e\.g\., o\-cresol,b\)but not others \(e\.g\., 3\-methylcyclohexanol,c\), and structurally dissimilar molecules \(e\.g\., a complex bicyclic ketone,d\)can show odor similarity\. Molecules sourced from the GF\-LS dataset\.[Luebke \(2019\)](https://arxiv.org/html/2609.05694#bib.bib36);[Leffingwell \(2005\)](https://arxiv.org/html/2609.05694#bib.bib37)
## 7S2\. Dataset Analysis

\(a\)\(b\)
Figure 7:Figure S2:Dataset insights: \(a\)Frequency distribution of the top 30 odor labels among 138\. \(b\) Number of labels per molecule in the GS\-LF dataset[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)\(total n = 4983\)\.Table 7:Table S1: Odor labels used in the curated dataset\.1. 1\.alcoholic
2. 2\.aldehydic
3. 3\.alliaceous
4. 4\.almond
5. 5\.amber
6. 6\.animal
7. 7\.anisic
8. 8\.apple
9. 9\.apricot
10. 10\.aromatic
11. 11\.balsamic
12. 12\.banana
13. 13\.beefy
14. 14\.bergamot
15. 15\.berry
16. 16\.bitter
17. 17\.black currant
18. 18\.brandy
19. 19\.burnt
20. 20\.buttery
21. 21\.cabbage
22. 22\.camphoreous
23. 23\.caramellic
24. 24\.cedar
25. 25\.celery
26. 26\.chamomile
27. 27\.cheesy
28. 28\.cherry
29. 29\.chocolate
30. 30\.cinnamon
31. 31\.citrus
32. 32\.clean
33. 33\.clove
34. 34\.cocoa
35. 35\.coconut
36. 36\.coffee
37. 37\.cognac
38. 38\.cooked
39. 39\.cooling
40. 40\.cortex
41. 41\.coumarinic
42. 42\.creamy
43. 43\.cucumber
44. 44\.dairy
45. 45\.dry
46. 46\.earthy
47. 47\.ethereal
48. 48\.fatty
49. 49\.fermented
50. 50\.fishy
51. 51\.floral
52. 52\.fresh
53. 53\.fruit skin
54. 54\.fruity
55. 55\.garlic
56. 56\.gassy
57. 57\.geranium
58. 58\.grape
59. 59\.grapefruit
60. 60\.grassy
61. 61\.green
62. 62\.hawthorn
63. 63\.hay
64. 64\.hazelnut
65. 65\.herbal
66. 66\.honey
67. 67\.hyacinth
68. 68\.jasmine
69. 69\.juicy
70. 70\.ketonic
71. 71\.lactonic
72. 72\.lavender
73. 73\.leafy
74. 74\.leathery
75. 75\.lemon
76. 76\.lily
77. 77\.malty
78. 78\.meaty
79. 79\.medicinal
80. 80\.melon
81. 81\.metallic
82. 82\.milky
83. 83\.mint
84. 84\.muguet
85. 85\.mushroom
86. 86\.musk
87. 87\.musty
88. 88\.natural
89. 89\.nutty
90. 90\.odorless
91. 91\.oily
92. 92\.onion
93. 93\.orange
94. 94\.orange flower
95. 95\.orris
96. 96\.ozone
97. 97\.peach
98. 98\.pear
99. 99\.phenolic
100. 100\.pine
101. 101\.pineapple
102. 102\.plum
103. 103\.popcorn
104. 104\.potato
105. 105\.powdery
106. 106\.pungent
107. 107\.radish
108. 108\.raspberry
109. 109\.ripe
110. 110\.roasted
111. 111\.rose
112. 112\.rummy
113. 113\.sandalwood
114. 114\.savory
115. 115\.sharp
116. 116\.smoky
117. 117\.soapy
118. 118\.solvent
119. 119\.sour
120. 120\.spicy
121. 121\.strawberry
122. 122\.sulfurous
123. 123\.sweaty
124. 124\.sweet
125. 125\.tea
126. 126\.terpenic
127. 127\.tobacco
128. 128\.tomato
129. 129\.tropical
130. 130\.vanilla
131. 131\.vegetable
132. 132\.vetiver
133. 133\.violet
134. 134\.warm
135. 135\.waxy
136. 136\.weedy
137. 137\.winey
138. 138\.woody

## 8S3\. Model Architecture

GraphNOSE model has been built along the lines of the GraphGPS architecture[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)usingPyTorch Geometric[Fey and Lenssen \(2019\)](https://arxiv.org/html/2609.05694#bib.bib18)\. The important difference in our implementation is that we don’t update the edge attributes\. Each node in the input graph has an attribute and positional and structural encoding \(PSE\) associated with it\. Each edge in the graph is associated with a edge attribute\. The PSE is projected using a multi\-layer perceptron \(MLP\) to the projection dimension\. The result is concatenated to the original node features and processed through a linear layer that maps it to a hidden dimension\. Each edge attribute is also mapped to the*hidden dimension*using a linear layer\. This graph with updated node and edge features is passed through 4 layers of GPS \(General, Powerful and Scalable architecture\)[Rampášek et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib59)layers\. A readout layer of global mean pooling is used on the final node embeddings\. This pooled embedding is finally passed through an MLP for the final multi\-label classification\.

During the hyperparameter search for GraphNOSE, the global attention type in GPS layer was fixed to Performer[Choromanski et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib34)\. The search results showed that GPSE pre\-trained on MoLPCBA was the optimal PSE setting\. This exact architecture was used while evaluating other PSEs described in the main text of this paper\.

More information about the hyper\-parameter tuning of GraphNOSE can be found in Section[S8](https://arxiv.org/html/2609.05694#S13)\. The exact values of the above\-mentioned hyperparameters can be found in Table[S3](https://arxiv.org/html/2609.05694#S13.T2)in that section\.\.

## 9S4\. Loss Function

Whenever we work on a multi\-label odor dataset, the distribution of the label sets and degree of data imbalance play vital roles in model generalizability and predictive accuracy\. In highly imbalanced datasets, models often exhibit high micro\-averaged ROC scores but suffer from a low macro\-averaged ROCs as the model fails to generalize to under\-represented descriptors\. Though various measures help to deal with this issue, such as the co\-occurrence level measure and the imbalance measures, this study employs imbalance ratio per label[Tarekegn et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib69)to analyze the label\-wise distribution\.

Formally, let𝒟\\mathcal\{D\}be a multi\-label dataset with a label setLL, whereYiY\_\{i\}represents the label set for theiithinstance\. TheIRLblfor a specific labelλ∈L\\lambda\\in Lis defined as the ratio of the frequency of the most common label to the frequency of labelλ\\lambda\.

I​R​L​b​l​\(λ\)=max⁡∑i=1mλ′∈L⁡h⁡\(λ′,Yi\)∑i=1mh⁡\(λ,Yi\)IRLbl\(\\lambda\)=\\frac\{\\max\_\{\\lambda^\{\\prime\}\\in L\}\\sum\_\{i=1\}^\{m\}h\(\\lambda^\{\\prime\},Y\_\{i\}\)\}\{\\sum\_\{i=1\}^\{m\}h\(\\lambda,Y\_\{i\}\)\}\(6\)where the indicator functionh⁡\(λ,Yi\)h\(\\lambda,Y\_\{i\}\)is defined as:

h⁡\(λ,Yi\)=\{1if​λ∈Yi0if​λ∉Yih\(\\lambda,Y\_\{i\}\)=\\begin\{cases\}1&\\text\{if \}\\lambda\\in Y\_\{i\}\\\\ 0&\\text\{if \}\\lambda\\notin Y\_\{i\}\\end\{cases\}\(7\)Under this metric, the most frequent label receives an IRLbl of 1; higher values signify a greater degree of imbalance for that specific descriptor\. Motivated by POM[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42), we implemented a loss which is weighted by ofl​o​g​\(1\+c​l​a​s​s​\_​i​m​b​a​l​a​n​c​er​a​t​i​o\)log\(1\+class\\\_imbalance\_\{r\}atio\)

## 10S5\. Baselines

### 10\.1OpenPOM

OpenPOM[Barsainyan et al\. \(2023b\)](https://arxiv.org/html/2609.05694#bib.bib51)is a publicly available version of the Principal Odor Map paper[Lee et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib42)and integrates with Deepchem[Barsainyan et al\. \(2023c\)](https://arxiv.org/html/2609.05694#bib.bib16)\. It uses a mono molecular odor dataset consisting of 4983 molecules[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53)curated by merging samples from two expert sources: GoodScents[Luebke \(2019\)](https://arxiv.org/html/2609.05694#bib.bib36)and Leffingwell PMP 2001[Leffingwell \(2005\)](https://arxiv.org/html/2609.05694#bib.bib37)\. OpenPOM addresses the problem of multi\-label classification for the 138 odor labels of molecules represented as SMILES strings\. The model predicts the probability for each of the 138 possible odor descriptors corresponding to a given molecule by utilizing a message\-passing neural network \(MPNN\), which maps chemical structures to perceptually meaningful latent space\. The MPNN sends messages between nodes to understand the local chemical neighborhood and then pools the node features into a global molecular embedding\. The embedding so obtained is fed into a feed\-forward network with a sigmoid activation layer at the end to output probabilities for the 138 odor labels\. OpenPOM uses a custom featurizer, with 134 node features, which include hydrogen count, valence, formal charge, degree, hybridization, atomic number; and 6 edge features, which include aromaticity, degree, and whether it is in a ring\. All these features are one\-hot encoded \(i\.e\., represented as binary vectors where a single active bit indicates the presence of a specific feature while all other elements are zero\)\. OpenPOM provides a standardized benchmark that bridges chemical structure to odor perception\.

### 10\.2Models Based on Atoms in Molecule Localization and Delocalization Matrices \(AIMLDM\)

This work by Sahaet al[Saha et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib73)is a paradigm shift where the quantum mechanical features are employed as the node and edge features along with the chemically grounded featurizers\. The model is trained on 4872 molecules, a subset of the curated OpenPOM dataset[Barsainyan et al\. \(2023a\)](https://arxiv.org/html/2609.05694#bib.bib53), using a summed binary cross\-entropy loss weighted by imbalance ratio per label \(IRLbl\)[Tarekegn et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib69)for training\.

#### 10\.2\.1LDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}

In this methodology the dataset goes through two featurization pipelines\. The first is constructed through the quantum features using the AIMLDM approach\. In this approach localization indices \(diagonal terms\) and delocalization indices \(off diagonal terms\) are taken as node features and edge features, respectively\. The second featurization pipeline employs DMPNN Featurizer,12 which captures 133 node and 14 edge features\. Consequently, the model incorporates the chemical features, bond connectivity, and atom types, as well as the quantum features, representing electron density distribution\. Of the two MPNNs, the one with chemical features is processed by utilizing the Set2Set readout, and the other with quantum features is processed through global sum pooling to obtain a global molecular embedding for each of the pipelines\. The embeddings from both branches are concatenated and fed into a feed\-forward neural network\. The training process employs the Adam optimizer with exponential learning rate decay and uses the AUROC metric to evaluate the performance\.

#### 10\.2\.2LDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}

This methodology also utilizes the same graph neural network \(GNN\) architecture as LDMDMPNN\. The architecture is same, except that it employs the MolGraph\-ConvFeaturizer[Kearnes et al\. \(2016\)](https://arxiv.org/html/2609.05694#bib.bib1)featurizer and has 30 node and 11 edge features\. As reported in the main text, LDMGCN achieves a better performance than the odor prediction model that employs the LDM matrix coupled with MPNN\.

### 10\.3MPNN model: Aroma Blend Prediction

The MPNN model proposed by Sissonet al[Sisson et al\. \(2025\)](https://arxiv.org/html/2609.05694#bib.bib41)utilizes exactly the same node and edge attributes employed by OpenPOM \(134 node and 6 edge attributes\)8\. Molecular structures for pairs of binary odor mixture molecules were grouped into a unified two\-component graph\. The aggregation layer treats all the atoms as if they belong to the same molecule and generates a final permutation invariant embedding\. The model aggregates molecular embeddings using a Set2Set readout function\. Training is performed using a summed binary cross\-entropy loss weighted by the imbalance ratio per label \(IRLbl\) to address class imbalance\. To evaluate the performance on the binary mixture molecules, it applies a “graph carving” technique that creates a 50:50 train/test split by maximizing the usable samples and minimizing the distribution shift\. It evaluates the model on 74 shared odor labels and strictly prevents data leakage by ensuring no individual molecule appears in both partitions\. Our work also uses the same split as proposed in the Sissonet al\.model\.

## 11S6\. Positional and structural encoding

We employ two complementary types of node\-level encodings to capture the local and global topology of the molecular graphs: random\-walk structural encodings \(RWSE\) and Laplacian positional encodings \(LapPE\)\.

### 11\.1Random\-walk structural encodings \(RWSE\)

RWSE characterizes the local neighborhood of a nodeiiby observing the probability that a random walker returns to the starting node afterkksteps\. For a graph with adjacency matrixAAand degree matrixDD, the random walk transition matrix is defined asP=D−1​AP=D^\{\-1\}A\. The RWSE for nodeiiis a vector defined by

RWSE​\(i\)=\[Pi​i1,Pi​i2,…,Pi​ik\]\\text\{RWSE\}\(i\)=\[P^\{1\}\_\{ii\},P^\{2\}\_\{ii\},\\dots,P^\{k\}\_\{ii\}\]\(8\)This encoding identifies local structural motifs, such as aromatic rings and functional groups, by capturing the density of local cycles\. For our experimentk=16k=16\.

### 11\.2Laplacian Eigenvector Positional Encodings \(LapPE\)

The Laplacian eigenvector positional encodings \(LapPE\) are derived from the eigen decomposition of the graph Laplacian matrixLL\.[Cantürk et al\. \(2023\)](https://arxiv.org/html/2609.05694#bib.bib23)For a simple undirected and unweighted graphG=\(V,E\)G=\(V,E\), the Laplacian is defined as

whereD∈ℕn×nD\\in\\mathbb\{N\}^\{n\\times n\}is the diagonal degree matrix withDi​i=deg​\(vi\)D\_\{ii\}=\\text\{deg\}\(v\_\{i\}\), andM∈\{0,1\}n×nM\\in\\\{0,1\\\}^\{n\\times n\}is the symmetric adjacency matrix\.

The real symmetric matrixLLadmits a full eigen decomposition:

L=U​Λ​U⊤L=U\\Lambda U^\{\\top\}\(10\)whereΛi​i=λi\\Lambda\_\{ii\}=\\lambda\_\{i\}andU\[:,i\]=uiU\_\{\[:,i\]\}=u\_\{i\}represent theii\-th eigenvalue and eigenvector, respectively\. Following standard convention, these are indexed such that0=λ1≤λ2≤⋯≤λn0=\\lambda\_\{1\}\\leq\\lambda\_\{2\}\\leq\\dots\\leq\\lambda\_\{n\}\.

We use the 8 non\-trivial eigenvectors of the graph Laplacian as the positional encoding\.

## 12S7\. Pre\-training datasets

Table 12\.1:Table S2: Summary of molecular datasets used in pretraining of GPSE\.DatasetPrimary SourceLicenseUnique GraphsMolPCBAMoleculeNet[Hu et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib7)MIT323,555ZINCZINC Database[Gómez\-Bombarelli et al\. \(2018\)](https://arxiv.org/html/2609.05694#bib.bib11)Apache 2\.0 License249,455GEOMAxelrod & Gomez\-Bombarelli[Axelrod and Gomez\-Bombarelli \(2022\)](https://arxiv.org/html/2609.05694#bib.bib8)CC0 1\.0169,925ChEMBLGaulton et al\.[Gaulton et al\. \(2012\)](https://arxiv.org/html/2609.05694#bib.bib9)CC BY\-SA 3\.0970,963PCQM4Mv2PubChemQC[Hu et al\. \(2021\)](https://arxiv.org/html/2609.05694#bib.bib10)CC BY 4\.0273,920
## 13S8\. Hyperparameter Search

For random forest \(RF\) baseline methods, we tuned the number of estimators, maximum depths of each tree, minimum number of samples required to split an internal node, minimum number of samples required to be a leaf node, and number of features to consider when looking for the best split\. We also tune the fingerprinting parameters such as radius and length\.

For the k\-nearest\-neighbor \(KNN\) models, we tuned the value ofkfor both GraphNOSE and Mordred embeddings setups\. For the Mordred setup, we additionally tuned the radius and length of the fingerprint\. The KNN predictions are distance weighted\.

For the LLM baselines, we derive the embeddings of SMILES using the pre\-trained weights of MoLFormer[Ross et al\. \(2022\)](https://arxiv.org/html/2609.05694#bib.bib17)and ChemBERTa[Chithrananda et al\. \(2020\)](https://arxiv.org/html/2609.05694#bib.bib12)\. We tune the hidden dimension of the MLP classifier along with the other common hyperparameters such as batch size, learning rate and weight decay\.

Table 13\.2:Table S3: Hyperparameters for GraphNOSEHyperparameterSearch SpaceOptimal ValueArchitecture ParametersHidden dimension\{64, 128, 256, 288, 352, 512\}64Classifier dimension\{128, 256, 512, 1024\}512Heads\{1, 2, 4\}1No\. of GPS layers\{4, 5, 6, 7\}4Dropout\[0\.0, 0\.5\]0\.440Local GNN\{gatv2, resgate, gine\}gineActivation function\{relu, gelu, elu, silu\}reluPositional Encoding ParametersPSE\{random\_walk\_pe, laplacian\_pe,gpse\_molpcbagpse\_zinc, gpse\_chembl,gpse\_molpcba, gpse\_pcqm4mv2,gpse\_geom\}Projection dimension\{4, 8, 16, 32\}16Training ParametersLearning rate\[1e\-5, 1e\-2\], log scale6\.82e\-4Weight decay\[1e\-6, 1e\-3\], log scale9\.43e\-5Batch size\{16, 32, 64, 128\}16Minimum learning rate\[1e\-7, 1e\-5\], log scale1\.98e\-7Optimizer\{adam, adamw, sgd\}adam
## 14S9\. Training data efficiency and comparative GNN performance

Table 14\.3:Table S4: AUROC on the held\-out test set across five random seeds and pairedtt\-testpp\-values comparing OpenPOM againstGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}across training data proportions \(30% – 90%\)\. Significance levels \(∗denote pairedtt\-test comparison\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.Sampled TrainSeed 1Seed 10Seed 100Seed 21Seed 42pp\-val\(vs\.GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\)pp\-val\(vs\.GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\)30%82\.3281\.8882\.6280\.5582\.401\.51×10−3∗∗1\.51\\times 10^\{\-3\\,\*\*\}1\.43×10−3∗∗1\.43\\times 10^\{\-3\\,\*\*\}40%83\.9183\.5284\.1284\.0184\.064\.39×10−4∗∗∗4\.39\\times 10^\{\-4\\,\*\*\*\}1\.55×10−4∗∗∗1\.55\\times 10^\{\-4\\,\*\*\*\}50%84\.3584\.3584\.0184\.9584\.562\.95×10−4∗∗∗2\.95\\times 10^\{\-4\\,\*\*\*\}1\.06×10−4∗∗∗1\.06\\times 10^\{\-4\\,\*\*\*\}60%85\.0185\.5485\.3885\.1585\.203\.00×10−5∗∗∗3\.00\\times 10^\{\-5\\,\*\*\*\}2\.80×10−5∗∗∗2\.80\\times 10^\{\-5\\,\*\*\*\}70%86\.7286\.9686\.5886\.9086\.517\.59×10−4∗∗∗7\.59\\times 10^\{\-4\\,\*\*\*\}1\.10×10−4∗∗∗1\.10\\times 10^\{\-4\\,\*\*\*\}80%87\.0387\.0786\.9287\.4886\.652\.87×10−4∗∗∗2\.87\\times 10^\{\-4\\,\*\*\*\}5\.70×10−4∗∗∗5\.70\\times 10^\{\-4\\,\*\*\*\}90%87\.5087\.0386\.9087\.2786\.489\.18×10−4∗∗∗9\.18\\times 10^\{\-4\\,\*\*\*\}1\.87×10−4∗∗∗1\.87\\times 10^\{\-4\\,\*\*\*\}

Table 14\.4:Table S5: AUROC on the held\-out test set across five random seeds and pairedtt\-testpp\-values comparingLDMGCN\\text\{LDM\}\_\{\\text\{GCN\}\}againstGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}across training data proportions \(30% – 90%\)\. Significance levels \(∗denote pairedtt\-test comparison\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.Sampled TrainSeed 1Seed 10Seed 100Seed 21Seed 42pp\-val\(vs\.GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\)pp\-val\(vs\.GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\)30%82\.7177\.3180\.9782\.0981\.561\.50×10−2∗1\.50\\times 10^\{\-2\\,\*\}1\.29×10−2∗1\.29\\times 10^\{\-2\\,\*\}40%83\.5783\.4582\.7483\.2483\.855\.10×10−5∗∗∗5\.10\\times 10^\{\-5\\,\*\*\*\}3\.60×10−5∗∗∗3\.60\\times 10^\{\-5\\,\*\*\*\}50%83\.8084\.8083\.8784\.9784\.722\.79×10−4∗∗∗2\.79\\times 10^\{\-4\\,\*\*\*\}2\.11×10−4∗∗∗2\.11\\times 10^\{\-4\\,\*\*\*\}60%84\.6884\.9183\.8384\.4385\.101\.38×10−4∗∗∗1\.38\\times 10^\{\-4\\,\*\*\*\}1\.80×10−4∗∗∗1\.80\\times 10^\{\-4\\,\*\*\*\}70%85\.0185\.3085\.8585\.3185\.561\.60×10−5∗∗∗1\.60\\times 10^\{\-5\\,\*\*\*\}2\.30×10−5∗∗∗2\.30\\times 10^\{\-5\\,\*\*\*\}80%86\.1686\.1886\.1386\.0886\.561\.00×10−5∗∗∗1\.00\\times 10^\{\-5\\,\*\*\*\}2\.31×10−7∗∗∗2\.31\\times 10^\{\-7\\,\*\*\*\}90%86\.3286\.4986\.5086\.7786\.682\.10×10−5∗∗∗2\.10\\times 10^\{\-5\\,\*\*\*\}6\.50×10−5∗∗∗6\.50\\times 10^\{\-5\\,\*\*\*\}

Table 14\.5:Table S6: AUROC on the held\-out test set across five random seeds and pairedtt\-testpp\-values comparingLDMDMPNN\\text\{LDM\}\_\{\\text\{DMPNN\}\}againstGraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}andGraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}across training data proportions \(30% – 90%\)\. Significance levels \(∗denote pairedtt\-test comparison\):∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗p<0\.05\{\}^\{\*\}p<0\.05\.Sampled TrainSeed 1Seed 10Seed 100Seed 21Seed 42pp\-val\(vs\.GraphNOSElin\\text\{GraphNOSE\}\_\{\\text\{lin\}\}\)pp\-val\(vs\.GraphNOSEtrans\\text\{GraphNOSE\}\_\{\\text\{trans\}\}\)30%81\.7682\.4082\.1081\.8181\.673\.00×10−6∗∗∗3\.00\\times 10^\{\-6\\,\*\*\*\}1\.66×10−6∗∗∗1\.66\\times 10^\{\-6\\,\*\*\*\}40%83\.9683\.9083\.3981\.8283\.691\.70×10−3∗∗1\.70\\times 10^\{\-3\\,\*\*\}2\.55×10−3∗∗2\.55\\times 10^\{\-3\\,\*\*\}50%83\.8783\.3082\.9583\.5884\.033\.71×10−4∗∗∗3\.71\\times 10^\{\-4\\,\*\*\*\}1\.62×10−4∗∗∗1\.62\\times 10^\{\-4\\,\*\*\*\}60%85\.4984\.7684\.7085\.0284\.865\.20×10−5∗∗∗5\.20\\times 10^\{\-5\\,\*\*\*\}3\.40×10−5∗∗∗3\.40\\times 10^\{\-5\\,\*\*\*\}70%85\.8885\.6385\.4685\.7483\.941\.46×10−3∗∗1\.46\\times 10^\{\-3\\,\*\*\}1\.06×10−3∗∗1\.06\\times 10^\{\-3\\,\*\*\}80%85\.9486\.0886\.3285\.7086\.127\.50×10−5∗∗∗7\.50\\times 10^\{\-5\\,\*\*\*\}1\.90×10−5∗∗∗1\.90\\times 10^\{\-5\\,\*\*\*\}90%86\.2786\.3686\.6486\.4286\.469\.00×10−6∗∗∗9\.00\\times 10^\{\-6\\,\*\*\*\}1\.50×10−5∗∗∗1\.50\\times 10^\{\-5\\,\*\*\*\}

## 15S10\. KDE Plot for True Labels

![Refer to caption](https://arxiv.org/html/2609.05694v1/labels_kde.png)Figure 15\.1:Figure S3: Analysis of odor space: KDE plot for true label\. Shaded and contoured areas are kernel density estimates of the distribution of labeled data\. Colors denote odor categories, with shaded areas for broad classes and contours for specific descriptors \(e\.g\., lavender, roasted\)\.
## 16S11\. Explainability Analysis of Compounds

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_alcohol_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_alcohol_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_alcohol_3.png)

\(a\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_sulphur_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_sulphur_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_sulphur_3.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_sulphur_4.png)

\(b\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_phenolic_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_phenolic_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_phenolic_3.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_phenolic_4.png)

\(c\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_fruity_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_fruity_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_fruity_3.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_fruity_4.png)

\(d\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_musk_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_musk_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_musk_3.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_musk_4.png)

\(e\)
![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_ketonic_1.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_ketonic_2.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_ketonic_3.png)

![Refer to caption](https://arxiv.org/html/2609.05694v1/sup_ketonic_4.png)

\(f\)

Figure 16\.2:Figure S4: Explainability through integrated gradients[Sundararajan et al\. \(2017\)](https://arxiv.org/html/2609.05694#bib.bib45)for different odor descriptors:\(a\) alcohol, \(b\) sulfurous, \(c\) phenolic, \(d\) fruity, \(e\) musk, and \(f\) ketonic\. Green indicates atom\-level contributions; red, bond\-level contributions\. Opacity indicates node and edge importance and is proportional to the values\.
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