Molecular Graphs

Graph Neural Networks (GNNs) achieve strong predictive performance on graph-structured data across domains such as chemistry, biology, and network analysis, yet they provide no intrinsic explanation of their predictions. This limits their adoption in high-stakes and safety-critical settings. Counterfactual explanations address this by revealing the minimal structural modifications that would change a model's prediction. On graphs, however, such a modification is hard to produce. The search space is discrete and combinatorial, and a valid answer must respect categorical node and edge types together with domain rules such as chemical valency in the case of molecular graphs. Existing explainers give up one of two things. Either edits are not held on the data manifold, or the search does not span the full edit space. We propose Graph Diffusion Counterfactual Explanation via Inversion (GDCE-I), which gives up neither. A discrete denoising diffusion model with a novel discrete inversion scheme enables distribution-aware edits leveraging the whole domain edit space. We further address the incomplete and inconsistent evaluation of graph counterfactuals by deriving a framework of explanation desiderata and applying it to every method under one shared protocol. Across four benchmarks, GDCE-I outperforms related work by a large margin on the defined framework. For the molecular domain, we further qualitatively show that GDCE-I attains interpretable in-distribution solutions.

Biomedical knowledge graphs (KGs) accelerate drug discovery, but standard pipelines assume query molecules already exist as graph entities, leaving unregistered molecules disconnected. We address this cold-start challenge, termed the out-of-graph molecule problem, by introducing MolBioKG. This two-layer system grounds unseen molecules in biomedical evidence via multi-resolution structural anchoring. It connects an index of 2.74 million molecules (represented by scaffolds, fragments, functional groups, and fingerprints) to a 9.6-million-edge KG. Given only a SMILES string, MolBioKG retrieves structurally related graph entities and traverses their biomedical neighborhoods without task-specific training. It features two inference mechanisms: static multi-anchor retrieval using Reciprocal Rank Fusion, and Adapt-KG, a tool-using LLM policy for adaptive traversal. Evaluated across in-graph link recovery, complex multi-hop reasoning, and out-of-graph generalization, MolBioKG outperforms strong baselines. Notably, it raises Hits@10 from 0.585 to 0.876 in multi-hop reasoning and out-of-graph target recall from 0.145 to 0.269, all while ensuring predictions retain traceable structural anchors and source-attributed KG evidence.

Machine learning is transforming molecular sciences by accelerating property prediction, simulation, and the discovery of new molecules and materials. Acquiring labeled data in these domains is often costly and time-consuming, whereas large collections of unlabeled molecular data are readily available. Standard semi-supervised learning methods often rely on label-preserving augmentations, which are challenging to design in the molecular domain, where minor changes can drastically alter properties. In this work, we show that semi-supervised methods that rely on an ensemble consensus can boost predictive accuracy across a diverse range of molecular datasets, task types, and graph neural network architectures. We find that training with an ensemble consensus objective increases robustness in models and exhibits an effect similar to knowledge distillation; an individual member of an ensemble trained this way outperforms a full ensemble trained in a traditional supervised fashion in almost all cases. In addition, this type of semi-supervised training reduces calibration error.

Masked discrete diffusion for molecular graph generation typically applies a uniform corruption schedule to all tokens in a lossless graph-to-sequence representation, implicitly treating structurally heterogeneous molecular components as equally difficult and equally important to reconstruct. However, different molecular graph token roles exhibit substantial variation in denoising difficulty and their influence on the decoded molecule, motivating role-specific corruption strategies. We introduce MotifRole-Diff, a role-aware corruption process that allocates masking rates according to empirically measured denoising difficulty and graph-level perturbation impact while preserving the model architecture, clean sequence space, and lossless molecular-graph decoder. We formulate schedule selection as the risk-optimal allocation of a fixed masking budget across token roles. Our theorem characterizes optimality for the modeled role-weighted residual risk, while downstream generation performance is evaluated empirically. Under matched architecture, training budget, and sampling compute, MotifRole-Diff improves validity on QM9 from 0.905 to 0.944 while reducing FCD from 1.701 to 1.609, and on MOSES improves validity from 0.920 to 0.938 while reducing FCD from 2.125 to 1.850. Role-wise diagnostics further show improved reconstruction across molecular graph token categories. Together, these matched-compute results indicate that structurally informed corruption is a more effective masking strategy than uniform schedules for serialized molecular graph diffusion.

Accurate prediction of ADMET (Absorption, Distribution, Metabolism, Excretion, and Toxicity) is important for drug discovery. Most predictors use undirected molecular graphs and pairwise edges. This choice misses asymmetric interactions, nonreversible dynamics, and motif level effects from functional groups and ring systems. We propose ChemHyperMag for multitask ADMET prediction under missing labels. ChemHyperMag builds a functional group hypergraph from rings, BRICS fragments, Bemis-Murcko scaffolds, and bonds. It also defines a potential driven nonreversible flow guided by electronegativity and Gasteiger partial charges. The resulting circulation is encoded by a Hermitian magnetic Laplacian and processed with a magnetic Chebyshev encoder. We perturb magnetic phases to form stochastic views and train with an InfoNCE objective. Experiments on multiple ADMET benchmarks show improvements over recent methods with fewer labeled samples and no conformers. ChemHyperMag is scalable and provides interpretable directional signals through its magnetic phases.

Drug discovery and development is time-consuming and resource-intensive, motivating computational approaches such as diffusion models for de novo drug design. Many such models follow the structure-based drug design (SBDD) paradigm, generating molecules to fit a target binding pocket. However, existing diffusion-based SBDD methods typically couple pocket and ligand representation learning, model interactions only at the atom level, and prioritize binding affinity over other developability properties. Here, we introduce conDitar-dev, a conditional diffusion-based SBDD framework for generating ligands with strong binding affinities and favorable ADMET properties. It consists of three modules: msPRL, a pretrained multi-scale pocket representation learning module; conDitar, a pocket-conditioned diffusion model guided by msPRL representations; and paOPT, a generation-time method for optimizing ligand developability. On a newly curated benchmark of human disease targets, conDitar outperforms state-of-the-art SBDD baselines, achieving an average binding score of -8.85 kcal/mol. Across five ADMET properties, conDitar-dev improves performance by up to 73% over conDitar. To further validate the abilities of conDitar-dev to generate developable molecules, we have applied it to two validated druggable targets: programmed death-ligand 1 (PD-L1) and colony-stimulating factor 1 receptor (CSF1R) proteins. Top-ranked generatively designed molecules and their analogs have been experimentally synthesized and biologically tested. Two molecules generated directly by conDitar-dev for PD-L1 exhibited SPR-derived $K_D$ values of 3.49 and 3.75 $μ$M, respectively. Hit expansion based on conDitar-dev-designed molecules identified selective CSF1R inhibitors with IC$_{50}$ values as low as 200 nM, while also uncovering opportunities for drug repositioning.

Aqueous solubility is a key property in early-stage drug discovery, but most predictive models merge physicochemical descriptors and molecular graph information into a single representation, obscuring whether a prediction is driven by global chemistry, molecular structure, or both. We present an additive deep-learning framework that keeps these two sources of information separate throughout training: physicochemical descriptors are encoded by a multilayer perceptron (the chemical branch) and molecular graph topology by a graph neural network (the structural branch), with the two outputs combined only at the prediction stage through an additive model with an optional multiplicative interaction. This design provides a direct decomposition of chemical and structural components that can be examined separately after training. Furthermore, pretraining on the larger AqSolDB dataset and fine-tuning on the smaller BigSolDB2 dataset substantially improve accuracy and reduce run-to-run variations, indicating generalizability of the learned features from the data-rich settings. We further interpret the fitted model using best linear projections of the branch outputs, molecule-level embedding summaries across solubility classes, and atom-level GNNExplainer masks aggregated over functional groups. These analyses show that the chemical branch aligns with familiar physicochemical descriptors, while the structural branch captures graph-topological and functional-group patterns associated with solubility. Across both datasets, the framework attains competitive predictive performance while making the distinct roles of chemical and structural information more transparent.

Using molecular large language models (LLMs) as a unified framework for understanding molecular structures and functions is emerging as a new trend in tasks such as molecular design and drug discovery. However, these models struggle to fully capture the visual representation of molecular structures, limiting their potential. While existing molecular vision-language models (VLMs) show promise, they still face challenges in structural alignment and lack the necessary topological modeling for accurate molecular understanding. To address this, we propose MolSight, a graph-aware vision-language model framework designed to enhance the understanding of molecular images by VLMs. MolSight integrates a Molecular Topology Module to inject chemical-bond adjacency information into vision tokens, and a Molecular Grounding Module to align visual features with chemical symbolic semantics. Our experiments demonstrate that MolSight significantly outperforms existing VLMs, molecular LLMs, and specialized tools across multiple chemical visual understanding tasks, achieving a new level of molecular image reasoning.

Predicting tandem mass spectra (MS/MS) from molecular structures represents a central task in analytical chemistry with direct relevance to clinical metabolomics, systems biology, and adjacent disciplines. In this work, we revisit the problem through the lens of object detection on molecular graphs. Molecular fragmentation, a central step in MS/MS prediction, can be approximated as detecting a set of subgraphs (i.e., fragments) and their associated spectral contributions. Existing fragment-based models follow a two-stage paradigm -- first generating candidate fragments and then scoring them -- analogous to two-stage R-CNNs in computer vision. Towards higher accuracy and faster inference, we introduce GLACIER, a single-stage transformer-based fragment detection neural network for molecular graphs. This unified formulation eliminates the need for candidate enumeration, enabling scalable and globally consistent modeling of molecular fragmentation. GLACIER is faster and more accurate than existing state-of-the-art by a significant margin, achieving 70.0% and 69.7% Top-1 retrieval accuracy with and without contrastive finetuning on the MassSpecGym dataset (from the previous SOTA of 64.0%) and 52.5% and 38.5% respectively on the NIST'20 dataset (from 33.2%). Furthermore, GLACIER provides nearly 8-fold inference speedup over our prior two-stage model. Code is available at https://github.com/coleygroup/ms-pred

Backdoor attacks on molecular graph neural networks (GNNs) are typically evaluated as abstract graph edits, but real molecular learning pipelines do not train on arbitrary graphs. Molecular records must first survive parsing, sanitization, canonicalization, and graph-string consistency checks. We formalize this overlooked admission stage as ChemGuard, an operational protocol for testing whether a submitted molecular record can enter a realistic learning pipeline, while complementing existing defenses. ChemGuard admits a record only when its molecular string is sanitizable and the graph reconstructed from that string matches the submitted molecular graph. Under this operational view, many existing graph-based backdoors lose much of their apparent efficacy because their poisons are chemically invalid or representation-inconsistent. We then show that admission checks alone are insufficient to rule out molecular backdoors. We propose ChemBack, an admission-aware molecular backdoor attack that constructs chemically feasible motif-anchor attachments and ranks admitted candidates by fingerprint-based Tanimoto similarity to clean target-class molecules. ChemBack is model-free during trigger selection, using molecular structures, target labels, fingerprints, and public validity checks, but no victim model, surrogate GNN, learned embedding, gradient, logit, or training-code access. Across molecular benchmarks, validators, architectures, and defenses, \textbf{ChemBack} achieves high attack success with fully admitted poisons while preserving clean accuracy. Our results reveal a two-sided lesson, chemistry-aware admission suppresses many graph-only backdoors, yet chemically valid and target-aligned molecular backdoors remain a practical threat.

Bayesian inference provides a principled framework for modeling epistemic uncertainty in neural networks by treating predictions as distributions rather than deterministic values. Meanwhile, diffusion-based models for 3D molecular graph generation operate on fragile geometric structures governed by strict chemical constraints, making inference highly sensitive to uncertainty miscalibration. A largely overlooked issue is that epistemic uncertainty arising from the learned denoiser interacts with the aleatoric uncertainty intentionally injected during reverse diffusion, leading to systematic variance inflation and a mismatch between the true distribution and the simulated distribution. This effect is particularly detrimental for high-precision molecular generation, where even small deviations can violate chemical validity. In this work, we provide a theoretical and empirical analysis of how epistemic uncertainty propagates through diffusion inference and degrades sampling quality. Building on this investigation, we propose UCD (Uncertainty-Calibrated Diffusion), a simple yet effective method that calibrates the reverse diffusion process to account for epistemic uncertainty. Extensive experiments on standard 3D molecular benchmarks demonstrate that UCD consistently improves sampling quality across diverse baseline methods, establishing new state-of-the-art performance for 3D molecular diffusion. The code is available at https://github.com/jiuguaiwf/UCD.

Polymers underpin applications across energy, healthcare, and materials science, yet their vast chemical space makes systematic discovery challenging. Most machine learning approaches represent polymers as molecular graphs of a single repeating unit, thereby missing both the periodicity of polymer chains and many-body interactions beyond pairwise bonds. We introduce Periodic-TDL, a deep learning framework built on periodic Vietoris-Rips complexes that capture many-body interactions across multiple spatial scales, followed by a hierarchical simplicial message-passing (HSMP) encoder that propagates information from long-range interactions to covalent bonds, yielding representations enriched by higher-order topological features. Periodic-TDL outperforms all state-of-the-art models across polymer property prediction tasks spanning electronic, optical, physical, and thermal targets. Furthermore, we quantitatively validate how ester-to-amide substitution and $α$-methylation enhance thermal stability. Using a computationally synthesized dataset of 48,208 structures-generated via systematic substitution of acrylate and acrylamide polymers-we observed a mean $T_g$ increase of $\sim 55^\circ$C for ester-to-amide substitutions and $\sim 14^\circ$C for backbone $α$-methylation across matched polymer pairs. To verify these predicted trends, we use our Periodic-TDL model to analyze six novel polymer pairs from independent experimental measurements, including three newly synthesized polymers previously unreported in the literature. The experimental data successfully confirmed the model's predictions. Ultimately, these findings demonstrate that Periodic-TDL captures the underlying physical effects of specific functional group modifications, rather than merely optimizing predictive performance on benchmark datasets.

Not all clinically relevant adverse effects are structurally inferable from molecular graphs - regardless of model quality or architectural complexity. This study introduces an operational taxonomy of the structural information limits that prevent structure-based toxicity prediction, independent of the learning algorithm employed. Graph Neural Networks (GNNs) have emerged as a natural approach for molecular toxicity prediction, operating directly on atomic connectivity without the information loss inherent to fixed-length fingerprints. However, the fraction of a drug's known pharmacological profile that is actually inferable from molecular structure remains systematically underexplored. A systematic case study using acetylsalicylic acid (ASA, Aspirin) - one of the most comprehensively characterized drugs in pharmacology - serves as model compound. A Message Passing Neural Network (MPNN) is trained on the Tox21 benchmark and GNNExplainer is applied to characterize atom-level attribution. Results indicate that molecular structure explains approximately 45% (5/11) of known ASA adverse effects. A four-category Gap Taxonomy (GAP-1 through GAP-4) is introduced distinguishing between principally non-encodable effects, data gaps arising from Missing Not At Random (MNAR) mechanisms, assay panel mismatches, and representation errors. The MNAR gap is empirically quantified via a systematic ChEMBL query (42 documented assays, 0 retrievable bioactivity entries). An attention pooling experiment localizes the representation error to the MPNN message passing layers rather than the aggregation step. The Gap Taxonomy has direct implications for drug safety signal detection and regulatory frameworks including Good Pharmacovigilance Practice (GVP) guidelines and New Approach Methodologies (NAMs). Structural limits identified are confirmed in a companion DDI ablation study.

Obtaining consistent explanations for machine learning on molecular graphs requires predictions and attributions to be aligned with chemical identity. However, chemically equivalent drawings of the same molecule can induce different molecular representations, leading to inconsistent predictions and explanations. Here, we introduce InChIfied Invariants, a class of node, edge, and graph features based on the International Chemical Identifier (InChI) and designed to be invariant under transformations that preserve chemical identity. Using one million molecular graphs from PubChem Substances, we show that InChIfied Invariants produce identical representations for chemically equivalent graphs in 99.62% of cases, whereas standard Daylight invariants do so in only 0.35% of cases. Across MoleculeNet tasks, InChIfied Invariants preserve predictive performance while significantly improving prediction consistency across alternative graph depictions of the same molecules. We further perform a quantitative attribution analysis and show that explanations produced with standard molecular featurization methods vary substantially across chemically equivalent graphs, while InChIfied Invariants enforce consistent attributions by construction. We release open-source software implementing InChIfied Invariants, which can be used as a drop-in replacement for standard molecular graph features.

Stochastic process-based molecular graph generators have become the state of the art for template-free single-step retrosynthesis. However, these models are typically trained only on product-reactant pairs, thereby acquiring chemistry-relevant representations in an indirect and implicit manner. Meanwhile, recent advances in computer vision demonstrate that offering representation guidance to a generator can effectively distill semantics from pretrained encoders into DiTs, substantially improving both convergence and generation quality. Whether similar gains extend to the retrosynthesis task, and what graph-specific design choices can make them work, remains an open question. To address these questions, we conduct a systematic empirical study over a unified design space spanning teacher molecular representations, endpoint and granularity choices, injection depths in the denoiser, correspondence strategies and guidance scheme. Guided by these considerations, we develop Graph-oriented Representation Guidance (GRG), which achieves 58.6 / 77.2 / 83.4 / 87.1 top-1 / 3 / 5 / 10 accuracy on USPTO-50k, while increasing diversity to 15.5, both substantially outperforming the adopted base generator. Notably, GRG consistently improves all top-k metrics in out-of-distribution settings, suggesting that representation guidance facilitates the acquisition of intrinsic chemical semantics. Meanwhile, the introduced representation guidance reduces the number of epochs by 35% and the wall-clock time by 30% to reach comparable performance. In addition, we introduce a simple yet effective representation-similarity-based reranking mechanism, which further improves the top of the ranked list without training an additional verifier.

Molecules are graphs, but large language models~(LLMs) are usually asked to reason about them through linear strings. The most popular molecular representation, SMILES, compresses atoms, bonds, branches and rings into a compact sequence in which topology is implicit, forcing LLMs to reconstruct molecular structure before performing the requested chemical operation. Here we introduce MoleCode, an LLM-native, training-free, graph-explicit molecular language in which all molecular components are represented as typed entities with persistent identifiers and explicit relations. MoleCode makes molecular topology directly readable, editable and auditable within the language context, allowing an LLM to operate on structure rather than recover it from syntax. Across molecular reasoning, editing, generation and analysis tasks, this representational shift improves frontier LLMs most strongly when structural access is limiting: unfamiliar molecules, topology-sensitive operations, larger structures and repetitive polymers. It also changes how inference is allocated, replacing long reasoning traces devoted to implicit structural reconstruction with shorter, more chemically directed reasoning over explicit atoms and bonds. In molecular optimization, this enables localized, property-aligned edits that preserve structural similarity to the starting compounds. The same Subgraph--Node--Edge grammar extends beyond small molecules to polymers, Markush structures, mechanism-style transformations and interleaved scientific documents, including research articles and patent disclosures in which chemical information is distributed across text and images. These results suggest that the interface between scientific objects and LLMs should not treat structure as something to be decoded from text. When the object of reasoning is relational, the structure itself should be part of the language.

Nuclear magnetic resonance (NMR) spectroscopy provides an experimental readout of local chemical environments, but its use in molecular representation learning has been constrained by heterogeneous data and incomplete atom-level assignments. Here we construct complementary high-fidelity experimental and computational 13C NMR resources, which reveal a recurrent form of representational collapse: atoms that are equivalent in molecular topology can remain experimentally distinct in their real chemical environments, whereas explicit 3D descriptions are further limited by static conformations in dynamic regimes. To alleviate this bottleneck, we develop CLAIM (Contrastive Learning for Atom-to-molecule Inference of Molecular NMR), a framework that aligns efficient topological molecular inputs with atom-resolved NMR observables. Through hierarchical chemical priors and cross-level contrastive learning, CLAIM restores lost chemical resolution and markedly improves atom-level molecule-spectrum retrieval. CLAIM remains robust in flexible and tautomeric systems for 13C NMR prediction, improves stereoisomer discrimination without explicit 3D modelling, and transfers to broader molecular property tasks including ADMET prediction and fluorescence estimation. These results establish physically grounded spectral alignment as an effective strategy for alleviating chemical-environment collapse and for guiding experimentally grounded molecular representation learning.

Every molecule ever synthesised can be drawn as a 2D skeletal diagram, yet in modern property prediction this universally available representation has received less focus in favour of molecular graphs, 3D conformers, or billion-parameter language models, each imposing its own computational and data-engineering overhead. We present $\textbf{MolSight}$, the first systematic large-scale study of vision-based Molecular Property Prediction (MPP). Using 10 vision architectures, 7 pre-training strategies, and $2\,M$ molecule images, we evaluate performance across 10 downstream tasks spanning physical-property regression, drug-discovery classification, and quantum-chemistry prediction. To account for the wide variation in structural complexity across pre-training molecules, we further propose a $\textbf{chemistry-informed curriculum}$: five structural complexity descriptors partition the corpus into five tiers of increasing chemical difficulty, consistently outperforming non-curriculum baselines. We show that a single rendered bond-line image, processed by a vision encoder, is sufficient for competitive molecular property prediction, i.e. $\textit{chemical insight from sight alone}$. The best curriculum-trained configuration achieves the top result on $\textbf{5 of 10}$ benchmarks and top two on $\textbf{all 10}$, at \textbf{\textit{80$\times$ lower}} FLOPs than the nearest multi-modal competitor.

Geometric representation-conditioned molecule generation provides an effective paradigm that decouples molecule representation modeling from structure generation. By decoupling molecule generation into two stages-first generating a meaningful molecule representation, and then generating a 3D molecule conditioned on this representation-the efficiency and quality of the generation process can be significantly enhanced. However, its effectiveness is fundamentally limited by the quality of the representation space: pretrained molecular encoders, such as UniMol, produce representations that are non-smooth and not fully exploited during the generative training process. In this work, we propose LENSEs, a framework that better exploits the potential of molecule representations in representation-conditioned generation methods. In particular, LENSEs introduces three complementary mechanisms: (1) a representation head, simultaneously trained during generative tasks, that extracts multi-level representations from the pretrained encoder; (2) a molecule perceptual loss that optimizes the generator in a semantic-informative representation space; and (3) a node-level representation alignment (REPA) loss that explicitly aligns the generator's hidden states with encoder representations, reducing the semantic gap between pretraining and generation. We demonstrate the effectiveness of these improvements through extensive molecule generation tasks. Specifically, on the challenging molecule generation dataset GEOM-DRUG, LENSEs achieves 97.28% validity and 98.51% molecule stability, surpassing existing advanced methods. Further analyses through Lipschitz constant reduction (4.6x) and QM9 probing tasks also demonstrate the smoother, more informative refined representations, establishing generative training with alignment objectives as a potential pretraining paradigm for molecular encoders.

Optical chemical structure recognition (OCSR) translates molecular images into machine-readable representations like SMILES strings or molecular graphs, but remains challenging in real-world documents due to inexhaustible variations in chemical structures, shorthand conventions, and visual noise. Most existing deep-learning-based approaches rely on teacher forcing with token-level Maximum Likelihood Estimation (MLE). This training paradigm suffers from exposure bias, as models are trained under ground-truth prefixes but must condition on their own previous predictions during inference. Moreover, token-level MLE objectives hinder the optimization towards molecular-level evaluation criteria such as chemical validity and structural similarity. Here we introduce Minimum Risk Training (MRT) to OCSR and propose COMO (Closed-loop Optical Molecule recOgnition), a closed-loop framework that mitigates exposure bias by directly optimizing over molecule-level, non-differentiable objectives, by iteratively sampling and evaluating the model's own predictions. Experiments on ten benchmarks including synthetic and real-world chemical diagrams from patent and scientific literature demonstrate that COMO substantially outperforms existing rule-based and learning-based methods with less training data. Ablation studies further show that MRT is architecture-agnostic, demonstrating its potential for broad application to end-to-end OCSR systems.

This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.

Predictive machine learning models generally excel on in-distribution data, but their performance degrades on out-of-distribution (OOD) inputs. Reliable deployment therefore requires robust OOD detection, yet this is particularly challenging for irregular 3D graphs that combine continuous geometry with categorical identities and are unordered by construction. Here, we present a probabilistic OOD detection framework for complex 3D graph data built on a diffusion model that learns a density of the training distribution in a fully unsupervised manner. A key ingredient we introduce is a unified continuous diffusion over both 3D coordinates and discrete features: categorical identities are embedded in a continuous space and trained with cross-entropy, while the corresponding diffusion score is obtained analytically via posterior-mean interpolation from predicted class probabilities. This yields a single self-consistent probability-flow ODE (PF-ODE) that produces per-sample log-likelihoods, providing a principled typicality score for distribution shift. We validate the approach on protein-ligand complexes and construct strict OOD datasets by withholding entire protein families from training. PF-ODE likelihoods identify held-out families as OOD and correlate strongly with prediction errors of an independent binding-affinity model (GEMS), enabling a priori reliability estimates on new complexes. Beyond scalar likelihoods, we show that multi-scale PF-ODE trajectory statistics - including path tortuosity, flow stiffness, and vector-field instability - provide complementary OOD information. Modeling the joint distribution of these trajectory features yields a practical, high-sensitivity detector that improves separation over likelihood-only baselines, offering a label-free OOD quantification workflow for geometric deep learning.

Flow matching models generate high-fidelity molecular geometries but incur significant computational costs during inference, requiring hundreds of network evaluations. This inference overhead becomes the primary bottleneck when such models are employed in practice to sample large numbers of molecular candidates. This work discusses a training-free caching strategy that accelerates molecular geometry generation by predicting intermediate hidden states across solver steps. The proposed method operates directly on the SE(3)-equivariant backbone, is compatible with pretrained models, and is orthogonal to existing training-based accelerations and system-level optimizations. Experiments on the GEOM-Drugs dataset demonstrate that caching achieves a twofold reduction in wall-clock inference time at matched sample quality and a speedup of up to 3x compared to the base model with minimal sample quality degradation. Because these gains compound with other optimizations, applying caching alongside other general, lossless optimizations yield as much as a 7x speedup.