Molecular Graph Generation

Graph-structured data jointly contain discrete topology and continuous geometry, which poses fundamental challenges for generative modeling due to heterogeneous distributions, incompatible noise dynamics, and the need for equivariant inductive biases. Existing flow-matching approaches for graph generation typically decouple structure from geometry, lack synchronized cross-domain dynamics, and rely on iterative sampling, often resulting in physically inconsistent molecular conformations and slow sampling. To address these limitations, we propose Equivariant MeanFlow (EQUIMF), a unified SE(3)-equivariant generative framework that jointly models discrete and continuous components through synchronized MeanFlow dynamics. EQUIMF introduces a unified time bridge and average-velocity updates with mutual conditioning between structure and geometry, enabling efficient few-step generation while preserving physical consistency. Moreover, we develop a novel discrete MeanFlow formulation with a simple yet effective parameterization to support efficient generation over discrete graph structures. Extensive experiments demonstrate that EQUIMF consistently outperforms prior diffusion and flow-matching methods in generation quality, physical validity, and sampling efficiency.

We present Rhizome OS-1, a semi-autonomous operating system for small molecule drug discovery in which multi-modal AI agents operate as a full multidisciplinary discovery team. These agents function as computational chemists, medicinal chemists, and patent agents: they write and execute analysis code (fingerprint clustering, R-group decomposition, substructure search), visually triage molecular grids using vision capabilities, formulate explicit medicinal chemistry hypotheses across three strategy tiers, assess patent freedom-to-operate, and dynamically adapt generation strategies based on empirical screening feedback. Powered by r1 - a 246M-parameter graph diffusion model trained on 800 million molecular graphs - the system generates novel chemical matter directly on molecular graphs using fragment masking, scaffold decoration, linker design, and graph editing primitives. In two oncology campaigns (BCL6 BTB domain and EZH2 SET domain), the agent team executed 26 seeds and produced 5,231 novel molecules. Across both targets, 91.9% of generated Murcko scaffolds are absent from ChEMBL, with median Tanimoto similarity of 0.56-0.69 to the nearest known active. Boltz-2 binding affinity predictions, calibrated against ChEMBL data, achieved Spearman correlations of -0.53 to -0.64 and ROC AUC values of 0.88-0.93. These results demonstrate that semi-autonomous agent systems, equipped with graph-native generative tools and physics-informed scoring, enable a new paradigm for early-stage drug discovery: scaled, rapid, and adaptive inverse design with embedded medicinal chemistry reasoning.

Autoregressive graph generators define likelihoods via a sequential construction process, but these likelihoods are only meaningful if they are consistent across all linearizations of the same graph. Segmented Eulerian Neighborhood Trails (SENT), a recent linearization method, converts graphs into sequences that can be perfectly decoded and efficiently processed by language models, but admit multiple equivalent linearizations of the same graph. We quantify violations in assigned negative log-likelihood (NLL) using the coefficient of variation across equivalent linearizations, which we call Linearization Uncertainty (LU). Training transformers under four linearization strategies on two datasets, we show that biased orderings achieve lower NLL on their native order but exhibit expected calibration error (ECE) two orders of magnitude higher under random permutation, indicating that these models have learned their training linearization rather than the underlying graph. On the molecular graph benchmark QM9, NLL for generated graphs is negatively correlated with molecular stability (AUC $=0.43$), while LU achieves AUC $=0.85$, suggesting that permutation-based evaluation provides a more reliable quality check for generated molecules. Code is available at https://github.com/lauritsf/linearization-uncertainty

Large Language Models (LLMs) have significantly advanced molecular discovery, but existing multimodal molecular architectures fundamentally rely on autoregressive (AR) backbones. This strict left-to-right inductive bias is sub-optimal for generating chemically valid molecules, as it struggles to account for non-local global constraints (e.g., ring closures) and often accumulates structural errors during sequential generation. To address these limitations, we propose MolDA (Molecular language model with masked Diffusion with mAsking), a novel multimodal framework that replaces the conventional AR backbone with a discrete Large Language Diffusion Model. MolDA extracts comprehensive structural representations using a hybrid graph encoder, which captures both local and global topologies, and aligns them into the language token space via a Q-Former. Furthermore, we mathematically reformulate Molecular Structure Preference Optimization specifically for the masked diffusion. Through bidirectional iterative denoising, MolDA ensures global structural coherence, chemical validity, and robust reasoning across molecule generation, captioning, and property prediction.

Generating chemically valid 3D molecules is hindered by discrete bond topology: small local bond errors can cause global failures (valence violations, disconnections, implausible rings), especially for drug-like molecules with long-range constraints. Many unconditional 3D generators emphasize coordinates and then infer bonds or rely on post-processing, leaving topology feasibility weakly controlled. We propose Hierarchy-Guided Latent Topology Flow (HLTF), a planner-executor model that generates bond graphs with 3D coordinates, using a latent multi-scale plan for global context and a constraint-aware sampler to suppress topology-driven failures. On QM9, HLTF achieves 98.8% atom stability and 92.9% valid-and-unique, improving PoseBusters validity to 94.0% (+0.9 over the strongest reported baseline). On GEOM-DRUGS, HLTF attains 85.5%/85.0% validity/valid-unique-novel without post-processing and 92.2%/91.2% after standardized relaxation, within 0.9 points of the best post-processed baseline. Explicit topology generation also reduces "false-valid" samples that pass RDKit sanitization but fail stricter checks.

Molecular generative models must jointly ensure validity, diversity, and property control, yet existing approaches typically trade off among these objectives. We present MOLPAQ, a modular quantum-classical generator that assembles molecules from quantum-generated latent patches. A \b{eta}-VAE pretrained on QM9 learns a chemically aligned latent manifold; a reduced conditioner maps molecular descriptors into this space; and a parameter-efficient quantum patch generator produces entangled node embeddings that a valence-aware aggregator reconstructs into valid molecular graphs. Adversarial fine-tuning with a latent critic and chemistry-shaped reward yields 100% RDKit validity, 99.75% novelty, and 0.905 diversity. Beyond aggregate metrics, the pretrained quantum generator, steered by the conditioner, improves mean QED by approx. 2.3% and increases aromatic motif incidence by approx. 10-12% relative to a parameter-matched classical generator, highlighting its role as a compact topology-shaping operator.

Generative modeling of discrete data, such as graphs, underpins many scientific and industrial applications, including molecular discovery and materials design. In these domains, probabilistic inference is particularly valuable, as it enables composable generation and principled incorporation of desired constraints, such as structural or functional properties. Energy-based models naturally support this goal by capturing relative likelihoods and enabling composable inference by directly enforcing constraints during inference. However, discrete energy-based models typically struggle with efficient and high-quality sampling, as off-support regions often contain spurious local minima, trapping samplers and causing training instabilities, resulting in a fidelity gap compared to discrete diffusion models. To address this gap, we introduce Graph Energy Matching (GEM), a discrete generative framework inspired by the Jordan-Kinderlehrer-Otto (JKO) transport-map optimization perspective. GEM learns a permutation-invariant potential energy that simultaneously guides discrete transport from noise toward high-likelihood graph regions and refines samples within these regions. We further introduce a sampling protocol leveraging an energy-based switching strategy, seamlessly bridging rapid, gradient-guided transport and a local mixing regime for effective exploration. On molecular graph benchmarks, GEM matches or surpasses strong discrete diffusion baselines on most reported metrics. Beyond improving generation quality, GEM's relative likelihood modeling enables targeted exploration, facilitating compositional generation, property-constrained sampling, and interpolation between graphs. Project page: https://michalbalcerak.ai/graph-energy-matching/.

Mass spectrometry (MS) stands as a cornerstone analytical technique for molecular identification, yet de novo structure elucidation from spectra remains challenging due to the combinatorial complexity of chemical space and the inherent ambiguity of spectral fragmentation patterns. Recent deep learning approaches, including autoregressive sequence models, scaffold-based methods, and graph diffusion models, have made progress. However, diffusion-based generation for this task remains computationally demanding. Meanwhile, discrete flow matching, which has shown strong performance for graph generation, has not yet been explored for spectrum-conditioned structure elucidation. In this work, we introduce FlowMS, the first discrete flow matching framework for spectrum-conditioned de novo molecular generation. FlowMS generates molecular graphs through iterative refinement in probability space, enforcing chemical formula constraints while conditioning on spectral embeddings from a pretrained formula transformer encoder. Notably, it achieves state-of-the-art performance on 5 out of 6 metrics on the NPLIB1 benchmark: 9.15% top-1 accuracy (9.7% relative improvement over DiffMS) and 7.96 top-10 MCES (4.2% improvement over MS-BART). We also visualize the generated molecules, which further demonstrate that FlowMS produces structurally plausible candidates closely resembling ground truth structures. These results establish discrete flow matching as a promising paradigm for mass spectrometry-based structure elucidation in metabolomics and natural product discovery.

Graph generation is a fundamental task with broad applications, such as drug discovery. Recently, discrete flow matching-based graph generation, \aka, graph flow model (GFM), has emerged due to its superior performance and flexible sampling. However, effectively aligning GFMs with complex human preferences or task-specific objectives remains a significant challenge. In this paper, we propose Graph-GRPO, an online reinforcement learning (RL) framework for training GFMs under verifiable rewards. Our method makes two key contributions: (1) We derive an analytical expression for the transition probability of GFMs, replacing the Monte Carlo sampling and enabling fully differentiable rollouts for RL training; (2) We propose a refinement strategy that randomly perturbs specific nodes and edges in a graph, and regenerates them, allowing for localized exploration and self-improvement of generation quality. Extensive experiments on both synthetic and real datasets demonstrate the effectiveness of Graph-GRPO. With only 50 denoising steps, our method achieves 95.0% and 97.5% Valid-Unique-Novelty scores on the planar and tree datasets, respectively. Moreover, Graph-GRPO achieves state-of-the-art performance on the molecular optimization tasks, outperforming graph-based and fragment-based RL methods as well as classic genetic algorithms.

Dual-target structure-based drug design aims to generate a single ligand together with two pocket-specific binding poses, each compatible with a corresponding target pocket, enabling polypharmacological therapies with improved efficacy and reduced resistance. Existing approaches typically rely on staged pipelines, which either decouple the two poses via conditional-independence assumptions or enforce overly rigid correlations, and therefore fail to jointly generate two target-specific binding modes. To address this, we propose FuseDiff, an end-to-end diffusion model that jointly generates a ligand molecular graph and two pocket-specific binding poses conditioned on both pockets. FuseDiff features a message-passing backbone with Dual-target Local Context Fusion (DLCF), which fuses each ligand atom's local context from both pockets to enable expressive joint modeling while preserving the desired symmetries. Together with explicit bond generation, FuseDiff enforces topological consistency across the two poses under a shared graph while allowing target-specific geometric adaptation in each pocket. To support principled training and evaluation, we derive a dual-target training set and use an independent held-out test set for evaluation. Experiments on the benchmark and a real-world dual-target system show that FuseDiff achieves state-of-the-art docking performance and enables the first systematic assessment of dual-target pose quality prior to docking-based pose search.

We study generative modeling of graphs with recurring subgraph motifs. We propose Flowette, a continuous flow matching framework that employs a graph neural network-based transformer to learn a velocity field over graph representations with node and edge attributes. Our model promotes topology-aware alignment through optimal transport-based coupling and preserves structural fidelity through endpoint- and chemistry-aware regularisation. To incorporate domain-driven structural priors, we introduce graphettes, a new probabilistic family of graph structure models that generalize graphons via controlled structural edits for motifs such as rings, stars, and trees. We theoretically analyze the coupling, invariance, and structural properties of the framework, evaluate it on synthetic and molecular benchmarks, and isolate the contributions of the structural prior, the optimal-transport coupling, and the regularisation terms through controlled ablations. Flowette achieves competitive performance overall, attaining state-of-the-art results on several metrics across multiple benchmarks, highlighting the effectiveness of combining structural priors with flow-based training for modeling complex graph distributions.

Existing insertion-based masked diffusion models that generate sequences by interleaving token insertion with unmasking use fixed schedules that are not dependent on the data. For structured sequences like graphs and molecules, learning data-dependent generation orders can improve generation quality by reducing uncertainty over the action space. We propose LoFlexMDM, an insertion-based masked diffusion model with learnable order dynamics that learns data-dependent insertion and unmasking rates. We generalize the discrete flow matching framework to work with variable-length sequences, propose a tractable schedule parameterization and a training objective for joint training of the generator and the target order dynamics. On De Novo and fragment-constrained molecule generation, LoFlexMDM improves sample quality over FlexMDM by up to 17.5% and 6.7%, respectively. These results show that learning the target generation order can improve insertion-based diffusion models without giving up tractable training. We open source the code at https://github.com/dhruvdcoder/LoFlexMDM.

Molecular generation with diffusion models has emerged as a promising direction for AI-driven drug discovery and materials science. While graph diffusion models have been widely adopted due to the discrete nature of 2D molecular graphs, existing models suffer from low chemical validity and struggle to meet the desired properties compared to 1D modeling. In this work, we introduce MolHIT, a powerful molecular graph generation framework that overcomes long-standing performance limitations in existing methods. MolHIT is based on the Hierarchical Discrete Diffusion Model, which generalizes discrete diffusion to additional categories that encode chemical priors, and decoupled atom encoding that splits the atom types according to their chemical roles. Overall, MolHIT achieves new state-of-the-art performance on the MOSES dataset with near-perfect validity for the first time in graph diffusion, surpassing strong 1D baselines across multiple metrics. We further demonstrate strong performance in downstream tasks, including multi-property guided generation and scaffold extension.

Many generative tasks in chemistry and science involve distributions invariant to group symmetries (e.g., permutation and rotation). A common strategy enforces invariance and equivariance through architectural constraints such as equivariant denoisers and invariant priors. In this paper, we challenge this tradition through the alternative canonicalization perspective: first map each sample to an orbit representative with a canonical pose or order, train an unconstrained (non-equivariant) diffusion or flow model on the canonical slice, and finally recover the invariant distribution by sampling a random symmetry transform at generation time. Building on a formal quotient-space perspective, our work provides a comprehensive theory of canonical diffusion by proving: (i) the correctness, universality and superior expressivity of canonical generative models over invariant targets; (ii) canonicalization accelerates training by removing diffusion score complexity induced by group mixtures and reducing conditional variance in flow matching. We then show that aligned priors and optimal transport act complementarily with canonicalization and further improves training efficiency. We instantiate the framework for molecular graph generation under $S_n \times SE(3)$ symmetries. By leveraging geometric spectra-based canonicalization and mild positional encodings, canonical diffusion significantly outperforms equivariant baselines in 3D molecule generation tasks, with similar or even less computation. Moreover, with a novel architecture Canon, CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.

Graph generation aims to sample discrete node and edge attributes while satisfying coupled structural constraints. Diffusion models for graphs often adopt largely factorized forward-noising, and many flow-matching methods start from factorized reference noise and coordinate-wise interpolation, so node-edge coupling is not encoded by the generative geometry and must be recovered implicitly by the core network, which can be brittle after discrete decoding. Bayesian Flow Networks (BFNs) evolve distribution parameters and naturally support discrete generation. But classical BFNs typically rely on factorized beliefs and independent channels, which limit geometric evidence fusion. We propose Variational Bayesian Flow Network (VBFN), which performs a variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions. Each Bayesian update reduces to solving a symmetric positive definite linear system, enabling coupled node and edge updates within a single fusion step. We construct sample-agnostic sparse precisions from a representation-induced dependency graph, thereby avoiding label leakage while enforcing node-edge consistency. On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods. The reproducible code is available at https://github.com/Cello2195/VBFN.

Molecular graph generation (MGG) is essentially a multi-class generative task, aimed at predicting categories of atoms and bonds under strict chemical and structural constraints. However, many prevailing diffusion paradigms learn to regress numerical embeddings and rely on a hard discretization rule during sampling to recover discrete labels. This introduces a fundamental discrepancy between training and sampling. While models are trained for point-wise numerical fidelity, the sampling process fundamentally relies on crossing categorical decision boundaries. This discrepancy forces the model to expend efforts on intra-class variations that become irrelevant after discretization, ultimately compromising diversity, structural statistics, and generalization performance. Therefore, we propose TopBF, a unified framework that (i) performs MGG directly in continuous parameter distributions, (ii) learns graph-topological understanding through a Quasi-Wasserstein optimal-transport coupling under geodesic costs, and (iii) supports controllable, property-conditioned generation during sampling without retraining the base model. TopBF innovatively employs cumulative distribution function (CDF) to compute category probabilities induced by the Gaussian channel, thereby unifying the training objective with the sampling discretization operation. Experiments on QM9 and ZINC250k demonstrate superior structural fidelity and efficient generation with improved performance.