Constrained Generative Modeling

Reconstructing complete 3D shapes from incomplete or noisy observations is a fundamentally ill-posed problem that requires balancing measurement consistency with shape plausibility. Existing methods for shape reconstruction can achieve strong geometric fidelity in ideal conditions but fail under realistic conditions with incomplete measurements or noise. At the same time, recent generative models for 3D shapes can synthesize highly realistic and detailed shapes but fail to be consistent with observed measurements. In this work, we introduce GG-Langevin: Geometry-Guided Langevin dynamics, a probabilistic approach that unifies these complementary perspectives. By traversing the trajectories of Langevin dynamics induced by a diffusion model, while preserving measurement consistency at every step, we generatively reconstruct shapes that fit both the measurements and the data-informed prior. We demonstrate through extensive experiments that GG-Langevin achieves higher geometric accuracy and greater robustness to missing data than existing methods for surface reconstruction.

We propose Score-based Relaxation-guided Generation (SRG), a generative framework based on an approximate formulation of relaxation-guided stochastic differential equations (SDEs) for mixed-integer linear programming. SRG employs a Transformer-based score network that incorporates feasibility and optimality signals into score modeling, encouraging the learned generative model to place more probability mass on feasible, high-quality regions of the solution space. At inference time, SRG directly samples diverse candidate solutions from the learned score model without requiring any additional guidance module. These candidates are then used to construct compact trust-region subproblems for standard MILP solvers. Across multiple public benchmarks, SRG matches or improves upon the solution quality of the strongest learning-based baselines, with particularly strong gains in challenging candidate-generation settings. Moreover, SRG shows promising zero-shot transferability to unseen cross-scale and cross-problem instances, improving solver objectives and reducing search time in several cases through higher-quality initial candidates and compact trust-region search.

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/.

Generating synthetic populations from aggregate statistics is a core component of microsimulation, agent-based modeling, policy analysis, and privacy-preserving data release. Beyond classical census marginals, many applications require matching heterogeneous unary, binary, and ternary constraints derived from surveys, expert knowledge, or automatically extracted descriptions. Constructing populations that satisfy such multi-way constraints simultaneously poses a significant computational challenge. We consider populations where each individual is described by categorical attributes and the target is a collection of global frequency constraints over attribute combinations. Exact formulations scale poorly as the number and arity of constraints increase, especially when the constraints are numerous and overlapping. Grounded in methods from statistical physics, we propose a maximum-entropy relaxation of this problem. Multi-way cardinality constraints are matched in expectation rather than exactly, yielding an exponential-family distribution over complete population assignments and a convex optimization problem over Lagrange multipliers. We evaluate the approach on NPORS-derived scaling benchmarks with 4 to 40 attributes and compare it primarily against generalized raking. The results show that MaxEnt becomes increasingly advantageous as the number of attributes and ternary interactions grows, while raking remains competitive on smaller, lower-arity instances.

Building virtual cells with generative models to simulate cellular behavior in silico is emerging as a promising paradigm for accelerating drug discovery. However, prior image-based generative approaches can produce implausible cell images that violate basic physical and biological constraints. To address this, we propose to post-train virtual cell models with reinforcement learning (RL), leveraging biologically meaningful evaluators as reward functions. We design seven rewards spanning three categories-biological function, structural validity, and morphological correctness-and optimize the state-of-the-art CellFlux model to yield CellFluxRL. CellFluxRL consistently improves over CellFlux across all rewards, with further performance boosts from test-time scaling. Overall, our results present a virtual cell modeling framework that enforces physically-based constraints through RL, advancing beyond "visually realistic" generations towards "biologically meaningful" ones.

Reticular chemistry has enabled the synthesis of tens of thousands of metal-organic frameworks (MOFs), yet the discovery of new materials still relies largely on intuition-driven linker design and iterative experimentation. As a result, researchers explore only a small fraction of the vast chemical space accessible to reticular materials, limiting the systematic discovery of frameworks with targeted properties. Here, we introduce Nexerra-R1, a building-block chemical language model that enables inverse design in reticular chemistry through the targeted generation of organic linkers. Rather than generating complete frameworks directly, Nexerra-R1 operates at the level of molecular building blocks, preserving the modular logic that underpins reticular synthesis. The model supports both unconstrained generation of low-connectivity linkers and scaffold-constrained design of symmetric multidentate motifs compatible with predefined nodes and topologies. We further combine linker generation with flow-guided distributional targeting to steer the generative process toward application-relevant objectives while maintaining chemical validity and assembly feasibility. The generated linkers are subsequently assembled into three-dimensional frameworks and are structurally optimized to produce candidate materials compatible with experimental synthesis. Using Nexerra-R1, we validate this strategy by rediscovering known MOFs and by proposing the experimental synthesis of a previously unreported framework, CU-525, generated entirely in silico. Together, these results establish a general inverse-design paradigm for reticular materials in which controllable chemical language modelling enables the direct translation from computational design to synthesizable frameworks.

Adeno-associated viral (AAV) vectors are widely used delivery platforms in gene therapy, and the design of improved capsids is key to expanding their therapeutic potential. A central challenge in AAV bioengineering, as in protein design more broadly, is the vast sequence design space relative to the scale of feasible experimental screening. Machine-guided generative approaches provide a powerful means of navigating this landscape and proposing novel protein sequences that satisfy functional constraints. Here, we develop a generative design framework based on protein language models and reinforcement learning to generate highly novel yet functionally plausible AAV capsids. A pretrained model was fine-tuned on experimentally validated capsid sequences to learn patterns associated with viability. Reinforcement learning was then used to guide sequence generation, with a reward function that jointly promoted predicted viability and sequence novelty, thereby enabling exploration beyond regions represented in the training data. Comparative analyses showed that fine-tuning alone produces sequences with high predicted viability but remains biased toward the training distribution, whereas reinforcement learining-guided generation reaches more distant regions of sequence space while maintaining high predicted viability. Finally, we propose a candidate selection strategy that integrates predicted viability, sequence novelty, and biophysical properties to prioritize variants for downstream evaluation. This work establishes a framework for the generative exploration of protein sequence space and advances the application of generative protein language models to AAV bioengineering.

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.

Motivation: Generative models for protein backbone design have to simultaneously ensure geometric validity, sampling efficiency, and scalability to long sequences. However, most existing approaches rely on iterative refinement, quadratic attention mechanisms, or post-hoc geometry correction, leading to a persistent trade-off between computational efficiency and structural fidelity. Results: We present Physics-Informed Mamba (PI-Mamba), a generative model that enforces exact local covalent geometry by construction while enabling linear-time inference. PI-Mamba integrates a differentiable constraint-enforcement operator into a flow-matching framework and couples it with a Mamba-based state-space architecture. To improve optimisation stability and backbone realism, we introduce a spectral initialization derived from the Rouse polymer model and an auxiliary cis-proline awareness head. Across benchmark tasks, PI-Mamba achieves 0.0% local geometry violations and high designability (scTM = $0.91\pm 0.03$, n = 100), while scaling to proteins exceeding 2,000 residues on a single A5000 GPU (24 GB).

High-quality human motion data is becoming increasingly important for applications in robotics, simulation, and entertainment. Recent generative models offer a potential data source, enabling human motion synthesis through intuitive inputs like text prompts or kinematic constraints on poses. However, the small scale of public mocap datasets has limited the motion quality, control accuracy, and generalization of these models. In this work, we introduce Kimodo, an expressive and controllable kinematic motion diffusion model trained on 700 hours of optical motion capture data. Our model generates high-quality motions while being easily controlled through text and a comprehensive suite of kinematic constraints including full-body keyframes, sparse joint positions/rotations, 2D waypoints, and dense 2D paths. This is enabled through a carefully designed motion representation and two-stage denoiser architecture that decomposes root and body prediction to minimize motion artifacts while allowing for flexible constraint conditioning. Experiments on the large-scale mocap dataset justify key design decisions and analyze how the scaling of dataset size and model size affect performance.

Deep learning models for chest X-ray diagnosis are constrained by limited coverage of clinically meaningful concept combinations in publicly available training datasets. While synthetic image generation has been explored to increase data diversity, existing methods rarely enforce clinical or anatomical constraints, limiting utility for improving model reliability. We propose CARPA, a clinically aware and anatomically grounded framework for synthetic chest X-ray generation that applies targeted perturbations to clinical concept vectors while preserving anatomical structure. By producing anatomically faithful synthetic images with controlled concept insertions and deletions, CARPA expands clinically relevant concept coverage. We evaluate CARPA across seven backbone architectures by fine-tuning models on synthetic subsets and testing on a held-out MIMIC-CXR benchmark. Compared to prior concept perturbation approaches, fine-tuning on CARPA-generated images consistently improves precision-recall performance, reduces predictive uncertainty, and improves model calibration. Structural and semantic analyses demonstrate high anatomical fidelity, strong concept alignment, and low semantic uncertainty. Evaluation by two expert radiologists further confirms realism and clinical agreement. Together, these results show that anatomically grounded concept perturbations enable more effective use of synthetic data, improving both performance and reliability of chest X-ray classification models and supporting safer clinical deployment.

Variational autoencoders (VAEs) frequently suffer from posterior collapse, where the latent variables become uninformative as the approximate posterior degenerates to the prior. While recent work has characterized collapse as a phase transition determined by data covariance properties, existing approaches primarily aim to avoid rather than eliminate collapse. We introduce a novel framework that theoretically guarantees non-collapsed solutions by leveraging spherical shell geometry and cluster-aware constraints. Our method transforms data to a spherical shell, computes optimal cluster assignments via K-means, and defines a feasible region between the within-cluster variance $W$ and collapse loss $δ_{\text{collapse}}$. We prove that when the reconstruction loss is constrained to this region, the collapsed solution is mathematically excluded from the feasible parameter space. Critically, we introduce norm constraint mechanisms that ensure decoder outputs remain compatible with the spherical shell geometry without restricting representational capacity. Unlike prior approaches, our method provides a strict theoretical guarantee with minimal computational overhead without imposing constraints on decoder outputs. Experiments on synthetic and real-world datasets demonstrate 100% collapse prevention under conditions where conventional VAEs completely fail, with reconstruction quality matching or exceeding state-of-the-art methods. Our approach requires no explicit stability conditions (e.g., $σ^2 < λ_{\max}$) and works with arbitrary neural architectures. The code is available at https://github.com/tsegoochang/spherical-vae-with-Cluster.

Constrained generative modeling is fundamental to applications such as robotic control and autonomous driving, where models must respect physical laws and safety-critical constraints. In real-world settings, these constraints rarely take the form of simple linear inequalities, but instead complex feasible regions that resemble road maps or other structured spatial domains. We propose a constrained generation framework that generates samples directly within such feasible regions while preserving realism. Our method fine-tunes a pretrained generative model to enforce constraints while maintaining generative fidelity. Experimentally, our method exhibits characteristics distinct from existing fine-tuning and training-free constrained baselines, revealing a new compromise between constraint satisfaction and sampling quality.

Translating informal requirements into formal specifications is challenging due to the ambiguity and variability of natural language (NL). This challenge is particularly pronounced when relying on compact (small and medium) language models, which may lack robust knowledge of temporal logic and thus struggle to produce syntactically valid and consistent formal specifications. In this work, we focus on enabling resource-efficient open-weight models (4B--14B parameters) to generate correct linear temporal logic (LTL) specifications from informal requirements. We present LTLGuard, a modular toolchain that combines constrained generation with formal consistency checking to generate conflict-free LTL specifications from informal input. Our method integrates the generative capabilities of model languages with lightweight automated reasoning tools to iteratively refine candidate specifications, understand the origin of the conflicts and thus help in eliminating inconsistencies. We demonstrate the usability and the effectiveness of our approach and perform quantitative evaluation of the resulting framework.

Large language models perform well in short text generation but still struggle with long text generation, particularly under complex constraints. Such tasks involve multiple tightly coupled objectives, including global structural consistency, local semantic coherence, and constraint feasibility, forming a challenging constrained optimization problem. Existing approaches mainly rely on static planning or offline supervision, limiting effective coordination between global and local objectives during generation. To address these challenges, we propose HiFlow, a hierarchical feedback-driven optimization framework for constrained long text generation. HiFlow formulates generation as a two-level optimization process, consisting of a planning layer for global structure and constraint modeling, and a generation layer for conditioned text generation. By incorporating constraint-aware plan screening and closed-loop feedback at both levels, HiFlow enables joint optimization of planning quality and generation behavior, progressively guiding the model toward high-quality, constraint-satisfying outputs. Experiments on multiple backbones confirm HiFlow's effectiveness over baseline methods.

High-stakes synthetic data generation faces a fundamental Quadrilemma: achieving Fidelity to the original distribution, Control over complex logical constraints, Reliability in uncertainty estimation, and Efficiency in computational cost -- simultaneously. State-of-the-art Deep Generative Models (CTGAN, TabDDPM) excel at fidelity but rely on inefficient rejection sampling for continuous range constraints. Conversely, Structural Causal Models offer logical control but struggle with high-dimensional fidelity and complex noise inversion. We introduce JANUS (Joint Ancestral Network for Uncertainty and Synthesis), a framework that unifies these capabilities using a DAG of Bayesian Decision Trees. Our key innovation is Reverse-Topological Back-filling, an algorithm that propagates constraints backwards through the causal graph, achieving 100% constraint satisfaction on feasible constraint sets without rejection sampling. This is paired with an Analytical Uncertainty Decomposition derived from Dirichlet priors, enabling 128x faster uncertainty estimation than Monte Carlo methods. Across 15 datasets and 523 constrained scenarios, JANUS achieves state-of-the-art fidelity (Detection Score 0.497), eliminates mode collapse on imbalanced data, and provides exact handling of complex inter-column constraints (e.g., Salary_offered >= Salary_requested) where baselines fail entirely.

Autoregressive image generation has seen recent improvements with the introduction of chain-of-thought and reinforcement learning. However, current methods merely specify "What" details to depict by rewriting the input prompt, yet fundamentally fail to reason about "How" to structure the overall image. This inherent limitation gives rise to persistent issues, such as spatial ambiguity directly causing unrealistic object overlaps. To bridge this gap, we propose CoR-Painter, a novel framework that pioneers a "How-to-What" paradigm by introducing Constrained Reasoning to guide the autoregressive generation. Specifically, it first deduces "How to draw" by deriving a set of visual constraints from the input prompt, which explicitly govern spatial relationships, key attributes, and compositional rules. These constraints steer the subsequent generation of a detailed description "What to draw", providing a structurally sound and coherent basis for accurate visual synthesis. Additionally, we introduce a Dual-Objective GRPO strategy that specifically optimizes the textual constrained reasoning and visual projection processes to ensure the coherence and quality of the entire generation pipeline. Extensive experiments on T2I-CompBench, GenEval, and WISE demonstrate that our method achieves state-of-the-art performance, with significant improvements in spatial metrics (e.g., +5.41% on T2I-CompBench).

Tabular synthetic data generators are typically trained to match observational distributions, which can yield high conventional utility (e.g., column correlations, predictive accuracy) yet poor preservation of structural relations relevant to causal analysis and out-of-distribution (OOD) reasoning. When the downstream use of synthetic data involves causal reasoning -- estimating treatment effects, evaluating policies, or testing mediation pathways -- merely matching the observational distribution is insufficient: structural fidelity and treatment-mechanism preservation become essential. We propose CausalWrap (CW), a model-agnostic wrapper that injects partial causal knowledge (PCK) -- trusted edges, forbidden edges, and qualitative/monotonic constraints -- into any pretrained base generator (GAN, VAE, or diffusion model), without requiring access to its internals. CW learns a lightweight, differentiable post-hoc correction map applied to samples from the base generator, optimized with causal penalty terms under an augmented-Lagrangian schedule. We provide theoretical results connecting penalty-based optimization to constraint satisfaction and relating approximate factorization to joint distributional control. We validate CW on simulated structural causal models (SCMs) with known ground-truth interventions, semi-synthetic causal benchmarks (IHDP and an ACIC-style suite), and a real-world ICU cohort (MIMIC-IV) with expert-elicited partial graphs. CW improves causal fidelity across diverse base generators -- e.g., reducing average treatment effect (ATE) error by up to 63% on ACIC and lifting ATE agreement from 0.00 to 0.38 on the intensive care unit (ICU) cohort -- while largely retaining conventional utility.

Diffusion models have gained prominence as powerful generative tools for solving inverse problems due to their ability to model complex data distributions. However, existing methods typically rely on complete knowledge of the forward observation process to compute gradients for guided sampling, limiting their applicability in scenarios where such information is unavailable. In this work, we introduce Constrained Particle Seeking (CPS), a novel gradient-free approach that leverages all candidate particle information to actively search for the optimal particle while incorporating constraints aligned with high-density regions of the unconditional prior. Unlike previous methods that passively select promising candidates, CPS reformulates the inverse problem as a constrained optimization task, enabling more flexible and efficient particle seeking. We demonstrate that CPS can effectively solve both image and scientific inverse problems, achieving results comparable to gradient-based methods while significantly outperforming gradient-free alternatives. Code is available at https://github.com/deng-ai-lab/CPS.

Ray tracing has become a standard for accurate radio propagation modeling, but suffers from exponential computational complexity, as the number of candidate paths scales with the number of objects raised to the interaction order. This bottleneck limits its use in large-scale or real-time applications, forcing traditional tools to rely on heuristics that reduce path candidates at the cost of potentially reduced accuracy. To overcome this limitation, we propose a machine-learning-assisted framework that replaces exhaustive path searching with intelligent sampling via Generative Flow Networks. Applying these generative models to this domain presents challenges, particularly sparse rewards due to the rarity of valid paths, which can lead to convergence failures and trivial solutions when evaluating high-order interactions in complex environments. To ensure robust learning and efficient exploration, our framework incorporates three key components. First, an experience replay buffer captures and retains rare valid paths. Second, a uniform exploratory policy improves generalization and prevents overfitting to simple geometries. Third, a physics-based action masking strategy filters out physically impossible paths before the model considers them. Validated on idealized street-canyon scenarios, our model achieves substantial speedups over exhaustive search -- up to $10\times$ faster on GPU and $100\times$ faster on CPU -- while maintaining high coverage accuracy and successfully uncovering complex propagation paths. However, out-of-distribution evaluations on real-world Manhattan street geometries reveal that generalizing to substantially different urban morphologies requires further advancement in model capacity or alternative training strategies. Source code, tests, and a tutorial are available at https://github.com/jeertmans/sampling-paths.

Safety mechanisms for diffusion and flow models have recently been developed along two distinct paths. In robot planning, control barrier functions are employed to guide generative trajectories away from obstacles at every denoising step by explicitly imposing geometric constraints. In parallel, recent data-driven, negative guidance approaches have been shown to suppress harmful content and promote diversity in generated samples. However, they rely on heuristics without clearly stating when safety guidance is actually necessary. In this paper, we first introduce a unified probabilistic framework using a Maximum Mean Discrepancy (MMD) potential for image generation tasks that recasts both Shielded Diffusion and Safe Denoiser as instances of our energy-based negative guidance against unsafe data samples. Furthermore, we leverage control-barrier functions analysis to justify the existence of a critical time window in which negative guidance must be strong; outside of this window, the guidance should decay to zero to ensure safe and high-quality generation. We evaluate our unified framework on several realistic safe generation scenarios, confirming that negative guidance should be applied in the early stages of the denoising process for successful safe generation.

In scientific and engineering domains, modeling high-dimensional complex systems governed by partial differential equations (PDEs) remains challenging in terms of physical consistency and numerical stability. However, existing approaches, such as physics-informed neural networks (PINNs), typically rely on known physical fields or coefficients and enforce physical constraints via external loss functions, which can lead to training instability and make it difficult to handle high-dimensional or unobservable scenarios. To this end, we propose PhysFormer, a generative modeling framework that is self-consistent at both the data and physical levels. PhysFormer leverages a low-dimensional, physically interpretable latent space to learn key physical quantities directly from data without requiring known high-dimensional physical field parameters, and embeds the physical process of radiative flux generation within the network to ensure the physical consistency of the generated spectra. In high-dimensional, degenerate inversion tasks, PhysFormer constrains generation within physical limits and enhances spectral fidelity and inversion stability under varying signal-to-noise ratios (SNRs). More broadly, this approach shifts the physical processes from external loss functions into the generative mechanism itself, providing a physically consistent generative modeling paradigm for complex systems involving unknown or unobservable physical quantities.

Synthetic data generation is an important capability for privacy-preserving data sharing, system benchmarking and test data provisioning. For mixed-type data, existing synthesizers largely target dense, fixed-schema tables, but many modern data systems store and exchange sparse, semi-structured JSON with nested objects, variable-length arrays and optional keys. Applying tabular synthesizers to such data requires flattening records into wide, sparse tables, turning nested structure and arrays into column-layout artifacts. We present ORiGAMi, an autoregressive transformer architecture for modeling and synthesizing semi-structured records without flattening. ORiGAMi serializes JSON records into key, value, and structural tokens, and encodes token positions by their path in the document tree. Grammar and schema constraints enforce syntactically valid JSON and dataset-consistent structure. We evaluate ORiGAMi against VAE, GAN, diffusion, and autoregressive baselines that operate on flattened representations across six datasets ranging from dense tabular benchmarks to large-scale semi-structured collections. Across fidelity, detection, and utility metrics, ORiGAMi achieves the best score in 17 of 18 benchmark comparisons, while maintaining high privacy scores above 96% across all settings. These results establish native record modeling as a strong alternative to tabular synthesis pipelines, preserving structure while achieving state-of-the-art benchmark performance.

Large language models (LLMs) are powerful tools that have found applications beyond human-machine interfaces and chatbots. Beside free-form generation, there has been an interest in constrained generation, a setting where LLMs are constrained to generate well-formed outputs with respect to the language defined by a formal grammar. Although appealing, this setting may be over restrictive for downstream applications. For example, many LLM tasks require the model to reason freely before generating its final response in a specific format. In this work, we introduce suffix-constrained generation, a constrained generation setting in which only the end of the response is constrained by a grammar, a scenario that is not supported by existing constrained generation methods. We introduce several suffix-constrained generation algorithms that are based on greedy search. We experiment on several datasets, and show that our approach allows to guarantee suffix constraints without having a negative impact on results, and even improving them in many settings.

We propose Light-Geometry Interaction (LGI) maps, a novel representation that encodes light-aware occlusion from monocular depth. Unlike ray tracing, which requires full 3D reconstruction, LGI captures essential light-shadow interactions reliably and accurately, computed from off-the-shelf 2.5D depth map predictions. LGI explicitly ties illumination direction to geometry, providing a physics-inspired prior that constrains generative models. Without such prior, these models often produce floating shadows, inconsistent illumination, and implausible shadow geometry. Building on this representation, we propose a unified pipeline for joint shadow generation and relighting - unlike prior methods that treat them as disjoint tasks - capturing the intrinsic coupling of illumination and shadowing essential for modeling indirect effects. By embedding LGI into a bridge-matching generative backbone, we reduce ambiguity and enforce physically consistent light-shadow reasoning. To enable effective training, we curated the first large-scale benchmark dataset for joint shadow and relighting, covering reflections, transparency, and complex interreflections. Experiments show significant gains in realism and consistency across synthetic and real images. LGI thus bridges geometry-inspired rendering with generative modeling, enabling efficient, physically consistent shadow generation and relighting.

Flow-based generative models, such as diffusion models and flow matching models, have achieved remarkable success in learning complex data distributions. However, a critical gap remains for their deployment in safety-critical domains: the lack of formal guarantees that generated samples will satisfy hard constraints. We propose a safety filtering framework that acts as an online shield for any pre-trained generative model. Our key insight is to cooperate with the generative process rather than override it. We define a constricting safety tube that is relaxed at the initial noise distribution and progressively tightens to the target safe set at the final data distribution, mirroring the coarse-to-fine structure of the generative process itself. By characterizing this tube via Control Barrier Functions (CBFs), we synthesize a feedback control input through a convex Quadratic Program (QP) at each sampling step. As the tube is loosest when noise is high and intervention is cheapest in terms of control energy, most constraint enforcement occurs when it least disrupts the model's learned structure. We prove that this mechanism guarantees safe sampling in discrete-time. The minimum-norm control synthesized at each step minimizes the per-step contribution to the KL divergence between the safe and original distributions. Across all experiments, we observe 100% constraint satisfaction. Our framework applies to any pre-trained flow-based sampling scheme requiring no retraining or architectural modifications. We validate the approach across constrained image generation, physically-consistent trajectory sampling, and safe robotic manipulation policies, achieving 100% constraint satisfaction while preserving semantic fidelity.

Point defects affect material properties by altering electronic states and modifying local bonding environments. However, high-throughput first-principles simulations of point defects are costly due to large simulation cells and complex energy landscapes. To this end, we propose a generative framework for simulating point defects, overcoming the limits of costly first-principles simulators. By leveraging a primal-dual algorithm, we introduce a constraint-aware diffusion model which outperforms existing constrained diffusion approaches in this domain. Across six defect configuration settings for Bi2Te3, the proposed approach provides state-of-the-art performance generating physically grounded structures.

For multivariate co-generation in scientific applications, we advocate pairwise block rather than joint modeling of all variables. This design mitigates the computational burden and data imbalance. To this end, we propose an Annealed Co-Generation (ACG) framework that replaces high-dimensional diffusion modeling with a low-dimensional diffusion model, which enables multivariate co-generation by composing pairwise variable generations. We first train an unconditional diffusion model over causal variables that are disentangled into pairs. At inference time, we recover the joint distribution by coupling these pairwise models through shared common variables, enabling coherent multivariate generation without any additional training. By employing a three-stage annealing process-Consensus, Heating, and Cooling-our method enforces consistency across shared common variables and progressively constrains each pairwise data distribution to lie on a learnable manifold, while maintaining high likelihood within each pair. We demonstrate the framework's flexibility and efficacy on two distinct scientific tasks: flow-field completion and antibody generation. All datasets and code will be made publicly available upon publication.

We present a generative modeling framework for synthesizing physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion-based flow generators either ignore physical constraints, impose soft penalties that do not guarantee feasibility, or specialize to fixed geometries, our approach integrates three complementary components: (1) a boundary-conditioned diffusion model operating on velocity fields; (2) a physics-informed training objective incorporating a divergence penalty; and (3) a projection-constrained reverse diffusion process that enforces exact incompressibility through a geometry-aware Helmholtz-Hodge operator. We derive the method as a discrete approximation to constrained Langevin sampling on the manifold of divergence-free vector fields, providing a connection between modern diffusion models and geometric constraint enforcement in incompressible flow spaces. Experiments on analytic Navier-Stokes data and obstacle-bounded flow configurations demonstrate significantly improved divergence, spectral accuracy, vorticity statistics, and boundary consistency relative to unconstrained, projection-only, and penalty-only baselines. Our formulation unifies soft and hard physical structure within diffusion models and provides a foundation for generative modeling of incompressible fields in robotics, graphics, and scientific computing.

Counterfactual explanations (CFs) provide human-interpretable insights into model's predictions by identifying minimal changes to input features that would alter the model's output. However, existing methods struggle to generate multiple high-quality explanations that (1) affect only a small portion of the features, (2) can be applied to tabular data with heterogeneous features, and (3) are consistent with the user-defined constraints. We propose CounterFlowNet, a generative approach that formulates CF generation as sequential feature modification using conditional Generative Flow Networks (GFlowNet). CounterFlowNet is trained to sample CFs proportionally to a user-specified reward function that can encode key CF desiderata: validity, sparsity, proximity and plausibility, encouraging high-quality explanations. The sequential formulation yields highly sparse edits, while a unified action space seamlessly supports continuous and categorical features. Moreover, actionability constraints, such as immutability and monotonicity of features, can be enforced at inference time via action masking, without retraining. Experiments on eight datasets under two evaluation protocols demonstrate that CounterFlowNet achieves superior trade-offs between validity, sparsity, plausibility, and diversity with full satisfaction of the given constraints.