Constrained Generative Modeling

Multimodal Sewing pattern generation aims to infer executable sewing patterns from design cues such as sketches and textual descriptions. As an interpretable and simulation-compatible representation, sewing patterns are particularly valuable for digital garment creation. However, existing methods often model garment specifications as flat long sequences, which entangles garment structure with detailed parameters and leads to redundant components, inaccurate local details, and poor simulation compatibility. In this paper, we present GarmentWeaver, a schema-aware framework for multimodal Sewing pattern generation. GarmentWeaver constructs compact hierarchical targets by activating garment-relevant structural branches and predicts executable Sewing patterns in a structured manner. Specifically, we introduce a schema-aware target construction strategy, build the generator on top of a pretrained vision-language model for multimodal garment understanding, and impose feasibility-aware regularization to encourage structurally valid and simulation-compatible outputs. Extensive experiments show that GarmentWeaver produces more accurate and more executable sewing patterns than strong baselines, while also yielding better simulation results. These findings demonstrate the effectiveness of schema-aware structured generation for reliable multimodal Sewing pattern prediction.

Large language models can generate fluent clinical case vignettes, but fluency alone does not ensure fidelity to a specifiable clinical structure. We introduce FORMA, a theory-grounded framework that compiles a cognitive model of a disorder into a directed weighted graph, samples a person-specific configuration of that graph, and validates whether the generated vignette preserves the specified components and causal links. We instantiate FORMA on Posttraumatic Stress Disorder using the Ehlers and Clark cognitive model, generating 16,500 vignettes across 500 personas, 11 generation models, and three ablation conditions. Evaluation combines an external edge-recovery probe, two clinical experts, a scaled LLM judge, and a clinician user study with 100 licensed practitioners. The cognitive graph is recoverable from full-condition vignettes (MCC = +0.41, AUC = 0.70) but not from zero-shot generation (MCC = +0.01, AUC = 0.50). Experts rate full vignettes substantially higher than zero-shot alternatives, and clinicians perceive them to be human-written 85% of the time, compared with 22% for zero-shot. FORMA also reduces demographic disparity in perceived quality by 1.5-7x. These results show that cognitive formulation can serve as an auditable specification for scalable synthetic clinical text generation. A repository with the data and code is available online: https://github.com/Amit-Oren/FORMA.

Generative AI models are primarily designed to imitate the data distribution, an objective that neither corrects diversity lost by a learned generator nor defines how generation should extend beyond the diversity of the data itself. We introduce Imaginative Generative AI (IGA), a framework that makes diversity part of the target-distribution design problem: among distributions close to a reference, IGA selects one whose spectral diversity reaches a prescribed level. Diversity is measured by the von Neumann entropy of the generated distribution's kernel covariance operator in a fixed representation space, providing a reference-free representation-guided measure of how broadly probability mass occupies embedding directions. The spectral entropy of the population data distribution defines an Entropy Wall. Below the wall, IGA performs diversity repair, recovering variation that a learned generator has lost while remaining within the diversity level of the data. Beyond the wall, the data distribution itself becomes infeasible, and IGA deliberately departs from it to produce distributions with greater representation-relative spectral diversity, an operational notion of imaginative generation. These regimes form a single regularization path from imitation to imagination and define an i.i.d. target distribution at each prescribed diversity level. We develop the theory of this entropy-constrained projection and show that, under a KL anchor to a pretrained generator, the optimum satisfies a self-consistent exponential-tilt relation. This characterization leads to IGA Guidance, a retraining-free inference-time method for score-based and diffusion models, including DDPM and DDIM samplers. Experiments on synthetic and vision benchmarks demonstrate diversity repair below the Entropy Wall and controlled spectral extrapolation beyond it.

Optimizing 3D shapes within the latent spaces of deep generative models is fundamental to computer assisted engineering, yet remains prone to a critical failure mode we term manifold drift: the tendency of gradient-based optimization to move latent vectors away from the manifold of valid shapes. This problem is exacerbated in state-of-the-art 3D shape generative models that operate in increasingly high-dimensional latent spaces where valid shapes occupy a vanishingly small fraction of the full space. Existing mitigation strategies, including latent regularization and flow-matching approaches, either sacrifice expressiveness, demand a difficult trade-off between objective guidance and generative fidelity that remains prone to manifold drift, or are computationally infeasible to scale to modern, large-capacity 3D shape models. We introduce a novel optimizer-corrector framework that alternates between gradient steps for objective minimization and guided flow matching to drive the latent state back to the valid shape manifold. By decoupling objective minimization from flow-based correction, optimizing freely and correcting strictly, this alternating design avoids inherent trade-offs, preserving geometric validity without sacrificing expressiveness while remaining computationally feasible on modern 3D shape models. We demonstrate its effectiveness across generative priors of varying complexity, from simple vector latent spaces to large-scale architectures across a variety of downstream optimization tasks, including aerodynamic drag reduction and object compliance optimization.

Synthetic 3D scene generation is increasingly used as a data source for computer vision and embodied AI, but existing generators often optimize perceptual realism without reliably satisfying task-critical functional constraints. This mismatch limits the usefulness of synthetic data for downstream training, where accessibility, traversability, and spatial rule compliance are often essential. We present iARCS, an iterative agentic reinforcement learning framework that adapts a pretrained scene generator to naturallanguage task requirements. iARCS uses a two-phase strategy: universal-reward pretraining to improve physical plausibility and layout quality, followed by task-specific finetuning with LLM-generated reward programs that are iteratively refined from training feedback. Experiments show improved constraint fidelity on walkability, reachability, and clearance-focused tasks, effective task-specific constraint optimization, and competitive scene diversity. We further show that data generated by iARCS improves a base generator, supporting its value as a practical synthetic data generation tool rather than only a controllable scene editing method.

Incentivized advertising allocates monetary or virtual rewards to drive user engagement, where a key challenge is optimizing continuous incentive magnitudes under strict global constraints. This problem is complicated by high-frequency interactions, delayed feedback, and non-Markovian user dynamics such as fatigue, which limit the effectiveness of existing uplift modeling and constrained reinforcement learning approaches. To address these challenges, we propose GOAL, a constraint-aware generative framework that formulates incentive allocation as a conditional sequence generation problem. GOAL directly generates incentive magnitudes conditioned on user histories and system-level global pressure, and integrates a hierarchical causal state encoder to capture both local behavioral dynamics and long-range dependencies. To enable flexible constraint control, we introduce \textbf{S}afe \textbf{C}onstrained \textbf{P}olicy \textbf{O}ptimization (SCPO), which learns a single generative policy that generalizes across a spectrum of ROI constraints without retraining. Experiments on large-scale real-world data and a synthetic fatigue-aware environment show that GOAL improves long-term revenue and user retention while substantially reducing ROI violation rates compared to strong baselines.

Synthetic power-grid scenarios are essential for planning, resilience assessment, contingency analysis, and data-driven power-system applications. Recent synthetic grid generation methods have improved structural realism and operational feasibility by incorporating engineering knowledge through post-generation validation, optimization, or physics-aware generation. However, generated scenarios may still exhibit low AC feasibility and robustness, limiting their practical value for downstream power-system studies. This paper proposes a feasibility-aware distribution-learning framework that learns the AC-operable joint distribution of network topology, branch electrical parameters, and time-varying load profiles. Instead of enforcing feasibility after generation, the proposed framework incorporates AC power-flow convergence and operational constraints into hierarchical diffusion-based distribution learning. This enables the generator itself to produce operationally feasible grid scenarios through efficient diffusion sampling. The hierarchical architecture decomposes the high-dimensional generation task into three engineering-motivated stages: topology and bus-attribute generation, branch-parameter generation conditioned on the generated structure, and load-profile generation conditioned on both network structure and electrical characteristics. Experiments on benchmark systems demonstrate that the proposed framework significantly improves operational feasibility and contingency robustness while maintaining strong statistical fidelity and eliminating optimization-based post-processing.

In demanding professional environments and meeting review scenarios, lengthy text often imposes a high cognitive load. To facilitate efficient information communication, transforming verbose text into logically clear diagrams is essential. Scalable Vector Graphics (SVG) provide an effective representation for this purpose due to their editability and resolution independence. However, current research on Text-to-SVG generation remains hindered by three major challenges: (1) the scarcity of datasets for complex, logic-rich diagrams; (2) the absence of explicit layout priors, which leads to chaotic spatial arrangements; and (3) the lack of fine-grained visual feedback to validate rendered outputs and correct aesthetic defects. To address these challenges, at the data level, we introduce DocMeetSVG-100K, a large-scale SVG dataset tailored for document authoring and meeting review scenarios. At the model level, we propose GVR-Coder, a novel framework designed to generate high-quality logical diagrams from lengthy professional texts. Specifically, we adopt a curriculum-driven rejection sampling fine-tuning to progressively enhance the model's capability in modeling complex structures, while explicitly incorporating layout constraint knowledge during training. In addition, we introduce reinforcement learning from dual rendering feedback, a mechanism that provides implicit feedback through reward signals to jointly optimize structural complexity and visual aesthetics. Furthermore, we design a generate-verify-repair agent loop, which improves generation quality through explicit, fine-grained feedback and targeted refinement. Extensive experiments demonstrate that GVR-Coder outperforms competitive baselines and reliably produces logically coherent and visually appealing diagrams. Code and data are available at https://github.com/CurryaNa/GVR-Coder.

Variational autoencoders (VAEs) transform high-dimensional, often noisy data into a compact latent representation, making downstream optimization more tractable. Three challenges persist in VAE-based constrained optimization: (i) sampling effectively within the latent space, (ii) identifying the active decision variables that actually influence the objective and constraints, and (iii) enforcing constraints without destabilizing training. We propose a Multi-stage Constrained Optimization Framework (MCOF). First, an entropy-constrained VAE (EC-VAE) coupled with a feature selector embeds objective and constraint information into a designated subset of latent variables, so that optimization proceeds over a low-dimensional subspace while the remaining coordinates supply solution diversity. Second, a Uniform Transformation (UT) module applies a per-dimension probability integral transform, replacing the irregular aggregate posterior with a uniform distribution over a bounded box and mitigating posterior collapse and Gaussian mixture bias. Third, a constraint-priority filter method (CPFM) solves the resulting surrogate problem by alternating violation-reduction and objective-reduction steps under a filter acceptance test, returning solutions that are feasible for the learned surrogate to a specified tolerance without requiring multiplier estimation. Finally, unselected latent coordinates are resampled to generate diverse decodings of a single optimized solution. We validate MCOF on a synthetic problem, where we ablate each stage and recover the analytic optimum, and on a ZINC250k drug design task, where the generated molecules satisfy the imposed constraints and are entirely novel relative to the training set.

AC optimal power flow determines the minimum-cost generation dispatch under nonlinear power balance constraints and is solved thousands of times daily in electricity market operations. Learning a direct mapping from load conditions to OPF solutions can accelerate this computation, yet with deepening renewable penetration, a single optimal dispatch is no longer sufficient. Operators require a characterization of the distribution of feasible near-optimal solutions for risk quantification, sensitivity analysis, and multi-objective trade-off assessment. Supervised neural networks provide fast point predictions but cannot capture this conditional distribution. Diffusion-based generative models can sample diverse solutions in principle, yet existing methods operating in the raw state space exhibit degraded solution quality and fail to scale beyond medium-sized systems. We identify the root cause as the conflation of two distinct tasks within a single model. Compressing the high-dimensional OPF solution manifold is one task, and learning the conditional mapping from loads to that manifold is another. This paper presents FMOPF, a framework that resolves this conflation by decoupling compression from generation through latent flow matching and by explicitly modeling load-state coupling through a Constraint-Aware Interaction Prior Network. Experiments on four IEEE test systems demonstrate that FMOPF provides the most effective Newton-Raphson warm starts, achieves the lowest tail risk among generative methods, and is the first such method to scale to systems with several hundred buses while preserving full feasibility. Ablation studies confirm that the latent generation pipeline is a necessary condition for physical feasibility and that the interaction prior functions as a late-stage tail-risk controller.

Data-driven generative models can extend partially observed simulation trajectories into ensembles of alternative future scenarios. However, consistency with a learned trajectory distribution does not ensure that generated continuations satisfy the structural conditions of the simulated system. This paper presents a method for learned scenario continuation and post-generation structural validation in dynamic graph simulations. A conditional diffusion model generates future graph-state trajectories from partial histories, while an external symbolic layer evaluates each continuation using a Boolean admissibility indicator and a continuous violation score. This information supports hard filtering and soft weighting, with optional projection considered as a deterministic repair baseline. The method is evaluated on two controlled dynamic-graph regimes sharing the same continuation architecture and training protocol but differing in dimensionality and dependency complexity. Evaluation considers invalid probability mass, scenario retention, effective sample size, diversity, robustness, and calibration. In the compact positive-control regime, unconstrained invalid mass is 0.002996, indicating near-complete overlap between the learned and admissible scenario spaces. In the medium-complexity regime, invalid mass rises to 0.155929. Hard filtering removes all invalid scenarios while retaining 84.4% of generated continuations. Soft weighting preserves an effective sample size ratio of 0.998764 but reduces invalid mass only to 0.148807. These results show that learned-distribution support, structural admissibility, and probability calibration can diverge and should therefore be assessed separately in learned simulation-scenario generation and management.

Score Entropy Discrete Diffusion (SEDD) parameterizes discrete reverse processes with unconstrained positive score ratios. While positivity guarantees nonnegative reverse jump rates, it does not ensure Bayes realizability: ratios at a noisy state need not be jointly induced by any clean-token posterior under the forward kernel. The score-entropy loss has the correct population optimum but does not enforce this constraint away from it. In a trained pure-uniform SEDD checkpoint, roughly one quarter of complete score vectors violate the coordinate box, while more than half lie inside it yet remain materially incompatible with any valid posterior. Such violations can produce negative pre-normalization weights in finite-step sampling. Projecting raw scores onto the bridge polytope removes all observed negative weights and improves external generative PPL from $203.6$ to $175.1$ without changing the sampler. We introduce \emph{mean-to-score} (M2S), which predicts a clean-token posterior mean and converts it to the score through an exact kernel-dependent linear map. The construction applies to any known coordinate-wise continuous-time Markov chain (CTMC) satisfying a mild support condition. For uniform corruption, it maps the probability simplex onto the bridge polytope; for absorbing-mask corruption, the resulting objective recovers MD4 exactly. In a controlled 28.4M-parameter CIFAR-10 comparison, M2S lowers test BPD from $3.173$ to $3.129$ and FID-50k from $\CifarSEDDFID$ to $\CifarMtwoSFID$. A 170M-parameter M2S model trained on about 262B OpenWebText token slots outperforms the evaluated pure-uniform SEDD, GIDD, and Neural CTMC checkpoints at every tested sampling budget, reaching generative PPL $143.3$ at 128 steps versus $183.6$ for the strongest pure-uniform baseline.

Generating realistic interior furniture layouts that strictly adhere to architectural constraints (e.g., walls, doors, and windows) remains a fundamental challenge in automated spatial design. Existing approaches, primarily based on one-shot generation using diffusion models or Large Language Models (LLMs), lack explicit mechanisms for intermediate geometric constraint verification, often resulting in structural collisions and functionally infeasible arrangements under complex room constraints. To address these challenges, we propose Agentic Designer, a progressive, multi-agent framework that formulates structure-aware interior layout generation as an iterative and constraint-verified decision process. By decomposing layout synthesis into modular stages of proposal, verification, and adjustment, the framework coordinates three specialized agents, a Generator, an Evaluator, and a Refiner, through a Progressive Consensus Mechanism. This mechanism enforces stepwise geometric validation and correction before each placement is committed, thereby preventing error accumulation. To facilitate this structure-aware paradigm and standardize evaluation, we establish InStruct, a comprehensive benchmark that integrates a dataset comprising over 18,000 high-quality, parametrically annotated samples with a novel suite of structure-centric metrics. Extensive quantitative evaluations, qualitative analyses, and user studies show that Agentic Designer significantly outperforms state-of-the-art methods, demonstrating substantial improvements in strict structural adherence and functional design coherence.

Current 3D generative models mostly produce a final surface: a visually strong but largely opaque mesh. Interactive 3D worlds need more than a surface. They need named parts, an assembly hierarchy, measurable constraints, local edit handles, and joints for articulation. We present Nova3D, a system that generates 3D assets as executable Blender source code; the compiled mesh, a binary glTF (GLB), is treated as the artifact, not the asset. Because the output is a program, semantic handles exist at generation time rather than being recovered afterward by segmentation or rigging. We evaluate on Nova3D-Bench, a frozen, spec-grounded benchmark of 54 items across six domains and three difficulty levels with text and image inputs, against eleven baselines in four families (mesh-native, part-structured, code-native, and CAD) plus a same-LLM ablation. Nova3D produces an executable program and a valid artifact for 54/54 items. Every asset exposes named parts organized in a parent-child assembly tree; no mesh-native, CAD, or segmentation baseline exposes either. It satisfies 51/52 prompt-stated numeric and count constraints (best baseline: 11/52), passes 14/18 blinded local edits with locality preserved in 18/18, and articulates 59 joints across 12 assets at 98.3% geometric validity, where every baseline exposes zero native joints. Its geometry is competitive: it wins the structured domains in a pairwise shape-quality tournament and is second only to the strongest mesh-native model, while conceding texture realism to baked-PBR systems. The central result is representational: code-native generation turns a generated 3D object from an opaque surface into a programmable asset that downstream systems can inspect, measure, edit, and animate.

We introduce Signed Rectified Flow (Signed RF), a generalization of Rectified Flow that targets the signed measure $π^{sign} = (1+α)π^+ - απ^-$, where $α>0$, $π^+$ is the distribution to promote, and $π^-$ is the distribution to suppress. Although direct sampling from a signed measure is not well-defined, Signed RF induces a valid generative process that concentrates probability in regions where the signed measure is positive while provably excluding regions dominated by its negative component. It therefore provides a principled framework for incorporating negative information and exclusion constraints into generative modeling. We analyze the signed continuity equation underlying Signed RF and use a charged-particle interpretation to explain how negative mass forms exclusion barriers. This theory further motivates practical adaptive guidance algorithms. Across several applications, Signed RF improves the fidelity-diversity trade-off on ImageNet, reduces nearest-neighbor similarity in anti-memorization experiments, and reduces nudity induced by adversarial prompts in Stable Diffusion 3.5 while preserving CLIP and aesthetic scores.

Structure-based drug design (SBDD) leverages the 3D structure of protein targets, often complemented by other spatial constraints, to generate candidate binding molecules. While diffusion models have dominated as a leading paradigm for high-quality 3D molecule generation, LLM-based methods are rapidly emerging in molecular design and have shown competitive performance in pocket-conditioned molecular generation. However, their ability to reason about physics and 3D spatial environments is largely underexplored. In this work, we systematically analyze whether current general-purpose LLMs are capable of navigating complex 3D constraints compared to established baselines such as specialized diffusion models. We consider 3D ligand generation conditioned on protein pockets together with ligand- and interaction-derived spatial constraints, including anchor fragments, pharmacophore points, and mandatory pocket-ligand interactions. To enable this evaluation, we introduce 3D-Fit - a token-efficient benchmarking strategy for assessing LLM performance on multi-conditioned spatial molecule generation. Our findings reveal a clear pattern in LLM spatial capabilities: while they still lag behind state-of-the-art approaches, they are promising and can handle multiple spatial constraints simultaneously, enabling scaling to heterogeneous setups.

Generative artificial intelligence is rapidly transforming materials design by enabling de novo exploration of immense chemical spaces. Yet a large proportion of AI-generated compositions remain implausible, violating established chemical principles, which limits the reliability and interpretability of generative materials design. Here, we introduce a chemical validity operator that recasts heuristic chemical rules as a configurable algorithmic prior for evaluating and guiding generative materials discovery. Built on the open-source SMACT package, a data-informed oxidation-state model exposes tunable thresholds, allowing users to interpolate continuously between permissive and conservative chemical constraints, while supporting both exploratory and conservative materials-design workflows. Benchmarking six state-of-the-art generative models for inorganic crystals shows that most reproduce stoichiometry but under-represent realistic oxidation-state combinations, and that filtering removes compositions reliant on rarely observed oxidation states while preserving low-energy compounds near the convex hull. Beyond screening, the same operator can also serve as a reinforcement-learning reward, steering a latent diffusion model towards chemically grounded compositions. By encoding chemical heuristics and observations, this work establishes a foundation for oxidation-state-aware generative models.

In this work we demonstrate that a single transformer-based generative model can capture Standard Model structure spanning five decades of invariant mass, from the sub-GeV regime to the TeV continuum, a range that no single Monte Carlo sample covers. To achieve this we design \textsc{ShellFlow}, a Riemannian conditional flow matching model that, given the recorded event composition, generates each particle on its on-shell manifold. Its only physics priors are the on-shell condition and the invariant-mass formula. The model is trained on $\sim 10^{9}$ real $pp$ collision events from the ATLAS Open Data 13~TeV release and told nothing else. From a single training run, the model learns to reproduce all of the following: intra-particle kinematics, the dilepton resonances ($J/ψ$, $Υ$, $Z$) at their PDG positions, the leptonic Weinberg angle, the $W$ and top-quark masses, and inter-particle correlations that enter no training objective. A substantial fraction of the Standard Model is thus learnable directly from recorded collision data.

In recent years Flow Matching has become a prominent method for generative modeling robot motion generation. In its generic form Flow Matching is an ODE-based neural sampler that is trained by regressing empirical flow fields associated with motion samples as data. However, in robot motion generation we often have additional constraints that might not be present in the collected data. The majority of current approaches train the flow on the available data and use inference-time guidance to enforce task-specific constraints. To address this mismatch, we propose \textbf{ConFlow}, a constraint-guided flow matching framework that incorporates constraint information directly into the training objective via differentiable barrier or cost functions. To address design specifications such as smoothness and boundary conditions, we propose replacing the standard Gaussian source distribution used in flow matching training with a conditional Gaussian Process. Our approach also uses infeasible demonstrations as negative supervision, improving constraint satisfaction without requiring additional expert data. Experiments on a two-robot navigation task demonstrate that ConFlow achieves lower collision rates and higher trajectory quality than standard flow matching baselines, with or without inference-time guidance. These results validate training-time constraint integration as an effective approach to closing the training--inference gap in generative motion models.

Constrained generative models aim to produce samples that satisfy complex feasibility constraints while remaining faithful to the data distribution. Existing constrained generation methods typically enforce constraints either through training-time optimization or sampling-time correction. Training-time optimization approaches optimize on states induced by the training distribution, which can differ substantially from those encountered during sampling. Sampling-time correction methods instead modify the sampling process at inference, introducing distribution shift and requiring expensive tuning, particularly for few-step sampling. We propose a fine-tuning framework that incorporates constraint guidance obtained through online rollout into the training process, which aligns training with sampling by differentiating through the fixed noise schedule used to numerically integrate the denoising process. This exposes the model to violations that arise along the denoising trajectory and aligns diffusion learning with the sampling process. Experiments across multiple tasks show that our method improves constraint satisfaction while maintaining competitive sampling quality compared to prior methods.

UV seam placement is a critical yet labor-intensive step in 3D content creation, requiring artists to balance chart shape, seam concealment, and alignment with semantic and geometric features. Existing automatic methods are primarily based on per-object optimization, relying on handcrafted objectives to avoid distortion or on proxies from pretrained models to inject semantic information. However, these strategies are not always well aligned with seams used in industrial production pipelines, often resulting in layouts that deviate from artist-preferred seam patterns and practical production requirements. To address these limitations, we propose SeamGen, a generative model for UV seam generation that aligns with artist preferences and production requirements. Instead of depending on manually designed objectives and constraints, SeamGen learns the distribution of per-edge seam labels from a large corpus of existing seam layouts using a flow-matching generative model. A key challenge is that typical Transformer architectures used in flow matching models are designed for sequential representations, such as point clouds, and cannot naturally account for mesh topology. To enable mesh-native learning, we design a Mesh Transformer backbone that interleaves local graph attention over mesh edges with global self-attention across vertices, capturing both fine-grained geometric cues and long-range topological coherence. To further improve inference-time controllability and quality, we exploit the training-free inpainting capability of flow models for both localized seam refinement and constraint-guided seam generation. Extensive experiments show that by learning priors from professional seam layout data, SeamGen produces UV layouts that better align with artist-authored preferences and achieve superior perceptual quality compared with distortion-based and semantic-proxy baselines.

Generative models often represent signals as dense grids of amplitudes, blurring sharp transients that are crucial for the correctness of physical signals. We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities, rooted in the classical pole-residue representation of meromorphic functions. We learn a latent space of physically constrained, per-signal singularity configurations to solve an inverse problem from degraded or partial observations. The framework has three key properties: interpretability, in which each generated singularity configuration corresponds to a set of physical parameters; structural stability, which mitigates Gibbs artifacts at discontinuities; and resolution-free output reconstruction on arbitrary grids without retraining or interpolation. Our framework employs a transformer-based diffusion model that directly predicts samples at complex-plane singularity coordinates, subject to geometric constraints during sampling. As a controlled test case for sharp-feature recovery, we evaluate our framework on 1D Burgers shocks, where each shock is represented by 32 predicted singularities (an $8\times$ reduction versus a 1024-point grid signal). Our framework preserves signal structure ($\text{TV ratio} \approx 1$) under unseen test-time observation noise, achieves a $4.2\times$ lower reconstruction error in zero-shot sub-resolution generalization than a grid-based baseline, and recovers physical parameters to $10^{-4}$ absolute error in-distribution. These results suggest that singularity-based representations may provide a practical foundation for other transient-dominated signals such as speech and biomedical signals, with potential extension to higher-dimensional domains.

Constrained decoding is essential for serving LLMs, ensuring that generated outputs follow specific structures such as JSON schema-formatted function calls. Existing systems are designed for autoregressive models and assume left-to-right generation, masking out invalid next tokens at each step. Diffusion language models, however, break this assumption: they sample multiple positions simultaneously from a fully-factorized mean-field distribution at each denoising step. In this paper, we present an exact and tractable algorithm for sampling from the constrained mean-field posterior under any constraint expressible as a finite automaton. Viewing finite automata as graphical models, we obtain tractable representations of the constrained distribution that enable efficient inference. The approach guarantees constraint satisfaction by construction, supports both greedy and sampling-based decoding, and is compatible with parallel and block-wise decoding under arbitrary remasking schedules. Applying depth-reduction techniques from arithmetic circuit theory, we further reduce sampling depth from linear to logarithmic in the sequence length. Empirical evaluations on Dream-7B and LLaDA-8B show substantial accuracy gains across various tasks including function calling (xLAM, BFCL), planning (Sudoku, Countdown), text-to-SQL (Spider), and math reasoning (GSM-Symbolic), with little inference overhead relative to unconstrained decoding. For example, on BFCL-Live, our approach improves Dream-7B's greedy decoding accuracy from 63.9% to 71.5%, and stochastic sampling accuracy from 22.3% to 69.0%, where the unconstrained baseline collapses, with under 5% wall-clock overhead.

Flow matching is a powerful tool for generative modeling, but emerging applications in robotics, planning, and control require inference-time constraints on generated outputs. Such constraints are often complex and highly nonlinear. As a result, methods designed for linear constraints like image inpainting are rarely sufficient, and projection or optimization-based alternatives can be prohibitively expensive. In this paper, we introduce Lagrangian Dual Flows, a new family of constrained generation techniques based on Lagrangian dual dynamics. By flowing a dual co-state alongside generated samples, we can guarantee nonlinear constraint satisfaction without expensive optimization subproblems, pseudoinverses, or projection steps during the denoising process. The resulting constrained generation algorithms are simple, effective, and open new theoretical connections between flow matching and primal-dual methods in numerical optimization.

While diffusion models have revolutionized image synthesis, their application to real-world inverse problems is often hampered by the need for massive datasets and the difficulty of imposing strict physical constraints. In this work, we introduce \textbf{SE-UNet} (Singular Equivariant UNet), a framework designed to solve ill-posed imaging tasks without extensive pre-training. By treating generation as an optimization problem constrained by geometric equivariance ($D_4$ group) and singular value gating, SE-UNet effectively standardizes the solution space. We demonstrate that these strong inductive biases allow for state-of-the-art zero-shot inpainting results (80% missing pixels) on CIFAR-10. Our method surpasses Deep Image Prior (DIP) baselines by over 4 dB in PSNR and exhibits a characteristic "singular snap" convergence -- rapidly locking into the signal manifold. SE-UNet thus offers a data-efficient pathway for constrained generation, aligning with the ReALM-GEN goal of bridging theoretical priors with practical deployment.

Training-free controllable motion generation has attracted growing interest for enabling flexible constraint enforcement without constraint-specific training. However, existing training-free methods require constraints to be continuous objective-based with differentiable losses, while many real-world requirements are criterion-based and provide only discontinuous, sparse, or even black-box feedback. In this paper, we propose Motion-Inference-as-Control (MIC), the first training-free motion generation framework that handles both continuous objective-based and criterion-based motion constraints under a shared mechanism. The key idea is to cast diffusion-based motion generation as a stochastic control problem. This perspective not only provides principled and practically effective step-wise control laws that support criterion-based constraints without requiring differentiability and naturally accommodate objective-based constraints as a special case, but also motivates a control-oriented constraint coordination mechanism that adaptively balances and reconciles motion constraints during generation. Experiments across diverse constraint settings demonstrate the effectiveness of our framework.

Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDE couplings induce in the projection subproblems -- namely, block-sparse Jacobian and KKT systems -- exposing this structure using ExaModels.jl and solving the resulting sparse nonlinear programs with MadNLP.jl and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and two-dimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.

Diffusion models have emerged as a dominant paradigm in generative modeling, enabling high-fidelity sampling from complex data distributions. Despite impressive capabilities, controlling diffusion models to produce outputs aligned with user intent remains an open challenge, especially when balancing global coherence with local precision. Existing control mechanisms vary in the granularity of their conditioning signals. For example, textual prompts guide generation globally through high-level semantics, while ControlNet-like approaches secure precise local structure via dense conditions. In this work, we introduce Histogram-constrained Image Generation (HIG), a novel control mechanism that falls into the middle ground of control granularity. Our framework enforces user-specified distributional constraints (e.g., color histograms or latent token distributions) during the generation process with exact precision. We model such control as an optimal transport (OT) problem and apply explicit guidance transformations during sampling, thereby driving the diffusion trajectory to align with the desired histogram. We demonstrate the versatility of HIG across diverse applications, including constrained generation via color/latent histograms and high-capacity information embedding through histogram-level encoding. Our findings underscore the promise of distributional control, a flexible and interpretable control scheme that is fully compatible with existing control mechanisms, diversifying the hybrid strategies for controllable image generation. Our project page is available at: https://maps-research.github.io/hig/.

Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-$N$ can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.

LLMs and agents can generate web scrapers from natural-language requirements, but direct generation remains unreliable because of dependency errors, broken selectors, schema mismatches, and heterogeneous page structures. We propose a constrained, verifiable agent framework that shifts LLM output from free-form code to typed JSON collector configurations, combining a six-type collector taxonomy, template and utility-function constraints, static Airflow DAG execution, rule-based quality checking, and structured feedback correction. Experiments on 138 tasks show that the taxonomy supports description-based requirement typing, while confirming that stable instantiation requires completing source, field, and execution constraints beyond the initial description. On 80 independently source-verified tasks, the framework runs with zero execution-stage LLM tokens and the lowest average wall-clock time, trading moderate one-shot quality for a reusable, deterministic, and verifiable execution path suited to repeated scheduled collection. These results position the framework as a reusable, low-cost, and verifiable execution path for repeated open-web data collection.