Conditional Flow Matching

Data assimilation estimates a dynamical state from partial and noisy observations. Classical ensemble filters are efficient but restrict analysis updates through finite sample covariance and affine Gaussian distribution. We introduce the Flow Ensemble Filter (FlowEF), which uses conditional flow matching to transport the forecast ensemble from a classical baseline filter to an analysis ensemble. FlowEF uses a localized Gaussian source during training, transports forecast ensemble members from a baseline filter at deployment, and conditions its velocity field on ensembles from that baseline filter and the observation. The proposed model therefore learns a nonlinear update while mapping each baseline ensemble independently. For sparsely observed dynamical systems, FlowEF improves both deterministic and probabilistic metrics over all four classical ensemble filters. It also achieves the best performance among the state-of-the-art generative data assimilation models.

Reward fine-tuning aims to update a pre-trained flow-based generative model to improve the downstream reward of its generated samples. Existing methods typically formulate this problem as sampling from a reward-tilted distribution, the solution to a KL-regularized reward-maximization problem. Here, we introduce an optimal transport regularizer built directly from the pre-trained drift. Unlike KL reward tilting, the resulting objective transports individual samples toward higher reward rather than reweighting the base distribution. We show that the resulting problem is equivalent to a deterministic optimal control problem on the flow. Given a pre-trained flow map, this equivalence yields a simulation-free reinforcement learning algorithm for fine-tuning generative flows. We call the resulting framework Wasserstein-Tilted Flow Maps (WTF), the first end-to-end fine-tuning recipe native to flow maps. The output is a fine-tuned flow map that retains strong reward-aligned performance at few-step inference budgets without post-hoc distillation. Experiments on ImageNet-256 and text-to-image show that WTF achieves higher reward with comparable or higher diversity than baselines, while requiring up to $280\times$ less training compute. More broadly, we argue that accelerated samplers such as flow maps are essential infrastructure for efficient post-training, and that the dominant KL-regularized formulation is only one of many choices worth revisiting.

This work studies prediction parameterization for stochastic generative dynamics in diffusion models. Existing velocity-based generative models provide the simplicity of learning a single transport field, but their standard formulation is deterministic, whereas stochastic extensions generally require additional score information or an intermediate velocity-to-score reconstruction. To retain single-field prediction while directly supporting stochastic reverse dynamics, this paper introduces Mean Velocity Matching (MVM). MVM constructs a Gaussian perturbation process for which the conditional expectation of a restoration-oriented velocity, $(x_0-x_t)/t$, directly forms the reverse-SDE drift. Consequently, a single learned field is sufficient to parameterize the stochastic reverse process without separately estimating or reconstructing the score. Because direct regression of this velocity becomes unbounded near $t=0$, MVM further introduces a $\sqrt{t}$-scaled parameterization that preserves the reverse dynamics while yielding a bounded training target. The same learned field also induces a deterministic probability-flow ODE, enabling stochastic and deterministic sampling to be studied within a unified formulation. Experiments with Transformer-based generative models achieve an FID of $\MVMImageNetThirtyTwoFID$ at \MVMImageNetThirtyTwoNFE\ NFE on ImageNet $32\times32$ and $\MVMImageNetTwoFiftySixFID$ at \MVMImageNetTwoFiftySixNFE\ NFE on ImageNet $256\times256$. Controlled SDE--ODE comparisons further show that the ODE performs better under very low NFE, whereas the stochastic reverse process achieves lower FID when sufficient function evaluations are available. These results demonstrate that MVM provides a direct single-field parameterization of stochastic reverse dynamics while maintaining competitive generation quality.

Bayesian filtering provides a principled framework for online state estimation under uncertainty, yet its application to systems with high-dimensional states and complicated posterior distributions remains challenging. Recent generative models, such as flow matching, have shown potential in Bayesian filtering. However, they still rely on particle-based representations of the posterior, which lose the rich information of the full distribution, or tackle a trajectory-level inverse problem that conflicts with the recursive structure of Bayesian filtering. To address this, we propose a new perspective of directly encoding the evolving distribution into flow matching model weights, namely, the Belief Flow Filter (BFF). It is a generative filtering framework that updates model weights via gradient descent at test time to track the posterior evolution. Thereby, BFF bypasses the scalability issue of particle representations or the flexibility limitation of Gaussian assumptions in conventional filters. We theoretically justify that the BFF design is structurally aligned with Bayesian filtering, and its training objective targets the recursive filtering operator. BFF is empirically verified across 5 different physical systems, including ones with chaotic dynamics and highly sparse, non-linear observations. The results show that BFF attains the best score in 8 of 9 metric-benchmark cells across the three standard 1D and 2D PDE benchmarks, and similarly leads on the extreme single-moving-sensor setting and on a real-world-grounded tokamak plasma estimation task, demonstrating its potential to accurately approximate the Bayesian filtering operator in high-dimensional probability space.

A splash lives for a fraction of a second: sheets tear into ligaments and droplets, appearance is view-dependent and nearly textureless, and little persists long enough to track. Reconstruction research has consequently focused on smoke, synthetic liquids, or gently deforming surfaces. To our knowledge, no synchronized multi-view dataset of splashing liquids exists. We therefore introduce a benchmark of 20 real scenes, from coherent streams to violent splashes, captured by seven synchronized, calibrated 4K cameras at 60 fps, with manually refined per-view liquid and container masks and fixed evaluation splits. We further present SplashSplat, built on a single principle: impose physical structure only where the observations can constrain it. Per-frame liquid SDFs fused from the masks provide the geometry, level-set transport between consecutive SDFs yields a coarse velocity field, and Lagrangian carriers advected along this flow, corrected against each new observation and reseeded where coverage is lost, decode local Gaussians for differentiable rendering. SplashSplat outperforms state-of-the-art dynamic Gaussian splatting methods on our real captures and on a synthetic benchmark, with physically more plausible motion and a lower training cost. The same representation supports temporal interpolation and style transfer without re-optimization.

The existing singularity-free guiding vector field (SF-GVF) with an additional virtual coordinate can eliminate singular points (i.e., points where the vector field vanishes) inherent in conventional GVFs and guarantee global convergence of robot trajectories to closed and self-intersecting desired paths. However, the desired speed given by the GVF along the desired path in the original lower-dimensional space cannot be arbitrarily specified but depends on path parameterizations. One possible workaround is to partially normalize the physical projection of the SF-GVF and assign a user-designed speed. However, we show that this workaround may introduce new singularities since the normalization denominator can become zero. To address this issue, we propose a new SF-GVF with prescribed physical speed (PPS). The integral curves of the new SF-GVF converge exponentially to the desired path from any initial condition in the higher-dimensional space (including virtual dimension); more importantly, the robot's physical speed converges to the PPS, while the path-error dynamics remain invariant under regular reparameterizations of the desired path. We further develop a saturated acceleration control law for second-order kinematic models. Finally, comparative simulations and 3D path-following experiments with a quadrotor under different PPS profiles validate the theoretical results and demonstrate the effectiveness of the proposed approach.

Antibody lead optimization calls for a small, bounded set of edits to an existing candidate: substitutions, but also insertions and deletions. Edit-based generative models are the only ones that allocate such an edit budget without fixing the edit positions, the edit count, or the output length in advance. However, the existing approaches Edit Flows and EvoFlows did not release code or complete training specifications. Here, we show that both methods follow the same underlying process -- edits firing one at a time, at learned rates, in continuous time -- the pure-jump case of generator matching over finite sequences. With EditJumps we introduce the first open implementation of this framework, with a single generalist antibody editor trained on 1.66M Observed Antibody Space homolog pairs to propose homolog-like variants of a seed sequence, editing unseen leads zero-shot, without the per-family retraining original approaches require. Replicating this system from scratch exposes why open code is essential for generative biology: reconciling published edit distributions required reverse-engineering an undocumented rate-scaling hyperparameter that dictates realized mutation counts. Moreover, we show that published evaluation metrics are highly sensitive to reference sample size, frequently flipping method rankings. We release our full codebase, automated test suite, and configurations at: https://github.com/VisiumCH/editjumps

In flow matching (FM), a velocity model $v_θ$ is trained using a predefined path $g_t$ that connects data and noise samples (e.g., $g_t(x_0, x_1) = (1 - t) x_0 + t x_1$). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class $G(p_t,v^\star_t)$ of paths that induce the same marginal distributions $p_t$ and marginal velocity field $v^\star_t$, and therefore the same FM objective. Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed $θ$ as the variance-minimization problem PathOpt$_θ$, constrained to $g_t\in G(p_t,v^\star_t)$. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint $g_t \in G(p_t,v^\star_t)$ cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.

City pedestrian counting systems now feed economic indicators, planning decisions and safety operations, yet the twins built on top of them treat the incoming stream as ground truth. We study what happens when it is not. We formalise stealthy false data injection for city-scale pedestrian sensing, where the map from latent flow to observation is far more rank deficient than in the power and water networks for which stealth has been characterised. Our twin estimates directed flows on the pedestrian street graph, assimilates counts through a learned graph-localised gain, and is trained against a flow conservation residual that couples metered and unmetered segments. Detection combines the innovation with that residual, and the alarm threshold is set by adaptive conformal calibration rather than by hand. To measure what the physics buys, we define the attack margin, the relative reduction in worst-case corruption of the estimated flow field, achieved against a white-box adversary that optimises directly through the twin. On six years of Melbourne data the margin reaches 0.54 against a single compromised device and falls to 0.19 when a third of the fleet is compromised, on a network where only 1.18 per cent of walkable segments are metered. Replacing the street graph by a distance graph collapses it to 0.09, which shows that the gain comes from the conservation law rather than from locality.

Cranial implant generation is an important task in medical imaging. Recent point cloud based generative methods, particularly flow matching, offer strong reconstruction quality and efficient sampling, but still require multiple neural function evaluations during inference. This limits rapid generation of multiple plausible implant candidates. We propose Teacher-guided Endpoint Distillation (TED), a simple one-step distillation framework for conditional cranial implant generation on point clouds. TED trains a one-step student using teacher-guided endpoint supervision and geometric matching losses, while avoiding explicit path straightening. We evaluate TED on the SkullFix and SkullBreak benchmarks. TED achieves the best overall performance on the SkullBreak dataset, remains competitive on SkullFix, and provides the strongest Chamfer distance performance among the compared one-step methods. In addition, TED generates implants in approximately 0.04s per sample. These results show that one-step distillation can substantially accelerate conditional point cloud implant generation without sacrificing reconstruction quality.

Trajectory curvature constraints are inherent in practical multi-robot systems due to the limited turning capabilities of the robots. Without properly accounting for these constraints, robots may fail to accomplish assigned tasks, and their trajectories may diverge from the intended paths. This paper proposes a distributed safe cooperative vector field approach for multi-robot systems subject to trajectory curvature constraints. The proposed approach is composed of a cooperative vector field and a safety-oriented collision avoidance vector field, aiming to address the problems of cooperative motion and safe collision avoidance in multi-robot path-following tasks. A safety-oriented collision avoidance vector field with adaptively adjustable reactive boundary is developed to accommodate the kinematic curvature constraints of robots, thereby ensuring the physical feasibility of collision avoidance maneuvers. The proposed vector field requires only a single virtual variable from each neighboring robot to achieve cooperative motion and ensure both obstacle avoidance and inter-robot collision avoidance. The effectiveness of the proposed approach is validated through both simulations and real-world experiments on an actual multi-robot platform.

Methods of Branched Optimal Transport (BOT) mimic the economy and efficiency of natural tree-like structures, such as those found in rivers and biological systems. These methods are widely applicable for designing efficient networks in society, from river basins and blood vessels to mail and gas distribution systems. However, they remain understudied in the context of designing deep generative models, particularly at a large scale. Standard continuous-time generative models, such as the flow matching approach, fail to capture the inherent hierarchical and branching patterns present in real-world data. Current models provide no mechanism for flows to merge or share pathways to minimize total transport cost. Inspired by the "economy of scale" principle in BOT, we introduce a novel, scalable branched flow-matching algorithm designed to solve the branched optimal transport problem in high dimensions. Our method adapts the Benamou-Brenier continuous-time optimal transport formulation to learn branched generative flows. These flows allow probability mass to aggregate along common pathways before branching out to diverse targets. Parametrized by neural networks, our method effectively learns complex branched generative processes. We demonstrate its effectiveness on challenging high-dimensional tasks in biology and image generation.

Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a separate data-consistency update. In these common formulations, the forward model is not explicitly enforced within the learned conditional velocity field. We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems. Under linear interpolation, we express the conditional velocity $v(x_t,t,y)$ in terms of the posterior mean $E[x_1 | x_t,y]$, and characterize that mean as the unique minimizer of a variational objective whose data-consistency term is explicit. We further prove that the velocity field defines a probability flow from the source distribution to the measurement-conditioned posterior. Splitting the variational objective yields a conditional velocity parameterization with operator-dependent data-consistency updates, which we train end-to-end under the flow-matching objective, with no additional guidance at inference. Our method achieves state-of-the-art PSNR with $50\times$ fewer function evaluations than the strongest flow baseline. Varying the sampling steps provides test-time control over the distortion-perception trade-off without retraining.

Humans and machines often solve harder problems by spending more time on computation. In deep learning, looped models implement this idea during inference by recurrently updating a hidden state. In practice, however, their training backpropagates through only one or a few updates, making it hard to train early updates to support future ones. We propose looped flows, an approach that sidesteps this issue by training the recurrence with local denoising objectives. By imposing temporal association across denoising objectives through progressively decreasing noise levels and shared noise, the model is incentivized to learn recurrent states that transfer useful computation over time, even when gradients cover only a few updates. We then formulate inference as integrating the velocity of a probability flow parameterized by the learned denoiser, coupled with recurrent states. This allows solving harder problems by spending more computation through a finer temporal grid and enables multiple valid predictions from different initial noise samples. Across six reasoning benchmarks including two multi-solution benchmarks, looped flows outperform prior state-of-the-art looped models overall, achieving 58.8% test accuracy on ARC-AGI-1 and 12.2% on ARC-AGI-2.

Text-to-image diffusion transformers (DiTs) are powerful generators, yet direct prompting provides limited control interface for style intensity and can fail to suppress unwanted concepts. To enable these controls, we introduce AcFlow, an inference-time controller that transports intermediate layer image-token activations through a learned concept-conditioned velocity field while keeping the base DiT frozen. A textual concept description specifies the desired intervention, while the integration horizon provides a continuous control parameter. The field produces token-varying, activation-dependent updates. With parameters shared across concepts within each task family, the field supports fine-grained descriptions and generalizes to concepts unseen during training without per-concept fitting. On style control, AcFlow achieves the best style--content trade-off among the evaluated baselines in the high-style-alignment regime. At a fixed operating point, AcFlow attains style--content alignment of 0.5365/0.2860, compared with 0.4397/0.2684 for the baseline with the highest style alignment. Qualitative results demonstrate suppression of diverse concepts, including cases where direct prompting fails. Our analyses support the learned velocity field as an adaptive control mechanism, with update directions varying across tokens and depend on their activation states. Our code is available at https://github.com/Nove1yst/AcFlow.

Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant paths with one-hot targets through a position-wise argmax yields discrete convex-interpolant paths. The result requires source laws with appropriate coordinate symmetry and boundary regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.

Robot actions are temporally correlated trajectories whose frequency components encode motion at different scales with highly non-uniform energy distributions. Yet Flow Matching--based vision-language-action (VLA) models typically generate actions in temporal coordinates, without explicitly modeling or systematically leveraging this frequency heterogeneity. We introduce \emph{FreqFM}, a frequency-conditioned Flow Matching framework for VLA models. It raises action frequency from an implicit trajectory property to an explicit conditioning dimension that spans the entire generation pipeline. Concretely, in DCT frequency coordinates, FreqFM constructs a spectrum-matched source distribution, adaptively balances the objective across frequencies, and constrains per-frequency guidance residuals using the corresponding reference transport scales. FreqFM integrates into existing Flow Matching action experts without changing the VLA backbone. Across LIBERO, LIBERO-Plus, and VLA-Arena, FreqFM consistently improves performance, including a 9.3-point gain on LIBERO-Plus, and further demonstrates its effectiveness on six real-robot tasks.

Molecular size is coupled to composition, structure, and function, yet most 3D molecular generators require a predefined atom count. We introduce Equivariant-Free Transformer-Autoencoded Latent Flow Matching, a two-stage framework that samples a fixed-dimensional latent vector using flow matching and uses an autoregressive Transformer to determine molecular size, atom types, coordinates, and chemical attributes. Canonical atom ordering and rigid-pose alignment enable Transformers without equivariant layers, while decoded attributes guide bond reconstruction. On PCQM4Mv2, unconditional generation yields 87.9% unique, novel molecules passing sanitization and PoseBusters checks, exceeding baselines with lower end-to-end training and sampling time and higher end-to-end throughput. Across ten target HOMO-LUMO gaps, internal ranking retains 30% of screened candidates and increases the density functional theory-verified hit rate within 0.1 eV from 25.0% to 52.4%, while largely preserving novelty and diversity. These results demonstrate fixed-dimensional latent generation with autoregressive decoding as a practical approach to molecular design without prespecifying size.

Uncertainty arising from inter-observer variability in medical image segmentation plays an important role in developing treatment plans. Research in this area is inhibited by the lack of multiple annotations for large-scale medical datasets, especially for volumetric data, which suffers from additional scaling and computational complexity challenges. Flow matching has emerged as a powerful framework for generative modelling and has also been demonstrated to maintain strong performance when working with latent representations of images. In this work, we introduce a latent-to-latent flow technique for stochastic segmentation of medical volumes via encoded representations of both the image and label space. We evaluate our method on two challenging applications covering delineation uncertainty for radiotherapy planning and multiple organ structure segmentation, improving efficiency up to 14x compared with full resolution models while maintaining clinically relevant performance.

Ground-penetrating radar (GPR) B-scan image synthesis is important for data augmentation, algorithm validation, and simulation acceleration, yet generating radargrams with both visual realism and physical consistency remains challenging. Existing learning-based generative models often emphasize visual appearance but provide limited control over response geometry. In this paper, we propose PCFlow, a physics-conditioned flow matching framework for fast GPR B-scan image synthesis. The core of PCFlow is a Maxwell-informed dense physical condition field constructed from the parameterized physical model used for electromagnetic simulation, including material properties, target geometry, propagation cues, and response-domain priors. This condition field provides an interpretable interface between physical scene parameters and radar response geometry, and guides conditional flow matching in the VAE latent space toward physically feasible generation paths. We evaluate PCFlow on a gprMax-based buried-pipeline dataset with both in-distribution and out-of-distribution test cases. Experimental results show that PCFlow generates images with more accurate response geometry and high visual fidelity, demonstrating its effectiveness for controllable and physically faithful radar image synthesis.

This paper develops a singularity-free guiding vector field (SF-GVF) for path following on the special orthogonal group SO(3). First, we lift the Euclidean SF-GVF construction to SO(3), integrating the augmented-state approach with the intrinsic Lie-group geometry and obtaining a closed-form geometric guidance law whose integral curves converge to a designer-specified attitude path. The field is defined on a dense open subset of SO(3), excluding only the measure-zero antipodal set - a manifestation of the topological obstruction to continuous global stabilization on SO(3). The construction requires no per-step optimization and produces a control input intrinsically in so(3) as body angular rates. Second, we formalize the progression behavior along the path as a designer-supplied function ν(ξ), promoting the parametric speed from an implicitly resolved degree of freedom to a first-class design specification. In contrast to the Euclidean condition v = 0, which excludes vehicles with minimum-speed constraints, the corresponding condition ω= 0 on SO(3) is physically admissible for most platforms with active attitude control, making the progression behavior a design freedom structurally available on SO(3) but absent in the Euclidean setting. The framework's structural results are established under a bi-invariant Riemannian metric and hold uniformly across choices of path, progression, and Lyapunov gain. The framework is illustrated in simulation on self-intersecting paths under both constant and point-convergence progression behaviors.

Synthetic medical time-series generation can alleviate data scarcity and support the development of reliable clinical prediction models. However, existing methods mainly focus on matching the overall distribution and temporal dynamics of real data, which does not necessarily ensure strong downstream utility on imbalanced medical datasets. Clinically informative patterns often occur at heterogeneous temporal scales, while rare minority-class characteristics can be obscured by dominant population patterns. To address these challenges, we propose MedFlow, a class-aware multi-scale flow matching framework for medical time-series synthesis. MedFlow employs a vector-quantized multi-scale tokenizer to represent medical sequences at complementary temporal resolutions, capturing both coarse clinical trends and fine-grained dynamics. We further introduce Token Marginal Guidance, which incorporates class-conditional token statistics directly into the flow matching process to steer generation toward class-specific regions of the learned tokens. This mechanism strengthens minority-class patterns, while preserving the global and tail distributions of real data. Experiments on four public datasets covering electronic health records, EEG, and ECG signals demonstrate that MedFlow consistently outperforms recent state-of-the-art diffusion-based baselines across downstream prediction tasks. On average, it improves AUPRC by 5.8%, reduces Context-FID by 88.6%, and achieves 3.8$\times$ higher sampling throughput.

Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.

Crystal generators can now propose periodic structures, but their control interfaces remain poorly matched to the mixed descriptors used in materials design. Text provides a compact way to combine composition, symmetry, prototype and property cues, yet it has not been clear whether such information can steer flow-based crystal generation. Here we introduce TFMat, a text-conditioned flow-matching framework that uses structured materials language as a semantic prior for a CrystalFlow generator. Across Perov-5, Carbon-24 and MP-20 crystal structure prediction benchmarks, TFMat improves one-candidate match rates over CrystalFlow and reaches a 92.04% MP-20 match rate with 20 candidates; in de novo generation, it improves element-count and density distribution alignment while retaining coarse property consistency in composition-selected outputs. These results position structured text as an inspectable control layer for translating human-readable materials intent into candidate crystals for downstream simulation and validation.

Magnetic resonance images of the same subject look markedly different across field strengths, which complicates the comparison and pooling of data across sites. We address cross-field brain-MRI translation for the MRIxFields2026 challenge, and in particular its Task~3: a single model that translates between any directed pair of the five field strengths and across three contrasts. We phrase the problem as a conditional flow matching path: because the source and target volumes are spatially registered, we learn a velocity field that carries the source slice directly to the target slice, rather than starting from noise. To learn this mapping from only three paired subjects, the unified model is trained in three stages: a degradation-bridge pretraining that distills a restoration prior from the abundant unpaired retrospective cohort, a cross-field finetuning over all directed pairs on the paired cohort, and an adversarial refinement that sharpens the output. At inference, we integrate the learned velocity with a second-order Heun solver in a handful of steps. A restoration prior learned without any paired data already reaches a mean SSIM of 0.837, and each subsequent training stage improves on it. A single 6.3M-parameter model thereby covers all 60 field-pair and contrast combinations, with inference in five solver steps per slice. On the challenge evaluation set the model reaches a mean SSIM of 0.909, averaged over the three contrasts, outperforming regression and diffusion baselines built on the identical network on all three challenge metrics.

Engineering inverse design is often limited by the high computational cost of iterative solvers for optimization problems constrained by partial differential equations (PDEs) and by their sensitivity to initialization. Deep generative models can produce candidate designs without rerunning the simulator at inference time. Generative adversarial networks (GANs) sample in one forward pass, whereas diffusion models require iterative reverse-time integration. In this work, we add conditional flow matching (CFM) to EngiOpt and compare it with a conditional diffusion model and a conditional generative adversarial network (cGAN) on structural (beams2d) and thermal (heatconduction2d) benchmarks from EngiBench using the same downstream optimization protocol. We use cumulative optimality gap (COG) and final optimality gap (FOG) as the primary metrics for evaluating the generated designs as warm starts for gradient-based refinement. On the evaluated EngiOpt implementations and two EngiBench tasks, CFM achieves the lowest measured COG, FOG, maximum mean discrepancy (MMD), and volume-fraction deviation on both tasks. CFM has mean volume-fraction deviations of 0.4% and 1.0% on beams2d and heatconduction2d, respectively, compared with 3.8% and 11.2% for diffusion. At Euler s = 16, CFM achieves 53.2 samples/s on beams2d, about 66 times the measured throughput of the evaluated diffusion baseline using 1000 network evaluations under the same timing protocol, with COG 1.182 +/- 3.126, compared with 1.173 +/- 3.100 for Euler s = 32. Across the two tasks, CFM produces warm starts with lower measured COG than both baselines and uses fewer network evaluations than diffusion.

Flow Matching provides an efficient generative prior for image restoration by learning continuous transport between source and data distributions. However, existing methods typically incorporate measurement constraints through local corrections. Such corrections may disrupt the source-clean endpoint coupling implicitly encoded by the pretrained flow, making the corrected endpoint pair incompatible with the current state. To address this issue, we propose ReBridge-Flow, a posterior bridge re-coupling method. Specifically, given the current state, ReBridge-Flow first decodes the corresponding local source and clean endpoints. It then incorporates measurement information through clean-side anchoring and synchronously re-couples the source endpoint, yielding a measurement-aware endpoint pair with improved local bridge compatibility. The re-coupled endpoints further define a posterior-informed transport direction for advancing the sampling process. We also introduce the Posterior Bridge Defect, which jointly characterizes measurement error, deviation from the flow prior, and bridge mismatch, and leads to explicit updates for clean-side anchoring and source-side re-coupling. Extensive experiments on multiple natural and medical image restoration tasks demonstrate that ReBridge-Flow effectively alleviates bridge mismatch and improves the structural consistency of restored images.

Controllers that guarantee energetic passivity with respect to the pair of external force and velocity realize safe interaction between the mechanical system and its physical environment. However, solely adhering to energetic passivity constraints may impose fundamental limitations on control performance and, in some cases, prevent the successful execution of controlled tasks. In addition, external disturbances from the physical environment can drive the system energy level and states beyond operational regions, thereby undermining task performance and safety. In this paper, we study a robust time-varying semi-passive velocity field control to aim to relax the inherently conservative nature of fully passive control methods in a controlled manner. Specifically, the proposed control method guarantees passivity of the closed-loop system with respect to the force-velocity input-output pair when the energy level exceeds a predefined level, while permitting non-passive behaviors to preserve task performance otherwise. Furthermore, the energy level and the states of the closed-loop system are proved to converge to bounded domains even in the presence of unpredicted disturbances. Additionally, the proposed method also enables constraining power flow between the closed-loop system and its physical environment to enhance safety in the interaction process. Numerical simulation examples demonstrate the effectiveness of the proposed method.

Understanding which training samples influence a generated image is an important problem in generative modeling. In flow matching, training samples influence the generated image through the velocity field along the generation trajectory. Removing samples to examine their counterfactual influence changes the velocity field, and the resulting effect on the final image depends on how the change propagates through the trajectory. Consequently, local changes in the velocity field do not necessarily predict the final counterfactual effect. This work investigates attribution in flow-matching models through a hybrid analytical--learned approach, and uses it to derive trajectory-based attribution scores at the cluster level. We evaluate these attribution scores using independently retrained leave-one-cluster-out (LOCO) models, and compare with several attribution baselines using two different flow-matching latent spaces. Our experiments show that semantic similarity constitutes a strong baseline, while the closed-form trajectory-based attribution is competitive in some metrics without requiring counterfactual retraining or model gradients. Our results show that attribution in flow matching depends not only on semantic similarity to training samples, but also on the latent representation, trajectory dynamics, and how influence is propagated to the final output.

Text-guided style editing of 3D assets is essential for adapting existing objects to diverse visual aesthetics in digital content creation. Despite rapid progress in 3D shape modeling, faithfully stylizing an existing asset remains challenging when the desired stylization involves fine-grained structural ornamentation, which requires the model to preserve the source geometry and object identity, while coherently integrating new style-specific details. We propose \textbf{OrnaStyler}, a zero-shot framework for text-guided ornament-aware 3D stylization. Built upon rectified flow-based generative modeling, OrnaStyler introduces an inversion-guided editing strategy that recovers content-aware latent representations at both geometry and appearance levels in a staged manner to facilitate faithful editing. Our core idea is to explicitly model the spatial configuration of stylistic elements, thereby mitigating the fundamental tension between content preservation and style expression in the voxel space. Specifically, at the geometry level, we manipulate voxel representations through flow inversion to synthesize ornament-enhanced structures while preserving the spatial identity of the source asset. Then, at the appearance level, we introduce an adjacency-aware feature inpainting mechanism to harmonize newly generated ornaments with the original content, yielding coherent geometry-appearance integration. Our approach operates solely in the inference phase and enables selective editing over geometric augmentation or appearance stylization. Extensive experiments on both generated and real-world 3D assets against prior methods demonstrate that OrnaStyler achieves state-of-the-art editing performance in terms of content preservation, style fidelity, and overall visual realism. Code is available at: https://github.com/tomohiro0427/OrnaStyler

Generating realistic 3D city environments from remote sensing data is important for simulation, urban planning, and mixed reality, yet existing point cloud generation methods are limited to single objects or bounded indoor scenes and cannot handle the scale, seamless tiling, and partial observability challenges of city-scale generation. We present \ours{}, a multi-stage framework that generates dense, colored point clouds ($10^5$ points per $150\text{m}{\times}150\text{m}$ tile) at city scale, conditioned on satellite imagery, semantic segmentation maps, and digital surface models (DSM). A \emph{Grid-Aligned VAE} encodes each tile into a topology-preserving latent grid where tokens correspond to fixed spatial regions, enabling spatially coherent multi-modal conditioning and compact latent-space edge consistency that implicitly aligns thousands of boundary points for seamless cross-tile generation. A conditional rectified flow model synthesizes geometry latents from the fused multi-modal conditions, and an orientation-aware diffusion colorizer separately handles satellite-visible horizontal surfaces and occluded vertical façades. To support standardized evaluation, we build on public 3D data sources to introduce \emph{City3D-MultiGen}, a benchmark of $163$K densely annotated tiles from Melbourne and London with aligned point clouds, satellite images, semantic maps, and elevation data. Experiments show that \ours{} outperforms adapted point cloud generation baselines across all geometry metrics and produces visually coherent colored point clouds with seamless boundaries over arbitrarily large urban extents. Our benchmark details are available at https://huggingface.co/datasets/e32/City3D-MultiGen

Current large language model development relies on massive, often non-permissible datasets, creating a high barrier for researchers committed to open-source and ethically sourced data. We introduce Mimir v1, a 1-billion-parameter language model based on the Hierarchical Reasoning Model (HRM) architecture, that is trained from scratch and delivers highly competitive performance for English and sets a new state of the art for Danish using only permissible post-training data. Trained on a mixture of 161 datasets, Mimir v1 outperforms the original HRM-Text 1B and competes with larger frontier models like Qwen 3.5 4B and Gemma 4 E2B, tested across 20 benchmarks for English, Math & Code and Danish. The model is available on the Hugging Face Hub: https://huggingface.co/danish-foundation-models/DFM-Mimir

High-fidelity image generation faces a trade-off between speed and quality. Diffusion models produce strong visuals but require costly iterative sampling. Existing efficient methods mainly distill pretrained models into few-step samplers, a challenging process that depends heavily on teacher-model quality. In this paper, we introduce XYZFlow, a framework that rethinks efficient generation through multidimensional scaling of flow matching. Unlike single-step mappings, XYZFlow enhances expressivity by making probability paths more identifiable and learnable through structured multidimensional conditioning. We view autoregressive modeling as implicit flow straightening, where richer context reduces trajectory ambiguity. XYZFlow realizes this idea through two orthogonal dimensions: temporal scaling, which uses non-Markovian conditioning on the full denoising history; and spatial scaling, enabled by Next Shortcut Prediction, which sequentially generates patches using preceding patches' denoising trajectories as priors. Experiments show that XYZFlow achieves state-of-the-art performance, with 7.2-8.5X teacher speedups and competitive FID, while Next Shortcut Prediction delivers superior quality-latency trade-offs over model scaling or step reduction.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

The rapid advancement of Large Language Models (LLMs) has facilitated their ubiquitous integration into various domains, leading to widespread adoption. However, this escalating trend has introduced significant security vulnerabilities, necessitating the identification and mitigation of flaws arising from malicious exploitation. Red teaming assessments, conducted to evaluate model robustness through diverse adversarial inputs, are essential for exposing security risks and implementing countermeasures. Currently, red teaming is performed either manually by experts or automatically using predefined attack datasets. Nevertheless, manual testing remains time-consuming, while existing automated methods suffer from limited creativity due to their inherent dependency on fixed datasets. In this study, we propose an automated, human-independent, and adaptive approach leveraging GFlowNets to identify LLM vulnerabilities by utilizing one large language model to test another. Within this framework, an attacker model is trained against a specified victim model to perform automated red teaming and provide a quantitative robustness score. This research aims to generate more effective adversarial attacks in English compared to existing benchmarks and, as a novel contribution to the literature, introduces a model capable of generating attack inputs in the Turkish language.

Diffusion-based super-resolution (SR) achieves strong perceptual quality but requires costly iterative denoising. Existing one-step distillation methods reduce inference time but depend on expensive pretrained teachers, whereas CTMSR avoids distillation through PF-ODE consistency training yet does not explicitly model the restoration dynamics from low-resolution (LR) inputs to high-resolution (HR) images. We propose MeanSR, a one-step perceptual SR method that learns an LR-conditioned average velocity field to directly capture the finite-time transition from degraded or noisy inputs to plausible HR outputs. We further reformulate distribution trajectory matching for average-velocity generation and introduce a Stage-Aware Temporal Sampling strategy to improve trajectory learning. Experiments on synthetic and real-world benchmarks show that MeanSR outperforms CTMSR on CLIPIQA, MUSIQ, and MANIQA while substantially reducing FLOPs and inference latency. MeanSR also reconstructs sharper structures and more realistic textures with fewer perceptual artifacts.

Flow-matching models are now a mainstream method to image generation, but its adaptation to diverse downstream scenarios typically relies on post-training, which may cause conflicts among task-specific optimization objectives. Reinforcement learning enables direct optimization of task-specific rewards beyond the original models, yet trajectory-level optimization may incur high-variance gradients and cross-task interference. On-policy distillation (OPD) offers dense and stable supervision on student rollouts, but conventional teacher matching remains imitation-based. We propose DreOPD, a Degraded-reference extrapolative OPD method for flow-matching models that bridges these two paradigms. Our DreOPD converts implicit reward extrapolation into closed-form velocity regression, enabling extrapolative post-training with the stability of OPD. It further uses a mildly degraded reference to strengthen the teacher-reference contrast, yielding a clearer extrapolation direction. Experiments on single- and multi-teacher settings show that DreOPD outperforms OPD and multi-task RL baselines in average performance, while surpassing specialized teachers on most metrics.

Instruction-driven editing of 3D human motion requires precise spatiotemporal localization, rich semantic grounding, and strict preservation of unmodified content. Existing methods either resort to training-free adaptation of generative models or rely solely on triplet supervision; however, adaptation often yields suboptimal control, and manually curated triplet datasets remain severely limited in scale and semantic diversity. To overcome this bottleneck, we ground motion editing directly within text-to-motion generation across data, architecture, and inference. At the data level, we develop a closed-loop synthesis-and-verification pipeline that produces Omni-MoEdit, a large-scale dataset spanning body-part, amplitude, temporal, action, and style edits. At the architectural level, we introduce UniMoFlow, a unified latent flow-matching model that shares broad semantic and kinematic knowledge between generation and editing. At the inference level, SAFE (Source-Anchored Flow Editing) complements UniMoFlow with controllable, source-anchored refinement. Furthermore, we augment standard evaluations with semantics-aware metrics to account for valid edits that inherently deviate from a single ground-truth reference. Extensive experiments demonstrate improved target-text alignment, edit effectiveness, and cycle consistency, while maintaining competitive source fidelity and text-to-motion generation quality.

Generating mixed-type tabular data requires jointly modeling diverse feature distributions and their complex cross-column dependencies. Variational flow matching handles distinct endpoints via factorized distributions, yet leaves feature-specific processing and cross-column interactions implicit within a shared backbone. We introduce Feature-wise Unified Specialization with cross-column Exchange (FUSE) to explicitly separate these roles. FUSE applies separate adaptive mixture modules to numerical and categorical features, allowing each feature to combine shared specialized subnetworks, while joint attention preserves information exchange across all columns. We also characterize the excess population risk from restricted conditioning contexts and bound the continuous Wasserstein generation error by endpoint-prediction risk. Comprehensive experiments on eight tabular datasets demonstrate that FUSE achieves strong and consistent performance across distributional fidelity and downstream utility metrics.

Bidirectional unpaired image translation must preserve source-specific structure while learning coherent transformations in both directions without paired supervision. Existing methods typically employ two direction-specific generators or train separate one-way models. Even when linked by cycle consistency, such models constrain only the round-trip endpoint reconstruction, without requiring the two directions to obey a common local transformation rule. We propose UniCycleFlow, a rectified-flow framework that represents bidirectional translation as forward and reverse integration of a single time-conditioned velocity field. This formulation organizes both directions within the same continuous dynamics, rather than coupling otherwise separate endpoint mappings. A key challenge is that unpaired data provide no meaningful source--target coupling from which rectified-flow trajectories can be constructed. UniCycleFlow addresses this challenge by learning deterministic source-conditioned endpoints whose marginal distributions are adversarially matched to the opposite domains. The resulting paths are regularized by stop-gradient self-flow matching for intermediate velocity supervision, discrete cycle closure for forward--reverse consistency, and representation path-velocity regularization for controlling localized feature changes along the trajectory. Across ten translation directions, UniCycleFlow achieves the lowest FID on 7 of 10 tasks using a single Euler evaluation and obtains the best average FID of 55.1.

Unpaired image-to-image translation must decide, per image, what to change and what to preserve without paired supervision. Many diffusion-based unpaired translators control preservation through a single global noise or guidance value applied across the image, which cannot separate content to keep from appearance to change. We present PRISM, a GAN-free flow-matching framework that replaces this global control with a learned per-feature gate. The gate's spatial prior is derived from each source feature's standardized distance to the target feature distribution, so features far from the target are freed while target-consistent features are preserved. The same gate controls both the initialization, which mixes the real source latent with a task-matched corruption, and the transport timing during Ordinary Differential Equation (ODE) integration. The corruption is matched to the task, content-anchored (AdaIN) for structure-preserving translation and partially anchored for structure-changing translation, and the gate can be overridden locally at inference from text or a detector without retraining, preserving important structures of the original image while still generating realistic results. We evaluate PRISM on five natural and biomedical benchmarks (AFHQ cat->dog, CelebA-HQ appearance translation, day->night relighting, virtual staining, and breast frozen->permanent histopathology). Among the evaluated methods under a shared same-split protocol, PRISM attains the best Inception FID and KID on four benchmarks and a competitive result on the fifth, and on histopathology yields the nuclei-count ratio closest to the ideal, supporting a favorable balance between target realism and structural preservation.

Pixel-space generative models bypass lossy latent compression, yet necessitate joint learning of global structure and fine-grained details in a high-dimensional space. Standard flow matching interpolates noise toward a fixed clean-image endpoint, leaving the spectral evolution to be learned implicitly. In this paper, we introduce Energy-Guided Flow Matching(EG-FM) that explicitly models a coarse-to-fine generative trajectory by moving endpoint. Specifically, EG-FM replaces the fixed endpoint with a heat-kernel-filtered endpoint that evolves smoothly from low-frequency image to clean image. The fraction of high-frequency signal in moving endpoint is released by an image-specific energy-guided scheduling, leading to the re-targeting of velocity in flow matching. Our framework requires no adaptation of the backbone and training data, bringing negligible cost on the training and inference stages. In our experiment, EG-FM consistently achieves lower FID on the ImageNet class-conditional image generation task at $256 \times 256$ with fewer epochs, reaching an FID of 1.55 at 200 epochs and 1.45 at 600 epochs. We continue training the generation task on the setting of $512 \times 512$ resolution, yielding a FID of 1.58 after only 40 high-resolution adaptation epochs. Furthermore, we transfer EG-FM on text-to-image generation and achieve 0.85 on GenEval score and 83.9 on DPG-Bench. Code is available at https://github.com/ysng123/EG-FM.

Multilingual text embedding models are commonly adapted using a single training objective across diverse tasks, despite different tasks requiring fundamentally different optimization strategies. We introduce Task-Conditional Flow Matching (TCFM), a multilingual embedding adaptation framework that selectively applies Flow Matching to translation tasks while optimizing retrieval, classification, and pair-classification tasks with objectives better aligned to their learning dynamics. TCFM further combines teacher-guided representation preservation with a three-stage curriculum to enable stable adaptation. Evaluated on the Indic Massive Text Embedding Benchmark, TCFM establishes a new state-of-the-art, consistently improving embedding quality across a diverse set of multilingual tasks and generalizing across embedding model families. We will publicly release the codebase and datasets upon acceptance of the paper.

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

Generating high-quality 3D point clouds requires capturing both global shape topology and local geometric details. Existing flow-based methods rely on continuous normalizing flows (CNFs) that demand expensive ODE solving and trace estimation during training, while diffusion models require hundreds of iterative denoising steps. Moreover, most approaches adopt single-level generation directly in point space, disregarding the hierarchical structure natural to 3D shapes. We propose Hierarchical Flow Matching (HFM) that extends flow matching to bilevel structure for unconditional 3D point cloud generation. HFM decomposes the task into two levels via optimal-transport flow matching: a \textit{Latent Flow Matching} models the global shape manifold in a compact latent space, and a \textit{Conditional Point Flow Matching} reconstructs detailed point clouds conditioned on the latent code. Both flows are trained with simple MSE regression losses. The resulting straight OT paths enable efficient sampling with as few as 15 Euler steps per flow, while the structured latent space supports downstream tasks including classification. Extensive experiments on ShapeNet and ModelNet benchmarks demonstrate that HFM achieves competitive or even best performance compared with prior state-of-the-art methods.

While many algorithms blend reinforcement learning (RL) with counterfactual regret (CFR) methods to leverage tradeoffs in computational speed and performance, there are fewer investigations into generative sampling frameworks in game theoretic applications in incomplete information games. We extend a generative flow network framework, Adversarial Flow Networks (AFlowNets), to incomplete information games, called Information Flow Networks (IFNs). We prove that previously established constraints for generative flow networks in complete information games are inadmissible for obtaining valid densities (corresponding to player strategies) and a valid training objective. We show that our proposed generalization, IFlowNets, alleviates this issue and strictly generalizes AFlowNets. In preliminary results for three standard game environments, IFlowNets perform comparably to or better than Outcome Sampling Monte Carlo Counterfactual Regret (OSMCCFR) and standard RL-based methods in performance and speed.

Vision Transformers (ViTs) achieve strong image-recognition performance, but their parameter count grows linearly with depth when each block is independently parameterized. Single-block recurrent ViTs (bViT) remove this growth by repeatedly applying one shared block. Rather than proposing a new architecture, we fix a bViT and provide a controlled empirical characterization of three training and inference regimes under a common CIFAR-100 protocol, asking: (i)~when does recurrence beat independently parameterized depth---at matched FLOPs or at matched parameter memory? (ii)~when a residual recurrent block is trained through an ODE solver, does solver order act as numerical refinement or as an architectural bias? and (iii)~what does robustness beyond the training horizon cost in nominal accuracy? We find that standard ViTs remain preferable when FLOPs are the primary constraint, whereas recurrent ViTs offer a better accuracy--parameter trade-off under memory constraints. Consistent with the standard view of residual networks as Euler discretizations of ODEs, the continuous-time analogue of a residual recurrent block is the state-subtracted vector field $\dot{z}=F_θ(z)-z$; although known in principle, this distinction is easy to violate when the block is wrapped as a black-box vector field, and we qualify the cost at few accuracy points. Because the vector field is learned jointly with the solver, higher-order solvers act as a solver-induced architectural bias rather than a numerical-accuracy improvement, and their gains are not uniform. Finally, stage-wise deep supervision traces an accuracy--robustness frontier: it does not improve nominal accuracy, but degrades gracefully far beyond the training horizon, where naive recurrence collapses to near-random performance.

Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier $0$ under projected restriction and $0.72$ under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a $\widetilde{O}(n^{-1})$ excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.

Generative Flow Networks (GFlowNets) have emerged as a flexible framework for amortised inference over discrete and mixed discrete-continuous objects, requiring only an unnormalised target density specified through a reward. In this work, we formulate forward-policy training in GFlowNets through the information geometry of the induced trajectory sampler. Treating the forward policy as an induced trajectory sampler, we show that its intrinsic first-order geometry is given by the Fisher-Rao metric of the trajectory family, and that the associated natural gradient provides the canonical local update whenever the corresponding Fisher information is computable or accurately approximable. We derive an exact decomposition of the trajectory Fisher into per-step conditional second moments, which clarifies when temporal score interactions vanish and when dense couplings remain under shared parameterisation. This leads to three computational regimes: settings with tractable exact Fisher information, settings where Monte Carlo estimators of the expected Fisher are sufficient, and structure-exploitable settings in which target locality or factorisation yields accurate approximations of the Fisher expectation. In the latter case, graphical-model tools such as exact marginalisation, separator methods, and belief propagation provide principled surrogates for natural-gradient updates. The resulting framework turns target structure into optimisation geometry and yields a tractable route to structure-aware forward-policy training in GFlowNets. We illustrate the framework empirically through examples comparing convergence and exploration behaviour under Riemannian and Euclidean optimisation.

3D multi-person motion prediction requires modeling both individual kinematics and inter-person interactions. While Flow Matching is effective for multi-hypothesis generation to improve prediction accuracy, directly predicting skeletal sequences from pure noise often compromises structural consistency and introduces unreliable cross-agent interactions during early noise-dominated integration steps. To address this, we propose a Prior-Guided Residual Flow Matching framework. First, a Deterministic Coarse Prior (DCP) establishes a kinematic anchor, formulating the generative process as a conditional flow over motion residuals to simplify the generative objective and preserve structural stability. Second, a Dynamic Cross-Interaction (DCI) mechanism temporally synchronizes inter-agent message-passing with the integration progress, ensuring the extraction of reliable social contexts and improving multi-person motion fidelity. Finally, a decoupled joint-motion architecture with bidirectional fusion effectively preserves fine-grained kinematic coherence. Extensive experiments demonstrate that our approach achieves state-of-the-art prediction accuracy across multiple datasets. Code is available at https://github.com/Wei-Wei-a/Residual-Flow-Matching-with-Dynamic-Cross-Interaction-for-3D-Multi-Person-Motion-Prediction.

Recent data-driven methods for synthesizing 6-DoF grasp poses use generative models to learn complex grasp pose distributions and generate diverse candidate poses. In particular, SE(3)-equivariant flow-based models generate grasp poses that transform consistently with object rotations and translations. However, these methods sample by iterative numerical integration, requiring tens of function evaluations per grasp and limiting their use in real-time manipulation. We propose GraspMeanFlow, an SE(3)-equivariant MeanFlow framework for few-step 6-DoF grasp generation. Our method learns the average velocity over a finite time interval, defined through the time-ordered exponential so that it reproduces exactly the rigid-body displacement accumulated over that interval. We prove that a point-cloud-conditioned distribution transported by an equivariant average-velocity flow map remains invariant, so equivariance is retained under few-step sampling, and we condition the field on a pair of times by lifting both to equivariant vectors, leaving the backbone otherwise unchanged. For stable training, we pair a flow-matching boundary term with either of two consistency terms: the differential MeanFlow identity, whose target requires a Jacobian-vector product, or an equivalent semigroup loss that avoids it. Experiments on ACRONYM show that a single function evaluation of GraspMeanFlow reaches the EMD that an iterative SE(3) flow model needs five steps to approach, that a second instantiation of the same framework improves grasp success by up to 24.3 points in the few-step regime, and that both generate grasp distributions transforming exactly with the object.

Motion priors improve reinforcement learning for physics-based humanoid tracking, but temporal priors require ordered motion clips, while pose priors provide limited guidance for policy-induced pose transitions. We present Pose Flow Matching for Humanoid Robots (PFM-HR), a reusable flow matching prior trained directly on large scale unordered pose data. PFM-HR introduces the Pose Geometry Score (PGS), which quantifies how joint coordinate changes during rollouts align with the local geometry of pose variation captured by the prior. Using PGS to modulate the tracking reward guides policy exploration toward structured pose changes while keeping the prior frozen across tracking tasks. Experiments demonstrate that PFM-HR improves both single motion and general motion tracking, especially for highly dynamic motions.

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying \textit{c}-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative \textit{c}-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative \textit{c}-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension $d \ge 3$ and a nearly parametric rate for $d=1,2$.

We propose an instantiation of flow matching that relies on a time-independent velocity field (an \emph{autonomous flow}) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.

Symbolic alpha factor discovery can score a completed expression, but it provides no direct label for the structural decisions that produced it. Generative flow networks (GFlowNets) preserve a diverse, reward-proportional distribution over complete expressions, yet their trajectory-level objective does not compare unchosen sibling actions at an intermediate state. We introduce AlphaG-OPD, a structural on-policy distillation framework that turns terminal factor evaluations into local action guidance. Its design separates three decisions. Component I determines where to teach by exposing grammar-valid siblings at partial abstract-syntax-tree (AST) states visited by the current forward policy. Component II determines what is reliable enough to teach: it evaluates three supported siblings under four shared suffixes and admits a KL-bounded target only when their matched comparisons exhibit sufficient winner agreement and a positive empirical lower confidence bound (LCB). Component III determines how strongly and for how long to teach by consolidating accepted targets through bounded replay, score-indexed expiry, and forward-gradient balancing, without additional factor evaluations. Terminal reward, Trajectory Balance, the backward policy, grammar, and factor-pool rules remain unchanged. An equal-physical-score four-arm ablation tests paired teaching, reliability gating, and consolidation. Across China's CSI300, CSI500, and CSI1000 and the U.S. S&P 500, the complete method delivers strong cross-market performance over multiple random seeds.

This article presents a modular inference framework that integrates Flow Matching generative models with formal Control Barrier Function (CBF) safety guarantees. Unlike existing methods that apply external safety filters to a model's final output, our approach modifies the Flow Matching denoising process within the model to inherently generate safe trajectories. By employing a smooth Log-Sum-Exponential aggregate barrier, we enforce safety over entire action chunks. This aggregate barrier ensures a minimal increase in computational overhead and does not alter the semantic intent of the model. We show that, within the proposed framework, the 2-Wasserstein distance between the generated distribution and the target distribution remains bounded. Our method eliminates the need for safety-specific datasets or costly model retraining, providing a versatile solution for safe inference. We validate the approach on two robotic manipulation platforms and a 2D navigation benchmark, verifying that our framework achieves reliable safety without degrading the success rate of the model.

Although world-action models (WAMs) enhance long-horizon robot control by predicting visual evolution before acting, long-horizon reliability demands repeated re-grounding in real observations--not recursive rollout. Existing WAMs address this by refreshing history or KV cache with ground-truth data between chunks. However, such chunk-wise feedback operates at a coarse temporal granularity and thus fails to correct prediction errors at the individual time-step level. To address this, we propose Feedback Flow Matching (FBFM), a training-free inference mechanism that pushes re-grounding inside the actively generated chunk. During flow matching, FBFM applies a masked pseudoinverse correction to the conditional velocity field: it leverages the preceding action chunk to guide generation of the next action chunk, and uses the image observed after executing that preceding chunk to guide the next frame prediction. This cross-chunk pairing--where feedback from one chunk arrives in time to shape the next--creates an asynchronous loop that corrects errors without waiting for chunk boundaries. Being training-free, the mechanism improves responsiveness to unexpected events and suppresses drift in long-horizon tasks. We evaluate FBFM on both a joint-generation WAM (DreamZero) and a stage-wise WAM (LingBot-VA). On selected LIBERO and RoboTwin2.0 tasks, it improves success rates by over 5% in favorable settings, and real-world robot observation-prediction diagnostics show notably better tracking. We argue that FBFM offers a new paradigm for fine-grained online correction, bridging open-loop flow generation with closed-loop real-world dynamics.

Text-to-motion generation must produce motions that are semantically correct, temporally coherent, and physically plausible. A natural approach is to first project motion data into a structured semantic space and then train a generative model within that space. Such a paradigm has been highly successful in image generation through Representation Autoencoders (RAEs), where a frozen self-supervised encoder provides semantic features for diffusion or flow models to learn from. However, direct transfer of such a paradigm to motion space using Motion-JEPA as the frozen encoder fails dramatically. We diagnose this failure geometrically and identify two motion-specific bottlenecks: (1) the JEPA feature space is spectrally ill-conditioned, making the Gaussian-to-data transport unstable; and (2) even with a well-conditioned spectrum, flow residuals tend to align with decoder-sensitive directions, where small latent errors are amplified into large motion artifacts after decoding. Based on these insights, we propose MoRAE. MoRAE addresses the two bottlenecks separately. A compact bottleneck distills the structured JEPA representation while removing weak and redundant directions, bringing the latent spectrum into a transport-stable regime. Motion-coupled training then aligns the retained latent geometry with the decoder, making characteristic flow errors less costly after decoding. With this flow-friendly latent, a standard non-autoregressive Flow-Matching DiT achieves state-of-the-art performance.

Dynamic matching markets require decisions about whom to match and when: matching now yields value but removes participants who may create better future opportunities. We develop a value-based reinforcement-learning framework for this problem on finite, evolving weighted graphs. We study an infinite-horizon continuous-time model with stochastic arrivals, node-type transitions, edge realizations, and exogenous exits. We prove an event-time reduction: without loss of optimality, the planner acts immediately after each exogenous event and then waits for the next one. We further show that the optimal edge-wise $Q$-function is characterized by a single continuation-value function on post-decision residual graphs, reducing the learned object from state-action values to graph values. Exact action selection still requires combinatorial matching optimization; we approximate the value with a graph neural network, train it by temporal-difference learning, and use it in a forward-greedy matching heuristic. In a binary-type benchmark, the learned policy substantially outperforms immediate and threshold-greedy rules by preserving common nodes for rare arrivals of valuable matches while forming lower-value matches only in thick pools. In a kidney paired donation benchmark, it performs similarly to immediate greedy when exits are unpredictable, recovers the logic of patient matching when warnings are reliable, and outperforms the better of Immediate Greedy and Patient Greedy across intermediate warning probabilities. These results show that residual-graph value learning yields state-dependent dynamic matching policies that adapt to realized connectivity and exit information.