Velocity Field

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.

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.

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.

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.

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.

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.

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.

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.

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.

In X-ray CT, metallic objects cause beam hardening, photon starvation, and scattering, leading to projection inconsistency, streaks, dark bands, and structural distortions that compromise clinical diagnosis and quantitative analysis. Existing metal artifact reduction (MAR) methods remain limited: optimization-based methods may leave residual artifacts or blur structures, regression networks may generalize poorly across scenarios, and generative models without sample-specific structural guidance and physical constraints may produce anatomically inconsistent structures. Flow Matching learns a continuous-time velocity field that deterministically transports a source distribution to a target distribution, providing a flexible MAR prior. However, standard unconditional Flow Matching does not exploit sample-specific structure, spatially nonuniform metal-induced degradation, or measured projections. To address these limitations, we propose SCMA, a structure-conditioned and metal-aware Flow Matching framework. First, a linear-interpolation-corrected image is fed into the velocity network with the intermediate state as a sample-specific structural condition, guiding inference toward artifact-free CT images while preserving anatomy. Second, time-varying spatial weights from the metal mask and its distance transform are incorporated into the Flow Matching loss to emphasize severe degradation within and around metal regions. Finally, conditional Flow Matching updates alternate with projection-consistency correction during inference, allowing reliable measurements outside metal traces to constrain predictions. Experiments on simulated and real CT data demonstrate that SCMA more effectively suppresses metal artifacts, preserves local anatomical structures, and reduces hallucination-like structures inconsistent with projection measurements than representative MAR methods.

Flow matching (FM) has become a popular action head paradigm for modern embodied models. However, as a conditional generative model, it does not explicitly expose its inherent uncertainty, producing faulty action chunks even when it misinterprets the scene or encounters out-of-distribution (OOD) inputs. Therefore, determining when an FM-generated action can be trusted is essential for safe deployment, yet existing uncertainty estimation methods on real-time control suffer from several issues: extra training budget, high computational overhead, and low generalization ability. In this work, we provide a geometric interpretation of FM uncertainty in the velocity field, showing that uncertainty manifests as deviation from an ideal affine-isotropic contraction field. Building on this observation, we introduce denoising acceleration ($\mathrm{accel}$), a highly-generalizable and cost-free uncertainty proxy that measures the bending of the denoising trajectory from a single forward pass, without additional model evaluations, training, or resampling. We theoretically and empirically demonstrate that $\mathrm{accel}$ is a faithful proxy for FM uncertainty and further test its utility in online failure detection. Results show that $\mathrm{accel}$ identifies failing rollouts well before termination, matching or even outperforming costly resampling- and training-based baselines across settings under realistic deployment budget. Code and demos available at: https://github.com/rrrrrrzy/fm-geometry.

In the physical world we inhabit, space and time are fundamentally continuous. However, existing machine learning paradigms for world modeling are largely confined to discrete-time prediction, thereby exhibiting significant inefficiency in capturing the dynamics of physical world. We introduce Physical-Time Flow (PT-Flow), a novel approach that learns a continuous latent velocity field operating in physical time. Crucially, the underlying dynamics of sequential data are parameterized by an ordinary differential equation (ODE) embedded in a well-structured representation space. Under this paradigm, the prediction of future can be recast as temporal integration via an ODE solver in the compressed latent space. Building upon PT-Flow, we construct ODEWorld, a continuous-time latent world model that is both efficient and versatile. By extracting time-variant features and enforcing ODE properties on both the dynamical representation space and the latent velocity field, ODEWorld effectively addresses the long-standing representation collapse issue in latent world model literature. This also enables high-quality image reconstruction even after long-horizon prediction. Moreover, its continuous nature allows for arbitrary temporal resolution and even backward prediction, which is impossible for most discrete-time models. Lastly, ODEWorld can provide rich planning-oriented information to facilitate downstream policy learning. Comprehensive experiments demonstrate that ODEWorld successfully reconciles planning-conducive dynamics abstraction with visual realism, excelling in both video generation and robotic control. Project page: https://odeworld.github.io/.

Modern generative models are increasingly trained using model-generated signals, creating both opportunities for self-improvement and risks of collapse. We study optimal self-distillation (SD) for rectified flow (RF): given a suboptimal teacher velocity field, can a student trained on a mixture of true RF velocities and teacher velocities provably improve the teacher? For linear RF with ridge regularization on fixed interpolation pairs, we prove an exact affine path identity, derive the optimal mixing coefficient in closed form, and show strict improvement in integrated velocity risk whenever the teacher risk is nonstationary along the regularization path. The optimal coefficient obeys a sign rule: positive mixing corrects under-regularized teachers, while negative mixing corrects over-regularized teachers. We also give one-shot generalized cross-validation (GCV) and validation tuning procedure that avoids grid search over mixing weights and repeated refitting. Combining this theorem with RF Wasserstein convergence bounds, we show that optimal self-distillation improves the velocity estimation terms controlling continuous-time and finite-step generation error. Experiments with Gaussian models, Gaussian mixtures, and image data show that optimal self-distillation improves velocity risk, mode recovery, and finite-step generation relative to both the teacher and pure distillation.

We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

Color transfer aims to align the color distribution of a source image with that of a reference image while preserving structural and semantic consistency. However, existing methods often suffer from inaccurate global mapping, semantic misalignment, and visual artifacts. To address these issues, we propose ColorFM, an optimization-to-learning framework. ColorFM connects online optimization to offline inference by reformulating color transfer as the transport of pixel distributions along velocity fields via Flow Matching. Specifically, we introduce ColorFM-O, an instance-specific optimization scheme that fits the velocity field through hierarchical color coupling guided by semantic priors. By numerically integrating the induced flow trajectories, ColorFM-O produces precise and semantically consistent color transfer results, while generating high-quality paired data as pseudo-supervision. Building upon this, we design ColorFM-L, an efficient feed-forward model trained on the generated pairs. Through implicit state modeling, ColorFM-L extracts deep semantic features to predict flow parameters for bidirectional linearized transport, ensuring accurate color transfer. Extensive experiments demonstrate that ColorFM-L outperforms state-of-the-art methods in visual quality, structural fidelity, and semantic consistency, successfully combining the accuracy of optimization with the speed of feed-forward inference.

We present FM-ChangeNet, a pathwise-supervised framework for change detection that reformulates bi-temporal reasoning as continuous transport in feature space rather than static endpoint comparison. Given encoded pre and post-temporal representations, we construct intermediate latent states and learn a time-conditioned velocity field $\hat{v}_θ(z_t,t)$ along the transformation trajectory. This pathwise formulation constrains the predictor over a continuum of intermediate states, providing a denser and less ambiguous supervision signal than conventional endpoint-only segmentation and enabling the model to capture temporal evolution explicitly. The learned velocity field is not only a transport mechanism but also an interpretable representation of change: its magnitude serves as a spatially localized change cue that helps distinguish true structural variation from nuisance effects such as illumination shifts and spatial misalignment. We develop a hierarchical multi-scale architecture with cross-temporal alignment, time-conditioned coarse-to-fine flow decoding, and a unified objective that couples flow supervision, trajectory consistency, spatial regularization, and segmentation loss. Experiments on remote sensing benchmarks show that the proposed framework produces more structured and robust change representations while achieving state-of-the-art performance.

Vision-Language Navigation has increasingly emphasized high-level instruction reasoning, memory, global map construction, and instruction decomposition, while the low-level action representation remains comparatively underexplored. We propose CoFL-S, a low-level vision-language-action framework that predicts a language-conditioned flow field over the robot's local visible sector and generates continuous trajectories by rolling out the predicted field. To train this low-level representation, we convert each VLN-CE episode, originally a whole-episode instruction paired with an action sequence, into frame-level local supervision with aligned sub-instructions and matched action, trajectory, and dense flow-field targets. For evaluation, we introduce a continuous-time Habitat benchmark that isolates low-level action interfaces from instruction decomposition and executes all methods through a shared velocity-command controller, enabling decomposition-independent closed-loop comparison across different planner frequencies rather than fixed discrete forward-and-turn transitions in VLN-CE. Under matched encoders and training settings, CoFL-S consistently outperforms action-token and action-chunk baselines across planner frequencies in the continuous-time Habitat benchmark, and zero-shot real-world closed-loop deployment further shows its advantage over both baselines beyond simulation.

Aligning generative flow models on continuous spaces via online reinforcement learning is constrained by intractable trajectory likelihoods. Existing density-approximated policy gradient methods rely on stochastic SDE samplers to construct tractable transition kernels, which introduce training-inference inconsistencies and necessitates Classifier-Free Guidance (CFG). While implicit frameworks such as DiffusionNFT directly optimize forward-process velocity fields, its heuristic fixed-magnitude corrections prevent optimization strength from relative intra-group quality. We propose \textit{Flow Advantage-Weighted Rectification} (\textbf{FlowAWR}), a paradigm that recasts continuous generative policy optimization as supervised regression toward a theoretically optimal velocity field. Starting from the optimal policy of a KL-constrained reward maximization, FlowAWR derives the optimal velocity field that admits a magnitude-aware, advantage-weighted rectification form, yielding SDE-free optimization and CFG-free generation. In comparative evaluations on SD3.5-Medium, FlowAWR achieves improved alignment performance alongside a 2$\times$ to 5$\times$ convergence acceleration over DiffusionNFT (e.g., reaching a 24.12 PickScore in 1.2k steps, versus 23.82 in 2.0k steps for DiffusionNFT and 23.50 in $>$4k steps for FlowGRPO). Under multi-reward constraints, FlowAWR sustains generation quality, satisfying structural rules while maintaining stable out-of-domain performance.

Reconstructing dynamic 3D scenes from monocular videos is challenging due to scene complexity and temporal dynamics. With the advancement of 3D Gaussian Splatting in novel view synthesis, existing methods extend 3D Gaussians to 4D domain with deformation fields, trajectories or spatiotemporal 4D volumes to model scene element deformation. However, these methods suffer from long training time, low rendering speed or high memory consumption for per-frame reconstruction of 4D volumes, without fully exploiting dense dynamic information. To address this issue, we propose Flow Splatting, which constructs the velocity field and enables the conventional splatting technique to render optical flow from the velocity field to supervise dynamics learning process from monocular videos. Specifically, we extend 4D volumes with time varying means and covariance to represent complex dynamics. Then, we construct and approximate the velocity field naturally based on this representations. While conventional volume rendering techniques support to render color fields, we extend the volume rendering strategy to splat the velocity field by considering the influence of camera motions. We conduct experiments on various benchmarks to demonstrate the efficiency and effectiveness of our method. Compared to the state-of-the-art methods, our model achieves better image quality with less time consumption and higher rendering speed.

Urban flow and air-quality simulations generate high-dimensional datasets describing velocity and pollutant transport across multiple spatial, temporal, and physical-variable dimensions. Reconstructing these fields from sparse sensor measurements is a fundamental challenge in environmental monitoring, digital twins, forecasting, and data assimilation. Existing low-cost reconstruction approaches are commonly based on matrix decompositions, which require multidimensional datasets to be flattened into two-dimensional snapshot matrices, thereby discarding important structural information. This work introduces the low-cost High-Order Singular Value Decomposition (lcHOSVD), a novel tensor-based sparse-sensing reconstruction framework for high-dimensional environmental fields. To the authors' knowledge, this is the first methodology that combines sparse sensing and HOSVD for field reconstruction. Unlike matrix-based approaches, lcHOSVD preserves the natural tensor structure of the data, enabling the exploitation of correlations across spatial, temporal, and physical-variable dimensions while substantially reducing the computational requirements of conventional HOSVD. The methodology is applied to urban flow and air-quality datasets, where three-dimensional velocity and pollutant concentration fields are reconstructed using only 1-4% of the available spatial locations. While lcSVD provides larger computational speed-ups, lcHOSVD consistently achieves lower reconstruction errors in configurations characterized by strong multidimensional coupling and heterogeneous dynamics across dimensions. Additional sensor-anisotropy analyses demonstrate that the tensor formulation is significantly more robust to uneven sensor distributions, a common situation in practical environmental monitoring networks.

A growing family of training-free solvers -- FlowDPS, FLOWER, PnP-Flow and their diffusion ancestors (DPS, DAPS) -- repurpose a pretrained flow-matching prior to solve imaging inverse problems by adding a measurement-guidance term to the deterministic probability-flow ODE. Despite strong empirical results, what these per-step corrections actually approximate -- and how far the resulting samples are from the true posterior $p(x\mid y)$ -- has not been characterized. We give a posterior-transport account of flow-based inverse problem solving. Our starting point is a simple but consequential fact: for a \emph{deterministic} flow prior, Bayesian conditioning is realized entirely by a \emph{reweighting of the source distribution}, not by a drift correction; pushing the reweighted source through the \emph{unmodified} velocity field yields exact posterior samples. From this we show that trajectory-guidance solvers can be read as the minimum-kinetic-energy \emph{correction} field needed to morph the unconditional source into the posterior, and that FlowDPS / FLOWER / PnP-Flow correspond to distinct zeroth-order / Gaussian / proximal approximations of this single object; we bound the resulting posterior bias in Wasserstein distance. A controlled $2$D study with a closed-form posterior confirms the theory decisively: source reweighting matches the true posterior to the Monte-Carlo floor on every metric, whereas trajectory guidance incurs $200$--$800\times$ larger error and collapses posterior modes, \emph{regardless of guidance strength}. Guided by the analysis we propose a cheap, principled velocity-correction solver that is competitive across two in-domain priors (AFHQ, CelebA) and two out-of-distribution settings while, unlike point-estimate source-space optimizers, producing diverse posterior samples with uncertainty that correlates with reconstruction error.

In this work, Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

Cross-embodiment data have become central to training robotic foundation models. To leverage such heterogeneous data, we focus on flow-based object manipulation, where robot flows (robot velocity fields) serve as embodiment-agnostic motion representations. Previous studies do not formulate robot flows as dense velocity fields, but as displacements of sparse keypoints, even though dense velocity fields better match the continuous-time nature of motions. To address this, we propose Flow as Flow, a framework that models robot flows as probability flows based on a flow matching formulation. By naturally modeling such velocity fields within this formulation, our method achieves efficient and high-quality robot flow generation. Across standard benchmarks, our method outperforms representative baseline methods on standard metrics, while achieving approximately 24$\times$ faster generation than standard flow matching. Furthermore, through real-world experiments evaluating 9 methods with 260 trials per method across 13 manipulation tasks, we show that our method achieves a higher average success rate than the baseline methods.

Diffusion models have achieved impressive empirical success in generative tasks, and their convergence theory is now relatively well understood. Motivated by privacy and scalability, recent decentralized diffusion architectures replace a single global velocity field with multiple local experts and a routing mechanism, yielding a sampling dynamics with stochastic expert switching that falls outside standard diffusion convergence analyses. In this work, We study a decentralized diffusion framework with stochastic velocity fields and ODE-based sampling. We establish a convergence guarantee in Wasserstein-2 distance, showing that the distribution of the $N$-step discretization converges to the analytical solution at rate $\mathcal{O}(N^{-1/2}+\varepsilon)$ in $W_2$, where $\varepsilon$ captures the neural approximation errors. To our knowledge, this is the first $W_2$ convergence result for decentralized diffusion models with an ODE-based sampling scheme.

In this work, we develop theoretical foundation for flow matching with neural-network-parameterized conditional velocity fields. We establish convergence guarantees for gradient descent in the over-parameterized 2-layered ReLU neural network regime. We derive generalization bounds for the conditional velocity-field matching objective. Building on these results, we provide Wasserstein-distance guarantees for the samples generated by the induced flow. Our analysis is based on generalization bound for multi-task representation learning with unbounded losses, which may be of independent interest beyond flow-based generative modeling. These theoretical results are validated through extensive experiments on both synthetic and real-world image benchmarks.

Wall shear stress (WSS) governs near-wall transport dynamics and is a key hemodynamic indicator in cardiovascular flows, yet remains difficult to infer accurately due to the need for precise computation of near-wall velocity gradients. Passive scalar fields, such as concentration or temperature, are advected by the same underlying velocity field and have the potential to uncover hidden flow physics metrics such as WSS. In this work, we demonstrate such reconstruction from spatially limited passive scalar observations using two fundamentally different inverse frameworks: a differentiable physics framework based on discrete adjoint, PDE-constrained optimization, which enforces the governing equations as hard constraints, and physics-informed neural networks (PINNs), which treat them as soft constraints. Benchmark problems include a 2D canonical backward-facing step (2D-BFS) and a 3D patient-specific stenotic coronary artery. For the 2D-BFS case, evaluated under three measurement scenarios (near-wall, far-field, and combined), PINN achieves high accuracy when near-wall data are available but fails when restricted to far-field measurements, whereas the differentiable physics approach recovers accurate WSS across all scenarios. In the 3D patient-specific case, the differentiable physics framework outperforms PINNs, yielding accurate WSS reconstruction. These results establish that measurement location and inverse formulation jointly determine reconstruction fidelity in scalar-based near-wall flow inference. The proposed framework opens a path toward estimation of near-wall hemodynamics from scalar transport data, with broader applicability to fluid flow problems where passive scalars can be observed.

We propose a method combining Multi-Agent Deep Reinforcement Learning (MARL) and eXplainable Deep Learning (XDL) to reduce drag in wall-bounded turbulent flows. Taking as a baseline the results of training agents directly targeting wall-shear stress and opposition control, three SHAP-guided approaches are compared. In the first, the reward is computed from SHAP attributions of a U-net predicting the future velocity field; in the second, from SHAP attributions of a U-net predicting the skin-friction coefficient; in the third, from a combination of SHAP attributions of two U-nets predicting the skin-friction coefficient and the wall pressure fluctuations, respectively. The combined SHAP strategy based on skin-friction coefficient and wall-pressure fluctuations achieves the best overall performance, achieving a DR of 34.44% and a NES of 34.01% with only 0.43% normalized input power. Relative to opposition control, drag reduction and net energy saving increase by 49.41% and 48.52%, respectively. Compared with the direct wall-shear-stress baseline, the proposed strategy simultaneously improves performance while reducing the normalized actuation cost from 5.90% to 0.43%. Analysis of the results reveals that the energetically efficient policy is consistent with pressure-gated actuation, activating predominantly at near-zero wall pressure, and operates on a temporal timescale comparable to the lifetime of the near-wall turbulent structures.

Diffusion and flow matching have emerged as expressive policy classes in reinforcement learning, but their reliance on multi-step denoising imposes substantial computational overhead at inference time, which is particularly problematic in online RL. MeanFlow offers a promising alternative by learning an average velocity field that maps noise to data in a single network evaluation. However, MeanFlow typically requires samples from the target distribution to construct its target velocity field, which are unavailable in online RL. We propose Score-Based One-step MeanFlow Policy Optimization (SOM), an actor-critic algorithm that resolves this by constructing the target velocity field directly from the Q-function via score estimation and a probability flow ODE, thereby concentrating probability mass on high-value modes. In the fully online RL setting, SOM achieves state-of-the-art performance on locomotion tasks with a single generation step, while substantially reducing both training and inference time compared to prior diffusion- and flow-matching-based policies.

We analyze finite-particle drifting models for one-step generative modeling. For a conservative velocity given by the difference of the kernel-smoothed data and model scores, a joint-entropy identity yields continuous-time bounds for the smoothed Fisher discrepancy and the squared particle velocity. The finite-particle correction involves reciprocal kernel density estimates. We give local-occupancy conditions and exact expectation bounds, and show that these expectations diverge for full-support initial densities in a shrinking-bandwidth regime. Keeping the bandwidth dependence of the quadrature constants explicit yields a conditional root residual-velocity rate of $N^{-1/(d+4)}$ under uniform regularity, with a corresponding rate under weaker quadrature growth conditions. We also analyze the original displacement field with the exact Laplace kernel. A companion kernel gives a weighted coercivity estimate that accommodates the unbounded empirical scale and the kernel's nondifferentiability. Localization controls the particle velocity, while relative scale alignment absorbs the mismatch into dissipation. Both analyses quantify the size of an additional drift correction under explicit regularity and stability assumptions.

Single-cell RNA sequencing (scRNA-seq) provides high-dimensional profiles of cellular states, enabling data-driven modeling of cellular dynamics over time. In practice, time-resolved scRNA-seq is collected at only a few discrete time points as unpaired snapshot populations, leaving substantial temporal gaps. This motivates trajectory inference at unmeasured time points. Existing methods mainly follow two directions, optimal-transport (OT) alignment provides distribution-level matching between observed snapshots, while continuous-time generative models support forecasting via learned dynamics. However, two challenges remain: (i) unpaired snapshots render local transitions between adjacent time points ambiguous, leading to unstable supervision; and (ii) long-horizon prediction relies on repeated integration, where small modeling errors compound and cause distribution drift. To address these challenges, we propose single-cell Flow Matching (scFM), a latent generative framework based on coupling-conditioned flow matching. First, we compute entropically regularized OT couplings between adjacent snapshots and use them to construct soft, weighted flow-matching targets for learning time-dependent velocity fields. Second, we learn bidirectional velocity fields and leverage their consistency to refine couplings and improve temporal coherence under sparse supervision. Third, we introduce distribution-level alignment and latent dynamic regularization to anchor long rollouts and mitigate drift. Experiments on real-world time-series scRNA-seq datasets show that scFM consistently improves distributional prediction performance for both temporal interpolation and extrapolation. Moreover, scFM yields more accurate trajectory reconstruction and temporally coherent visualizations where intermediate time points are absent, indicating a more faithful recovery of underlying temporal gene expression dynamics.

Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

We develop a unified theoretical framework for data-free one-step sampling from unnormalized target distributions based on Wasserstein gradient flows. For a broad class of standard f-divergence objectives, we show that the induced velocity field admits the universal form $\mathbf{V}(x)=w(r(x))\,β(x)$, where $β(x)=\nabla \log (p(x)/q(x))$ is shared across objectives and $w$ is determined solely by the choice of divergence. This decomposition shows that standard f-divergence drifts share the same asymptotic target distribution $p$ and differ primarily in how they redistribute transient repair effort across under-covered regions. To formalize this distinction, we derive a one-step regional-response theory for a soft under-coverage functional and obtain a compression--elasticity identity that links divergence choice to the geometry of mass transport into under-covered regions. We further extend the framework beyond the f-divergence family to the Log-Variance (LV) divergence, analyze how the reference distribution alters the resulting drift structure, and motivate a practical LV-inspired surrogate for data-free training. Based on this theory, we instantiate the framework with a KDE-based implementation and describe a complementary normalizing-flow route, enabling one-step inference after training. Experiments on multimodal Gaussian-mixture benchmarks are consistent with the theoretical predictions and demonstrate effective one-step sampling on these targets.

Classifier-free guidance (CFG) is the primary control over how strongly text semantics move a flow-based sampler, yet standard practice holds its scale fixed across the entire ODE trajectory. This is a fundamental mismatch: early steps are noise-dominated and carry weak semantic signal, while late steps commit image structure and demand stronger directional commitment; more critically, the value of any guidance strength depends on whether the guided velocity is consistent with the model's current dynamics or working against them. We propose \textit{Velocity-Adaptive Guidance Scale} (VAGS), a training-free replacement that multiplies the nominal scale by a bounded factor combining a temporal signal-level term with the cosine similarity between task-relevant velocity fields. For inversion-free editing, VAGS measures the alignment between source- and target-guided velocities, so edit strength at each step reflects local compatibility between preservation and transformation. For generation, VAGS-Gen uses the alignment between unconditional and conditional velocities as the analogous signal. Neither variant requires fine-tuning, auxiliary networks, or extra forward passes, and fixed CFG is recovered as a special case. On PIE-Bench and DIV2K for editing, and COCO17, CUB-200, and Flickr30K for generation, VAGS consistently improves structural fidelity and generation quality over fixed CFG and recent training-free guidance variants. The code is publicly available at https://github.com/Harvard-AI-and-Robotics-Lab/Velocity_Adaptive_Guidance_Scale.

Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path itself. Existing constructions, including rectified and optimal-transport-based paths, transport samples along straight lines between coupled endpoints and thus cover only a narrow class of dynamics. We observe that this corresponds to the simplest case of the least-action principle in classical mechanics, in which the kinetic Lagrangian yields free-particle straight-line trajectories. Building on this observation, we propose Lagrangian flow matching, a physics-based framework in which the probability path and velocity field are determined by minimizing the action of a general Lagrangian subject to the continuity equation and the prescribed endpoints. We show that this dynamic problem admits an equivalent static optimal transport (OT) formulation, yielding a family of simulation-free training objectives that recover OT-based flow matching as the kinetic special case and the trigonometric variance-preserving diffusion path as the harmonic-oscillator case. More general Lagrangians give rise to new probability paths and velocity fields, and numerical experiments show that they induce meaningful changes in the learned dynamics while remaining competitive with existing conditional flow matching models.

Generative models are often conditioned on a small set of examples via cross-attention. Under the Gaussian optimal-transport path, we show that the exact velocity field induced by a finite support set is a Nadaraya--Watson kernel smoother whose bandwidth decreases with flow time, from broad averaging at early steps to nearest-neighbor at late steps. A single Gaussian-kernel attention head exactly computes this field, connecting cross-attention conditioning to classical kernel theory. The theory predicts three failure regimes: nearest-neighbor collapse of the kernel at high dimension, mismatch between the isotropic kernel and the data geometry, and insufficient support for nonparametric estimation. Experiments on Gaussian mixtures, spherical shells, and DINOv2 ImageNet features confirm that learned conditioning improves in precisely these regimes, and that IP-Adapter's cross-attention implements approximate NW smoothing in practice.

Flow-based generation in high-dimensional spaces is difficult because velocity prediction requires modeling high-dimensional noise, even when data has strong low-rank structure. We present Asymmetric Flow Modeling (AsymFlow), a rank-asymmetric velocity parameterization that restricts noise prediction to a low-rank subspace while keeping data prediction full-dimensional. From this asymmetric prediction, AsymFlow analytically recovers the full-dimensional velocity without changing the network architecture or training/sampling procedures. On ImageNet 256$\times$256, AsymFlow achieves a leading 1.57 FID, outperforming prior DiT/JiT-like pixel diffusion models by a large margin. AsymFlow also provides the first-ever route for finetuning pretrained latent flow models into pixel-space models: aligning the low-rank pixel subspace to the latent space gives a seamless initialization that preserves the latent model's high-level semantics and structure, so finetuning mainly improves low-level mismatches rather than relearning pixel generation. We show that the pixel AsymFlow model finetuned from FLUX.2 klein 9B establishes a new state of the art for pixel-space text-to-image generation, beating its latent base on HPSv3, DPG-Bench, and GenEval while qualitatively showing substantially improved visual realism.

Mobile manipulation requires coordinated control of high-dimensional, bimanual robots. Imitation learning methods have been broadly used to solve these robotic tasks, yet typically ignore the bilateral morphological symmetry inherent in such systems. We argue that morphological symmetry is an underexplored but crucial inductive bias for learning in bimanual mobile manipulation: knowing how to solve a task in one configuration directly determines how to solve its mirrored counterpart. In this paper, we formalize this symmetry prior and show that it constrains optimal bimanual policies to be ambidextrous and equivariant under reflections across the robot's sagittal plane. We introduce a $\mathbb{C}_2$-equivariant flow matching policy that enforces reflective symmetry either via a regularized training loss or an equivariant velocity network. Across planar and 6-DoF mobile manipulation tasks, symmetry-informed policies consistently improve sample efficiency and achieve zero-shot generalization to mirrored configurations absent from the training distribution. We further validate this zero-shot generalization capability on a real-world manipulation task with a TIAGo++ robot. Together, our findings establish morphological symmetry as an effective, generalizable, and scalable inductive bias for ambidextrous generative policy learning.

Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan $\overlineπ_{k}$, obtained by averaging empirical OT plans over independent minibatches of size $k$. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of $\overlineπ_{k}$ to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments.

Flow matching models generate samples by numerically integrating a learned velocity field, with each integration step requiring a neural network evaluation. Fast generation therefore requires using a small fixed evaluation budget effectively: the key question is not only how to integrate the flow, but where the sampler should spend its steps. We propose SharpEuler, a training-free sampler that profiles a pretrained model offline by estimating where the learned velocity field changes most rapidly along calibration trajectories. This finite-difference estimate defines a solver-aware sharpness profile, which is smoothed and converted by a quantile transform into a timestep grid for any desired inference budget. At test time, sampling remains ordinary Euler integration with the same number of model evaluations as a uniform schedule. We justify SharpEuler using three principles: a numerical principle identifying trajectory acceleration as the leading source of Euler discretization error, a variational principle deriving sharpness-based power-law timestep densities, and a statistical guarantee showing that the finite-sample calibrated sampler is stable at the terminal distribution level. Our experiments show that SharpEuler improves sample quality at fixed budgets, reducing inter-mode leakage and increasing mode coverage.

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.

In this paper, we aim to jointly model the geometry, appearance, and physical information of 3D scenes solely from dynamic multi-view videos, without relying on any physical priors. Existing works typically employ physical losses merely as soft constraints or integrate physical simulations into neural networks; however, these approaches often fail to effectively learn complex motion physics. Although modeling velocity fields holds the potential to capture authentic physical information, due to the lack of appropriate physical constraints, current methods are unable to correctly learn the interaction mechanisms between rigid and non-rigid particles. To address this, we propose VeloGauss, designed to learn the physical properties of complex dynamic 3D scenes without physical priors. Our method learns the velocity field for each Gaussian particle by introducing a Physics Code and a Particle Dynamics System, and ultimately incorporates Global Physical Constraints to ensure the physical consistency of the scene. Extensive experiments on four public datasets demonstrate that our method outperforms achieves state-of-the-art performance in both Novel View Interpolation and Future Frame Extrapolation tasks.

Reconstructing 3D fluid velocity fields from sparse 2D video observations is a highly ill-posed inverse problem, demanding both transport consistency with observed motion and physical validity under fluid laws. Existing methods typically impose these constraints through soft penalties, often leading to compromised accuracy and convergence issues. We introduce a reconstruction framework that structurally enforces both constraints. Specifically, we parameterize the reconstructed velocity using a continuous Divergence-Free Kernel representation, driving the advection of a Lagrangian 3D Gaussian Splatting representation. This formulation intrinsically guarantees both flow incompressibility and long-range transport coherence by construction. To enable the efficient optimization of such a constrained system, we introduce a novel Sliding Window scheme that propagates gradients over meaningful temporal horizons while maintaining tractable training costs. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art baselines in both transport consistency and physical accuracy, enabling applications such as high-quality re-simulation and flow analysis.

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near $β\!=\!0.94$, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.

Flow matching policies learn continuous velocity fields that transport noise to actions, enabling fast deterministic inference for robot manipulation. However, standard training optimizes a pointwise velocity objective while inference requires numerical integration of that field -- a mismatch that causes compounding trajectory errors. We propose four complementary remedies: (1) auxiliary rectified flow velocity regression that provides uniform temporal supervision across the full time interval; (2) multi-step trajectory consistency training that supervises the integrated displacement of the velocity field over trajectory segments, directly closing the train-inference gap; (3) velocity field regularization that enforces temporal smoothness, preventing oscillations that destabilize integration; and (4) fourth-order Runge-Kutta (RK4) inference that reduces global discretization error by orders of magnitude over Euler methods. Critically, these components are not independently sufficient -- RK4 without a smooth velocity field fails, and smoothness without trajectory-level supervision still drifts, as our ablation study confirms. We further pair these with a dual-view 3D point cloud encoder using two independent PointNet encoders for complementary spatial perception. On four real-robot tasks across a Franka arm and a Boston Dynamics Spot, our method achieves 70% and 60% overall success on two long-horizon multi-phase tasks where both baselines score 0%, and reaches 100% on precision tool placement. Three MetaWorld simulation tasks confirm consistent improvements, validating that trajectory-level supervision is essential for reliable policy execution.

Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property, which we call Symmetry Rupture, serves two purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.

Flow matching has recently emerged as a flexible and efficient framework for generative modelling by learning deterministic transport dynamics between probability measures. In this work, we extend flow matching to the space of probability measures over probability measures, introducing a Wasserstein-on-Wasserstein (WoW) formulation. Leveraging the nested Wasserstein geometry, we show that measures over transport plans naturally induce velocity fields that realize metameasure flows. This yields a principled generalization of Wasserstein flow matching via coupled outer and inner transport plans. To address the substantial computational cost of WoW transport, we propose scalable approximations based on sliced and linear Wasserstein distances, enabling efficient training while promoting numerically stable, near-straight trajectories. Our framework unifies and extends existing approaches to point cloud and set generation, providing a practical and theoretically grounded method for generative modelling in WoW spaces.

While generative modeling has achieved remarkable success on tasks like natural language-conditioned image generation, enabling model adaptation from example data points remains a relatively underexplored and challenging problem. To this end, we propose Function Projection for Flow Matching (FP-FM), an algorithm that directly conditions generation on samples from the target distribution. FP-FM learns basis functions to span the velocity fields corresponding to a set of training distributions, and adapts to new distributions by computing a simple least-squares projection onto this basis. This enables efficient generation of samples from diverse target distributions without additional training at inference time. We further introduce multiple variants of FP-FM that provide a trade-off in expressivity and compute by enriching the coefficient calculation, e.g., by making the coefficients dependent on time. FP-FM achieves greatly improved precision and recall relative to baselines across synthetic and image-based datasets, with especially strong gains on unseen distributions.

Activation steering has emerged as a promising alternative for controlling language-model behavior at inference time by modifying intermediate representations while keeping model parameters frozen. However, large-scale evaluations such as AxBench show that existing steering methods are often outperformed by simple in-context prompting and generalize poorly to unseen concepts. We hypothesize that these limitations arise from unvalidated simplifying assumptions shared across prior methods, which typically restrict steering interventions to fixed, single-step, position-invariant transforms. We propose FLAS (Flow-based Activation Steering), which learns a general, concept-conditioned velocity field $v_t(h,t,c)$ that transports unsteered activations to steered ones without relying on these assumptions. On AxBench, FLAS is the first learned method to consistently outperform prompting, reaching held-out harmonic means of $1.015$ on Gemma-2-2B-IT and $1.113$ on Gemma-2-9B-IT without per-concept tuning. Analysis of the learned flow shows curved, multi-step, token-varying trajectories, which suggests that previous hypotheses on activation space geometry might be incomplete.

The task of capturing and rendering 3D dynamic scenes from 2D images has become increasingly popular in recent years. However, most conventional cameras are bandwidth-limited to 30-60 FPS, restricting these methods to static or slowly evolving scenes. While overcoming bandwidth limitations is difficult for general scenes, recent years have seen a flurry of computational imaging methods that yield high-speed videos using conventional cameras for specific applications (e.g., motion capture and particle image velocimetry). However, most of these methods require modifications to a camera's optics or the addition of mechanically moving components, limiting them to a single-view high-speed capture. Consequently, these methods cannot be readily used to capture a 3D representation of rapid scene motion. In this paper, we propose a novel method to capture and reconstruct a volumetric representation of a high-speed scene using only unaugmented low-speed cameras. Instead of modifying the hardware or optics of each individual camera, we encode high-speed scene dynamics by illuminating the scene with a rapid, sequential color-coded sequence. This results in simultaneous multi-view capture of the scene, where high-speed temporal information is encoded in the spatial intensity and color variations of the captured images. To construct a high-speed volumetric representation of the dynamic scene, we develop a novel dynamic Gaussian Splatting-based approach that decodes the temporal information from the images. We evaluate our approach on simulated scenes and real-world experiments using a multi-camera imaging setup, showing first-of-a-kind high-speed volumetric scene reconstructions.

Single-image point cloud reconstruction must infer complete 3D geometry, including occluded parts, from a single RGB image. While diffusion-based reconstructors achieve high accuracy, they typically require many denoising iterations, resulting in slow and expensive inference. We propose Point-MF, a Mean-Flow-based framework for low-NFE single-image point cloud reconstruction that couples a Mean-Flow-compatible architecture with an auxiliary loss. Specifically, Point-MF operates directly in point-cloud space to learn the mean velocity field and enables one-step reconstruction with a single network function evaluation (1-NFE), without relying on VAE-based latent representations. To make Mean Flow effective under large interval jumps, Point-MF employs a Diffusion Transformer tailored to the Mean-Flow setting, conditioned on frozen DINOv3 image features via a lightweight token adapter and equipped with explicit interval/time conditioning. Moreover, we introduce Denoised Space Anchor, a set-distance auxiliary loss on the denoised-space estimate $x_θ$ induced by the predicted velocity field, to stabilize large-step generation and reduce outliers and density artifacts. On ShapeNet-R2N2 and Pix3D, Point-MF strikes a strong balance between reconstruction quality and inference speed compared to multi-step diffusion baselines and competitive feedforward models, while generating high-quality point clouds with millisecond-level latency.

Autonomous driving (AD) is evolving towards end-to-end (E2E) frameworks through two primary paradigms: monolithic models exemplified by Vision-Language-Action (VLA), and specialized modular architectures. Despite their divergent designs, both paradigms increasingly rely on Transformer backbones for complex reasoning, potentially causing a shared vulnerability: visually imperceptible perturbations can manipulate E2E AD models into hazardous maneuvers by targeting the Transformer module. Most existing adversarial attack approaches against AD systems operate under white-box or black-box settings; yet, they typically necessitate full model transparency, or suffer from either prohibitive query latency or limited attack transferability. In this paper, we propose Adversarial Flow Matching (AFM), a novel gray-box attack framework that exploits Transformer structural vulnerabilities in E2E AD models. AFM enables efficient one-step generation of adversarial examples via a neural average velocity field. Additionally, the proposed technique yields effective and visually imperceptible attacks by synergistically perturbing the generative latent space and the neural average velocity field. Extensive experiments demonstrate that AFM achieves a superior trade-off between attack effectiveness and imperceptibility: it substantially degrades the performance of both VLA and modular AD agents across various scenarios compared to baselines, while maintaining state-of-the-art visual imperceptibility. Furthermore, adversarial examples generated by AFM exhibit robust cross-model transferability, indicating that AFM closely approximates a black-box attack setting while requiring only the prior knowledge that the target AD model incorporates a Transformer-based module.

Flow matching generates data by integrating a learned velocity field, where the number of integration steps (NFE) directly determines inference cost. We analyze which properties of the velocity field govern integration error by decomposing the velocity Jacobian into its symmetric part S (strain rate) and antisymmetric part Omega (vorticity). We prove that strain and vorticity play different roles: strain controls exponential error amplification through the logarithmic norm, while vorticity contributes only linearly to the local truncation error. We further show that the optimal transport velocity field is irrotational and has zero material derivative, implying second-order Euler accuracy; for exact displacement interpolation, the associated Lagrangian particle dynamics are integrated exactly by Euler. Motivated by this analysis, we study weighted Jacobian regularization with strain weight alpha and vorticity weight beta. Experiments on 2D synthetic data confirm the main theoretical predictions, showing up to 2.7x lower integration error at NFE=5. Preliminary CIFAR-10 experiments show consistent trends, with a lightweight fine-tuning procedure improving FID by 14 percent at NFE=10 while preserving high-NFE quality.

Medical image denoising (MID) lacks absolutely clean images for supervision, leading to a noisy reference problem that fundamentally limits denoising performance. Existing simulated-supervised discriminative learning (SimSDL) and simulated-supervised generative learning (SimSGL) treat noisy references as clean targets, causing suboptimal convergence or reference-biased learning, while self-supervised learning (SSL) imposes restrictive noise assumptions that are seldom satisfied in realistic MID scenarios. We propose \textbf{RelativeFlow}, a flow matching framework that learns from heterogeneous noisy references and drives inputs from arbitrary quality levels toward a unified high-quality target. RelativeFlow reformulates flow matching by decomposing the absolute noise-to-clean mapping into relative noisier-to-noisy mappings, and realizes this formulation through two key components: 1) consistent transport (CoT), a displacement map that constrains relative flows to be components of and progressively compose a unified absolute flow, and 2) simulation-based velocity field (SVF), which constructs a learnable velocity field using modality-specific degradation operators to support different medical imaging modalities. Extensive experiments on Computed Tomography (CT) and Magnetic Resonance (MR) denoising demonstrate that RelativeFlow significantly outperforms existing methods, taming MID with noisy references.

We study dynamic measure transport for generative modeling, focusing on transport maps that connect a source measure $P_0$ to a target measure $P_1$ by integrating a velocity field of the form $v_t(x) = \mathbb{E}[\dot X_t \mid X_t = x]$, where $X_\bullet = (X_t)_t$ is a stochastic process satisfying $(X_0,X_1)\sim{P_0}\otimes{P_1}$ and $\dot X_t$ is its time derivative. We investigate when $X_\bullet$ induces a \emph{straight-line flow}: a flow whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. First, we develop multiple characterizations of straight-line flows in terms of PDEs involving the conditional statistics of the process. Then, we prove that straight-line flows under endpoint independence exhibit a sharp dichotomy. On the one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show that straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this obstruction through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process' flow map. Taken together, these results provide a structural theory of when straight-line generative flows can, and cannot, exist.

Flow-based models learn a target distribution by modeling a marginal velocity field, defined as the average of sample-wise velocities connecting each sample from a simple prior to the target data. When sample-wise velocities conflict at the same intermediate state, however, this averaged velocity can misguide samples toward low-density regions, degrading generation quality. To address this issue, we propose the Flow Divergence Sampler (FDS), a training-free framework that refines intermediate states before each solver step. Our key finding reveals that the severity of this misguidance is quantified by the divergence of the marginal velocity field that is readily computable during inference with a well-optimized model. FDS exploits this signal to steer states toward less ambiguous regions. As a plug-and-play framework compatible with standard solvers and off-the-shelf flow backbones, FDS consistently improves fidelity across various generation tasks including text-to-image synthesis, and inverse problems.

Generative models, particularly Diffusion Models (DM), have shown strong potential for Computed Tomography (CT) reconstruction serving as expressive priors for solving ill-posed inverse problems. However, diffusion-based reconstruction relies on Stochastic Differential Equations (SDEs) for forward diffusion and reverse denoising, where such stochasticity can interfere with repeated data consistency corrections in CT reconstruction. Since CT reconstruction is often time-critical in clinical and interventional scenarios, improving reconstruction efficiency is essential. In contrast, Flow Matching (FM) models sampling as a deterministic Ordinary Differential Equation (ODE), yielding smooth trajectories without stochastic noise injection. This deterministic formulation is naturally compatible with repeated data consistency operations. Furthermore, we observe that FM-predicted velocity fields exhibit strong correlations across adjacent steps. Motivated by this, we propose an FM-based CT reconstruction framework (FMCT) and an efficient variant (EFMCT) that reuses previously predicted velocity fields over consecutive steps to substantially reduce the number of Neural network Function Evaluations (NFEs), thereby improving inference efficiency. We provide theoretical analysis showing that the error introduced by velocity reuse is bounded when combined with data consistency operations. Extensive experiments demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. The codebase is open-sourced at https://github.com/EFMCT/EFMCT.

Rigid link flapping mechanisms remain the most practical choice for flapping wing micro-aerial vehicles (MAVs) to carry useful payloads and onboard batteries for free flight due to their long-term durability and reliability. However, MAVs with these mechanisms require significant weight reduction to achieve high agility and maneuverability. One approach involves using single-DOF planar rigid linkages, which are rarely optimized dimensionally for high lift and low power, considering their sweeping kinematics and the unsteady aerodynamic effects. We integrated a mechanism simulator based on a quasistatic nonlinear finite element method with an unsteady vortex lattice method-based aerodynamic analysis tool within an optimization routine. We optimized three different mechanism topologies from the literature. Significant power savings were observed up to 34% in some cases, due to increased amplitude and higher lift coefficients resulting from optimized asymmetric sweeping velocity profiles. We also conducted a robustness analysis to quantify performance sensitivity to manufacturing tolerances. It provided a trade-off between performance and reliability and revealed the need for tight manufacturing tolerances and careful material selection. Finally, the analysis helped select the best mechanism topology, as we observed significant variation in sensitivity to manufacturing tolerances and peak input torque values across different topologies for a given design lift value. The presented unified computational tool can find application in flapping mechanism topology optimization, as it can simulate any generic single-DOF planar rigid linkage without supplying kinematics manually.