Organizations: Inria, Université de Lorraine, CNRS, Loria Nancy, France
Abstract
The ability of Flow Matching (FM) to model complex conditional distributions has established it as the state-of-the-art for prediction tasks (e.g., robotics, weather forecasting). However, deployment in safety-critical settings is hindered by a critical extrapolation hazard: driven by smoothness biases, flow models yield plausible outputs even for off-manifold conditions, resulting in silent failures indistinguishable from valid predictions. In this work, we introduce Diverging Flows, a novel approach that enables a single model to simultaneously perform conditional generation and native extrapolation detection by structurally enforcing inefficient transport for off-manifold inputs. We evaluate our method on synthetic manifolds, cross-domain style transfer, and weather temperature forecasting, demonstrating that it achieves effective detection of extrapolations without compromising predictive fidelity or inference latency. These results establish Diverging Flows as a robust solution for trustworthy flow models, paving the way for reliable deployment in domains such as medicine, robotics, and climate science.
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.
While flow-based generative models have demonstrated strong performance across a wide range of domains, deploying them in safety-critical physical systems remains challenging due to strict constraint requirements. Existing approaches typically enforce safety through post-hoc corrections, which incur substantial computational overhead and may distort the learned distribution. We propose PolyFlow, a polytope-constrained flow matching framework that embeds constraints directly into the model and flow dynamics. PolyFlow introduces a discrete-time flow formulation and a projection-free architecture, which eliminate the discretization error and guarantee strict satisfaction of arbitrary polyhedral constraints, without the need for expensive iterative solvers. Experimental results show that PolyFlow achieves zero constraint violation while maintaining high distributional fidelity across a range of planning and control tasks. Compared to state-of-the-art constrained generation baselines, PolyFlow significantly reduces inference latency and demonstrates a favorable trade-off between safety, efficiency, and generative quality. Code is available on https://github.com/MJianM/PolyFlow.
Generative motion prediction must satisfy three simultaneous requirements for real-world autonomy: high accuracy, diverse multimodal futures, and strictly bounded latency. Diffusion models meet the first two but violate the third, requiring tens to hundreds of denoising steps. We identify a conditioning strategy that resolves this tension: \textit{single-step integration is accurate when the underlying transport problem is local}. A model that must both discover the correct behavioral mode and traverse a long displacement in one step accumulates large discretization errors; conditioning the base distribution to lie near plausible futures reduces the problem to short-range refinement, the regime where a single Euler step suffices. We instantiate this \emph{local transport conditioning} in FlowS, a conditional flow matching framework with two mechanisms. First, an online, scene-conditioned learned prior emits K calibrated anchor trajectories per agent, each already near a plausible future, converting mode discovery into local correction. Second, a step-consistent displacement field enforces semigroup self-consistency, guaranteeing that a single step inherits multi-step accuracy. Crucially, anchoring this field at learned priors along straight-line paths yields a {stable, low-variance} training target, unlike prior self-consistency methods that suffer from {high-variance bootstrap} signals on curved diffusion paths. On the Waymo Open Motion Dataset, FlowS achieves state-of-the-art Soft mAP {(0.4804) and mAP (0.4703) with ensemble at 75,FPS} with single-step inference, demonstrating that local transport conditioning makes one-step generative motion prediction practical for safety-critical autonomy. Code and pretrained models will be released upon acceptance.