We organize our supplementary material as follows: In Section A, we present more analytical results on VSPW dataset. In Section B, we provide more ablation studies on Cityscapes dataset. In Section D, we provide computational cost analysis and training details. In Section E, we provide comparison with bi-directional optical flow. A.1 Explanation of Video Consistency Following [3], we use Video Consistency (VC) to evaluate the category consistency among adjacent frames in the videos.

Video Semantic Segmentation (VSS) involves assigning a semantic label to each pixel in a video sequence. Prior work in this field has demonstrated promising results by extending image semantic segmentation models to exploit temporal relationships across video frames; however, these approaches often incur significant computational costs. In this paper, we propose an efficient mask propagation framework for VSS, called MPVSS. Our approach first employs a strong querybased image segmentor on sparse key frames to generate accurate binary masks and class predictions. We then design a flow estimation module utilizing the learned queries to generate a set of segment-aware flow maps, each associated with a mask prediction from the key frame.

The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative integration. Flow Maps bypass this bottleneck by compressing generative trajectories into single-step mappings, theoretically enabling the generation of full text sequences from noise in a single forward pass. However, standard formulations rely on Euclidean regression losses that are geometrically ill-suited for discrete data. In this work, we resolve this conflict with Discrete Flow Maps, a framework that reconciles trajectory compression with the geometry of the probability simplex. We recast standard flow map training for the discrete domain, aligning the training dynamics with the discrete nature of language. Empirically, this strict geometric alignment allows our method to surpass previous state-of-the-art results in discrete flow modeling.

Mohamed bin Zayed University of AI We organize our supplementary material as follows: In Section A, we present more analytical results on VSPW dataset. In Section B, we provide more ablation studies on Cityscapes dataset. In Section D, we provide computational cost analysis and training details. In Section E, we provide comparison with bi-directional optical flow. This observation demonstrates that our mask propagation framework implicitly captures the long-range temporal relationships among video frames.

Simulating parameter-dependent stochastic differential equations (SDEs) presents significant computational challenges, as separate high-fidelity simulations are typically required for each parameter value of interest. Despite the success of machine learning methods in learning SDE dynamics, existing approaches either require expensive neural network training for score function estimation or lack the ability to handle continuous parameter dependence. We present a training-free conditional diffusion model framework for learning stochastic flow maps of parameter-dependent SDEs, where both drift and diffusion coefficients depend on physical parameters. The key technical innovation is a joint kernel-weighted Monte Carlo estimator that approximates the conditional score function using trajectory data sampled at discrete parameter values, enabling interpolation across both state space and the continuous parameter domain. Once trained, the resulting generative model produces sample trajectories for any parameter value within the training range without retraining, significantly accelerating parameter studies, uncertainty quantification, and real-time filtering applications. The performance of the proposed approach is demonstrated via three numerical examples of increasing complexity, showing accurate approximation of conditional distributions across varying parameter values.

A common recipe to improve diffusion models at test-time so that samples score highly against a user-specified reward is to introduce the gradient of the reward into the dynamics of the diffusion itself. This procedure is often ill posed, as user-specified rewards are usually only well defined on the data distribution at the end of generation. While common workarounds to this problem are to use a de-noiser to estimate what a sample would have been at the end of generation, we propose a simple solution to this problem by working directly with a flow map. By exploiting a relationship between the flow map and velocity field governing the instantaneous transport, we construct an algorithm, Flow Map Trajectory Tilting (FMTT), which provably performs better ascent on the reward than standard test-time methods involving the gradient of the reward. The approach can be used to either perform exact sampling via importance weighting or principled search that identifies local maximizers of the reward-tilted distribution. We demonstrate the efficacy of our approach against other look-ahead techniques, and show how the flow map enables engagement with complicated reward functions that make possible new forms of image editing, e.g. by interfacing with vision language models. Figure 1: Test-time search can overcome model biases and reliably sample from regions of the distribution (e.g., precise clock times) that baselines fail to capture. Large scale foundation models built out of diffusions (Ho et al., 2020; Song et al., 2020) or flow-based transport (Lipman et al., 2022; Albergo & V anden-Eijnden, 2022; Albergo et al., 2023; Liu In this paradigm, performing generation amounts to numerically solving an ordinary or stochastic differential equation (ODE/SDE), the coefficients of which are learned neural networks.

Fine-grained urban flow inference is crucial for urban planning and intelligent transportation systems, enabling precise traffic management and resource allocation. However, the practical deployment of existing methods is hindered by two key challenges: the prohibitive computational cost of over-parameterized models and the suboptimal performance of conventional loss functions on the highly skewed distribution of urban flows. To address these challenges, we propose a unified solution that synergizes architectural efficiency with adaptive optimization. Specifically, we first introduce PLGF, a lightweight yet powerful architecture that employs a Progressive Local-Global Fusion strategy to effectively capture both fine-grained details and global contextual dependencies. Second, we propose DualFocal Loss, a novel function that integrates dual-space supervision with a difficulty-aware focusing mechanism, enabling the model to adaptively concentrate on hard-to-predict regions. Extensive experiments on 4 real-world scenarios validate the effectiveness and scalability of our method. Notably, while achieving state-of-the-art performance, PLGF reduces the model size by up to 97% compared to current high-performing methods. Furthermore, under comparable parameter budgets, our model yields an accuracy improvement of over 10% against strong baselines. The implementation is included in the https://github.com/Yasoz/PLGF.

In this short note, we characterize stability of the Kim--Milman flow map -- also known as the probability flow ODE -- with respect to variations in the target measure. Rather than the Wasserstein distance, we show that stability holds with respect to the relative Fisher information

Geometric data and purpose-built generative models on them have become ubiquitous in high-impact deep learning application domains, ranging from protein backbone generation and computational chemistry to geospatial data. Current geometric generative models remain computationally expensive at inference -- requiring many steps of complex numerical simulation -- as they are derived from dynamical measure transport frameworks such as diffusion and flow-matching on Riemannian manifolds. In this paper, we propose Generalised Flow Maps (GFM), a new class of few-step generative models that generalises the Flow Map framework in Euclidean spaces to arbitrary Riemannian manifolds. We instantiate GFMs with three self-distillation-based training methods: Generalised Lagrangian Flow Maps, Generalised Eulerian Flow Maps, and Generalised Progressive Flow Maps. We theoretically show that GFMs, under specific design decisions, unify and elevate existing Euclidean few-step generative models, such as consistency models, shortcut models, and meanflows, to the Riemannian setting. We benchmark GFMs against other geometric generative models on a suite of geometric datasets, including geospatial data, RNA torsion angles, and hyperbolic manifolds, and achieve state-of-the-art sample quality for single- and few-step evaluations, and superior or competitive log-likelihoods using the implicit probability flow.