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.

Flow map models such as Consistency Models (CM) and Mean Flow (MF) enable few-step generation by learning the long jump of the ODE solution of diffusion models, yet training remains unstable, sensitive to hyperparameters, and costly. Initializing from a pre-trained diffusion model helps, but still requires converting infinitesimal steps into a long-jump map, leaving instability unresolved. We introduce mid-training, the first concept and practical method that inserts a lightweight intermediate stage between the (diffusion) pre-training and the final flow map training (i.e., post-training) for vision generation. Concretely, Consistency Mid-Training (CMT) is a compact and principled stage that trains a model to map points along a solver trajectory from a pre-trained model, starting from a prior sample, directly to the solver-generated clean sample. It yields a trajectory-consistent and stable initialization. This initializer outperforms random and diffusion-based baselines and enables fast, robust convergence without heuristics. Initializing post-training with CMT weights further simplifies flow map learning. Empirically, CMT achieves state of the art two step FIDs: 1.97 on CIFAR-10, 1.32 on ImageNet 64x64, and 1.84 on ImageNet 512x512, while using up to 98% less training data and GPU time, compared to CMs. On ImageNet 256x256, CMT reaches 1-step FID 3.34 while cutting total training time by about 50% compared to MF from scratch (FID 3.43). This establishes CMT as a principled, efficient, and general framework for training flow map models.

Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary interactions. Despite the success of machine learning-based methods in learning SDEs, existing learning methods are not applicable to SDEs in bounded domains because they cannot accurately capture the particle exit dynamics. We present a unified hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network to capture both interior stochastic dynamics and boundary exit phenomena. Our ML model consists of two major components: a neural network that learns exit probabilities using binary cross-entropy loss with rigorous convergence guarantees, and a training-free diffusion model that generates state transitions for non-exiting particles using closed-form score functions. The two components are integrated through a probabilistic sampling algorithm that determines particle exit at each time step and generates appropriate state transitions. The performance of the proposed approach is demonstrated via three test cases: a one-dimensional simplified problem for theoretical verification, a two-dimensional advection-diffusion problem in a bounded domain, and a three-dimensional problem of interest to magnetically confined fusion plasmas.