Flow Map Learning

Also known as MAP

Follow this topic Ask about this topic

Flow map learning studies how to predict a system’s finite-time transition—from one state or distribution to another—rather than repeatedly integrating a local velocity field. Approaches include semigroup-consistency objectives such as MeanFlow, optimal-transport flow matching, stochastic flow maps for SDEs, and distillation of slow teachers into few-step generators; neural operators and physical or geometric constraints adapt the idea to PDEs, molecular dynamics, and control. Applications range from one-step image and text generation to long-horizon forecasting, inverse problems, and molecular simulation. A central tension is making large, few-step transitions accurate and stable while preserving physical structure and generalization across step sizes or unseen conditions; theory for training stability and out-of-distribution behavior remains uneven.

Latest papers

49 papers11 papers in the last four weeks, up 22% on the four weeks before. 0.1% of all new papers.

Jun 27, 2026

Few-Step Boltzmann Generators via Scalable Likelihood Flow Maps

Recent progress in flow-based generative modeling has led to models that output high-quality samples while using only a small number of function evaluations. However, at present, there is a lack of similar advances in estimating the model likelihood. In particular, most existing methods either rely on restrictive architectures that enable exact calculations, or use stochastic approximations such as Hutchinson's trace estimator that introduce substantial variance. In this work, we introduce SCAlable LikeLihood distillation of flOw maPs (SCALLOP). SCALLOP builds on the recently proposed F2D2, a likelihood flow map model that can generate samples and their densities in a small number of function evaluations. While F2D2 uses Hutchinson's estimator during training, we introduce an alternative and more scalable likelihood distillation objective that is Hutchinson-free and admits a vectorized formulation. Empirically, we demonstrate the effectiveness of SCALLOP as a Boltzmann generator in molecular science, and further validate its benefit on image datasets. SCALLOP significantly reduces both training variance and training time while consistently improving performance compared to F2D2, and is competitive with the state-of-the-art while achieving up to 10x inference speedup over the fastest baseline.

RuiKang OuYang, Hanlin Yu, Xinyue Ai +7

University of Cambridge · University of Helsinki · Peking University +3

Flow Map Learning Generative Modeling Molecular Generation Flow-Based Generative Modeling

Jun 18, 2026

Flow Map Denoisers: Traversing the Distortion-Perception Plane for Inverse Problems

Image restoration faces a fundamental tradeoff: methods that minimize error produce blurry reconstructions, while those that maximize perceptual quality yield sharp but less faithful images. Existing approaches either commit to a single operating point on this distortion perception (DP) frontier or require paired-data supervision, auxiliary models, or hyperparameter tuning of the sampler to access different points. We show that flow map models, a recent extension of flow matching for few-step sampling that learns an average field, implicitly define a one-parameter family of denoisers that continuously spans the DP frontier. The lookahead parameter t acts as a control knob between the MMSE and perceptual regimes. For Gaussian targets, we prove that varying t exactly recovers the optimal DP frontier; for natural images, we observe similar behavior empirically. Within a Plug-and-Play solver, the same mechanism extends to general inverse problems, where it controls a tradeoff between perceptual alignment and data consistency. Despite the lack of exact optimality guarantees in this setting, a single trained flow map spans the DP tradeoff, matching or exceeding specialized baselines at both extremes. Extensive experiments on CelebA (

128\times 128

) and AFHQ (

256\times 256

) across several linear and nonlinear inverse tasks validate our findings.

Nicolas Zilberstein, Morteza Mardani, Santiago Segarra

Rice University · NVIDIA Inc.

Flow Matching Flow Map Learning Image Restoration Distortion-Perception Tradeoff Image Inverse Problems

Jun 9, 2026

Itô maps for any-step SDEs

Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.

Zhengkai Pan, Peter Potaptchik, Wenxi Yao +2

Harvard University · University of Oxford · Kempner Institute

Stochastic Differential Equations Diffusion Distillation Posterior Sampling Flow Map Learning

Jun 3, 2026

Multimarginal flow matching with optimal transport potentials

Flow matching (FM) has emerged as a powerful framework for learning dynamic transport maps between two empirical distributions. However, less explored is the setting with intermediate observed marginals that can help constrain the flows between the endpoints. This "multimarginal" regime is central to modeling temporal evolution in dynamical systems in many scientific domains that can sample sequential distributions. We tackle this problem with a novel approach that leverages the connection between FM and dynamic optimal transport (OT), softly steering the flow towards the intermediate marginals through potential terms in the dynamic OT action. By extending the conditional FM learning target to incorporate these potentials, we derive an efficient, simulation-free algorithm for multimarginal FM that offers considerable flexibility in the spatiotemporal dynamics of the learned flows. We demonstrate state-of-the-art performance and training efficiency of OT-potential FM (OTP-FM) on diverse single-cell RNA sequencing, oceanographic, and meteorological datasets. Our code is available at https://github.com/Bexorg-Inc/OTP-FM.

Raghav Kansal, David Crair, Nghia Nguyen +2

Bexorg, Inc., New Haven, CT, USA.

Flow Matching Flow Map Learning Optimal Transport

May 31, 2026

Strong Stochastic Flow Maps

Flow and diffusion models generate high-quality samples in many modalities; however, many network evaluations are required during inference due to numerical integration of an underlying differential equation. Flow maps alleviate this problem by learning the solution map of the differential equation directly, enabling few-step sampling. Yet, current methods are restricted to approximating the solution map of ODEs. These methods can be used to learn the transition kernel of an SDE, thereby obtaining a solution map that recovers the marginal distributions of the process (weak convergence) rather than the solution path (strong convergence). We propose Strong Stochastic Flow Maps (SSFMs) as a novel framework for learning the strong solution map of additive-noise SDEs, directly generalizing deterministic flow maps to the stochastic setting. Further, a polynomial approximation to Brownian motion is introduced and shown to converge pathwise. These results enable a simulation-free training objective for the solution map of diffusion models. We demonstrate that SSFMs outperform previous stochastic flow map methods on image generation and enable few-step sampling of molecular systems.

Sam McCallum, Zander W. Blasingame, Timothy Herschell +3

University of Bath · 2AITHYRA

Stochastic Differential Equations Few-Step Diffusion Sampling Flow Map Learning Diffusion-Based Image Generation Flow-Based Generative Modeling

May 29, 2026

Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

Markus Gross

Institute for AI Safety and Security, German Aerospace Center (DLR), Sankt Augustin and Ulm, Germany

Koopman Operator Learning Nonlinear System Identification Reservoir Computing Flow Map Learning Chaotic Time-Series Prediction Autoregressive Modeling

May 23, 2026

LC-Flow: Learning Local Continuous Optical Flow and Confidence from events

Event cameras capture brightness changes asynchronously with microsecond resolution, yet existing optical flow methods fail to fully exploit this temporal continuity. Frame-based approaches impose artificial accumulation latency and suffer from domain overfitting, while model-based local methods operate statelessly, discarding temporal history between predictions and yielding inaccurate flows. We propose LC-Flow, the first temporally continuous, learning-based optical flow estimator that operates purely from local events. At its core, a Continuous Local Recurrent Network maintains persistent hidden states per spatial grid, incrementally accumulating temporal context as events arrive. Unlike frame-based methods constrained to fixed accumulation windows, and unlike stateless model-based methods that recompute motion from scratch at each step, LC-Flow produces sparse local flow estimates at arbitrary timestamps with full motion history. To address the inherent ambiguity of local observations, we jointly learn a confidence score that quantifies the reliability of each prediction, explicitly handling event sparsity and the aperture problem. This confidence serves a dual role: filtering unreliable estimates for downstream tasks such as visual odometry, and providing principled weights for a multi-scale confidence-guided aggregation that reconstructs globally consistent flow from the sparse local outputs. LC-Flow achieves state-of-the-art performance among local methods on both MVSEC and DSEC, while the confidence-guided aggregation establishes a new overall state-of-the-art on the MVSEC benchmark, surpassing heavy frame-based networks that rely on global spatial priors.

Gunwoo Jeon, Chaesong Park, Jongwoo Lim

IPAI, Seoul National University

Optical Flow Event-Based Vision Event-Based Optical Flow Recurrent Neural Networks Confidence Estimation Flow Map Learning

May 13, 2026

Path-independent Flow Matching for Multi-parameter Generative Dynamics

Flow Matching is a powerful framework for learning transport maps between probability distributions. Yet its standard single-parameter formulation is not designed to capture multi-parameter variations where the resulting transport should be path-independent. Path independence is crucial because it ensures that transformations depend only on the initial and target distributions, not on the specific path. In this work, we introduce Path-independent Flow Matching (PiFM), a method for learning vector fields whose induced flows yield path-independent transport between distributions. We show that PiFM generalizes Flow Matching to higher-dimensional parameter domains while enforcing structural conditions that ensure consistency of composed transformations. In addition, we show that, under suitable assumptions, PiFM approximates the Wasserstein barycenter, linking the framework to a notion of distributional interpolation. To enable practical training, we propose a tractable, simulation-free objective that regresses onto multi-parameter conditional probability paths. We showcase empirically that PiFM outperforms other approaches on both synthetic and real world data in interpolating path-independent trajectories and generating desired out of distribution samples.

Francisco Téllez, AmirHossein Zamani, Philippe Martin +5

Université de Montréal, Mila · Concordia University, Mila · Concordia University +2

Flow Matching Flow Map Learning Generative Modeling Flow-Based Generative Modeling Optimal Transport

May 8, 2026

Scaling Categorical Flow Maps

Continuous diffusion and flow matching models could represent a powerful alternative to autoregressive approaches for language modelling (LM), as they unlock a host of advantages currently reserved for continuous modalities, including accelerated sampling and tilting. Recently, several works have demonstrated the possibility of generating discrete data continuously by a simple flow matching process between a Gaussian and the one-hot encoded data distribution. They have further shown the feasibility of accelerated sampling via Categorical Flow Maps (CFMs), resulting in competitive sample quality in the few-step regime. However, this method had only been evaluated at relatively modest scales (

<1

B), leaving the question of its scalability completely open. In this article, we train a

1.7

B-parameter base flow model on

2.1

T tokens and self-distill it into a CFM that generates diverse, high-quality text in as few as

4

inference steps while maintaining near-data-level token entropy. Furthermore, we introduce a likelihood bound for CFMs in the semi-discrete setting, and show that they can be used to score the model on standard LM benchmarks, achieving results in the same range as discrete diffusion methods. Finally, we uncover some of the challenges that arise from training these models at scale, and we provide prescriptive insights on loss weighting and time scheduling.

Oscar Davis, Anastasiia Filippova, Pierre Ablin +4

Apple · University of Oxford

Language Model Scaling Flow Map Learning Flow-Based Generative Modeling Continuous Diffusion Language Models Text Generation Discrete Flow Matching

Apr 14, 2026

Semantically Stable Image Composition Analysis via Saliency and Gradient Vector Flow Fusion

The reliable computational assessment of photographic composition requires features that are discriminative of spatial layout yet robust to semantic content. This paper proposes a low-level representation grounded in the assumption that composition can be understood as the flow of visual attention across geometric structure. We introduce VFCNet, which fuses saliency and edge information into a gradient vector flow (GVF) field. The model computes dual-stream GVF representations, integrates them via attention, and extracts multi-scale flow features with a DINOv3 backbone. VFCNet achieves state-of-the-art performance on the PICD benchmark (CDA-1: 0.683, CDA-2: 0.629), improving by 33.1% and 36.1% over the previous best method. We also show that a simple classifier on self-supervised DINOv3 features substantially outperforms more sophisticated, composition-specialized models. Code is available at https://github.com/ADadras/VFCNet

Armin Dadras, Robert Sablatnig, Franziska Proksa +1

Computer Vision Lab, TU Wien Vienna, Austria · UAS St. Pölten St. Pölten, Austria

Flow Map Learning

Apr 10, 2026

Discrete Flow Maps

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.

Peter Potaptchik, Jason Yim, Adhi Saravanan +3

Harvard University · University of Oxford · Independent +3

Flow Map Learning Non-Autoregressive Text Generation One-Step Generative Modeling

Apr 7, 2026

ODE-free Neural Flow Matching for One-Step Generative Modeling

Diffusion and flow matching models generate samples by learning time-dependent vector fields whose integration transports noise to data, requiring tens to hundreds of network evaluations at inference. We instead learn the transport map directly. We propose Optimal Transport Neural Flow Matching (OT-NFM), an ODE-free generative framework that parameterizes the flow map with neural flows, enabling true one-step generation with a single forward pass. We show that naive flow-map training suffers from mean collapse, where inconsistent noise-data pairings drive all outputs toward the data mean. We prove that consistent coupling is necessary for non-degenerate learning and address this using optimal transport pairings with scalable minibatch and online coupling strategies. Experiments on synthetic benchmarks and image generation tasks (MNIST and CIFAR-10) demonstrate competitive sample quality while reducing inference to a single network evaluation.

Xiao Shou

Baylor University

One-Step Generative Modeling Flow Matching Flow Map Learning Image Generation

Apr 4, 2026

SBF: An Effective Representation to Augment Skeleton for Video-based Human Action Recognition

Many modern video-based human action recognition (HAR) approaches use 2D skeleton as the intermediate representation in their prediction pipelines. Despite overall encouraging results, these approaches still struggle in many common scenes, mainly because the skeleton does not capture critical action-related information pertaining to the depth of the joints, contour of the human body, and interaction between the human and objects. To address this, we propose an effective approach to augment skeleton with a representation capturing action-related information in the pipeline of HAR. The representation, termed Scale-Body-Flow (SBF), consists of three distinct components, namely a scale map volume given by the scale (and hence depth information) of each joint, a body map outlining the human subject, and a flow map indicating human-object interaction given by pixel-wise optical flow values. To predict SBF, we further present SFSNet, a novel segmentation network supervised by the skeleton and optical flow without extra annotation overhead beyond the existing skeleton extraction. Extensive experiments across different datasets demonstrate that our pipeline based on SBF and SFSNet achieves significantly higher HAR accuracy with similar compactness and efficiency as compared with the state-of-the-art skeleton-only approaches.

Zhuoxuan Peng, Yiyi Ding, Yang Lin +1

The Hong Kong University of Science and Technology · The Hong Kong University of Science and Technology (Guangzhou)

Video Action Recognition Skeleton-Based Action Recognition Motion Representation Learning Flow Map Learning

Mar 18, 2026

RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

We propose RHYME-XT, an operator-learning framework for surrogate modeling of spatiotemporal control systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with localized rhythmic behavior. RHYME-XT uses a Galerkin projection to approximate the infinite-dimensional PIDE on a learned finite-dimensional subspace with spatial basis functions parameterized by a neural network. This yields a projected system of ODEs driven by projected inputs. Instead of integrating this non-autonomous system, we directly learn its flow map using an architecture for learning flow functions, avoiding costly computations while obtaining a continuous-time and discretization-invariant representation. Experiments on a neural field PIDE show that RHYME-XT outperforms a state-of-the-art neural operator and is able to transfer knowledge effectively across models trained on different datasets, through a fine-tuning process.

Marijn Ruiter, Miguel Aguiar, Jake Rap +2

Digital Futures and Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. · Control Systems Group, EE Dept., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

PDE Surrogate Modeling PDE Operator Learning Neural Surrogate Modeling Operator Learning Flow Map Learning

Mar 7, 2026

Variational Flow Maps: Make Some Noise for One-Step Conditional Generation

Flow maps enable high-quality image generation in a single forward pass. However, unlike iterative diffusion models, their lack of an explicit sampling trajectory impedes incorporating external constraints for conditional generation and solving inverse problems. We put forth Variational Flow Maps, a framework for conditional sampling that shifts the perspective of conditioning from "guiding a sampling path", to that of "learning the proper initial noise". Specifically, given an observation, we seek to learn a noise adapter model that outputs a noise distribution, so that after mapping to the data space via flow map, the samples respect the observation and data prior. To this end, we develop a principled variational objective that jointly trains the noise adapter and the flow map, improving noise-data alignment, such that sampling from complex data posterior is achieved with a simple adapter. Experiments on various inverse problems show that VFMs produce well-calibrated conditional samples in a single (or few) steps. For ImageNet, VFM attains competitive fidelity while accelerating the sampling by orders of magnitude compared to alternative iterative diffusion/flow models. Code is available at https://github.com/abbasmammadov/VFM

Abbas Mammadov, So Takao, Bohan Chen +4

Department of Statistics, University of Oxford · California Institute of Technology · University of Toronto +1

Conditional Generative Modeling One-Step Generative Modeling Flow Map Learning Image Inverse Problems

Mar 3, 2026

CoFL: Continuous Flow Fields for Language-Conditioned Navigation

Existing language-conditioned navigation systems typically rely on modular pipelines or trajectory generators, but the latter use each scene--instruction annotation mainly to supervise one start-conditioned rollout. To address these limitations, we present CoFL, an end-to-end policy that maps a bird's-eye view (BEV) observation and a language instruction to a continuous flow field for navigation. CoFL reformulates navigation as workspace-conditioned field learning rather than start-conditioned trajectory prediction: it learns local motion vectors at arbitrary BEV locations, turning each scene--instruction annotation into dense spatial control supervision. Trajectories are generated from any start by numerical integration of the predicted field, enabling simple real-time rollout and closed-loop recovery. To enable large-scale training and evaluation, we build a dataset of over 500k BEV image--instruction pairs, each procedurally annotated with a flow field and a trajectory derived from semantic maps built on Matterport3D and ScanNet. Evaluating on strictly unseen scenes, CoFL significantly outperforms modular Vision-Language Model (VLM)-based planners and trajectory generation policies in both navigation precision and safety, while maintaining real-time inference. Finally, we deploy CoFL zero-shot in real-world experiments with BEV observations across multiple layouts, maintaining feasible closed-loop control and a high success rate.

Haokun Liu, Zhaoqi Ma, Yicheng Chen +5

Vision-Language Navigation Flow Map Learning

Feb 27, 2026

A Monte Carlo estimator of flow fields for sampling and noise problems

Learned field transformations may help address ubiquitous critical slowing down and signal-to-noise problems in lattice field theory. In the context of an annealed sequence of distributions, field transformations are defined by integrating flow fields that exactly solve a local transport problem. These proceedings discuss a new Monte Carlo approach to evaluating these flow fields, which can then be used directly in such contexts or as a means of generating unbiased training data for machine learning approaches. By defining the Monte Carlo estimator using coupled Langevin noise, the statistical noise in the required integrals is significantly mitigated. Demonstrations of the method include a U(1) transport problem and an SU(N) glueball correlator.

Michael S. Albergo, Gurtej Kanwar

Society of Fellows, Harvard University 78 Mount Auburn St, Cambridge, MA United States · Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom

Lattice Field Theory High-Energy Physics Flow Map Learning

Feb 24, 2026

A Long-Short Flow-Map Perspective for Drifting Models

This paper provides a reinterpretation of the Drifting Model~\cite{deng2026generative} through a semigroup-consistent long-short flow-map factorization. We show that a global transport process can be decomposed into a long-horizon flow map followed by a short-time terminal flow map admitting a closed-form optimal velocity representation, and that taking the terminal interval length to zero recovers exactly the drifting field together with a conservative impulse term required for flow-map consistency. Based on this perspective, we propose a new likelihood learning formulation that aligns the long-short flow-map decomposition with density evolution under transport. We validate the framework through both theoretical analysis and empirical evaluations on benchmark tests, and further provide a theoretical interpretation of the feature-space optimization while highlighting several open problems for future study.

Zhiqi Li, Bo Zhu

School of Interactive Computing, Georgia Institute of Technology

Drifting Models Flow Map Learning Generative Modeling Flow-Based Generative Modeling

Jan 29, 2026

Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.

Winfried Ripken, Michael Plainer, Gregor Lied +5

Technical University Berlin · BIFOLD · Free University of Berlin +6

Dynamical Systems Flow Map Learning Mean Flow Molecular Dynamics Simulation Physics-Based Simulation Hamiltonian Dynamics