We propose \emph{Euler Mean Flows (EMF)}, a flow-based generative framework for one-step and few-step generation that enforces long-range trajectory consistency with minimal sampling cost. The key idea of EMF is to replace the trajectory consistency constraint, which is difficult to supervise and optimize over long time scales, with a principled linear surrogate that enables direct data supervision for long-horizon flow-map compositions. We derive this approximation from the semigroup formulation of flow-based models and show that, under mild regularity assumptions, it faithfully approximates the original consistency objective while being substantially easier to optimize. This formulation leads to a unified, JVP-free training framework that supports both $u$-prediction and $x_1$-prediction variants, avoiding explicit Jacobian computations and significantly reducing memory and computational overhead. Experiments on image synthesis, particle-based geometry generation, and functional generation demonstrate improved optimization stability and sample quality under fixed sampling budgets, together with approximately $50\%$ reductions in training time and memory consumption compared to existing one-step methods for image generation.
Trajectory Consistency Distillation: Improved Latent Consistency Distillation by Semi-Linear Consistency Function with Trajectory Mapping
South China University of Technology（华南理工大学）；Nanyang Technological University（南洋理工大学）；Beijing Institute of Technology（北京理工大学）；The University of Sydney（悉尼大学）
MagicMotion: Controllable Video Generation with Dense-to-Sparse Trajectory Guidance
Shanghai Key Lab of Intell. Info. Processing, School of CS, Fudan University（上海智能信息处理关键实验室，复旦大学计算机学院）；Shanghai Collaborative Innovation Center of Intelligent Visual Computing（上海智能视觉计算协同创新中心）；Microsoft Research Asia（微软亚洲研究院）
TRACE: Trajectory Representation and Consistency Estimation for AI-Generated Video Detection
Zhejiang University（浙江大学）；University of the Chinese Academy of Sciences（中国科学院大学）；Nanjing University of Information Science and Technology（南京信息工程大学）；Zhejiang Gongshang University（浙江工商大学）
Trajectory Constraints for Imaging Inverse Problems
Department of Computational Mathematics, Science, & Engineering, Michigan State University（密歇根州立大学计算数学、科学与工程系）；Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气工程与计算机科学系）；Department of Biomedical Engineering, Michigan State University（密歇根州立大学生物医学工程系）
Elucidating the Preconditioning in Consistency Distillation
Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Center, THBI Lab（计算机科学与技术系，人工智能研究所，BNRist中心，THBI实验室）；Tsinghua-Bosch Joint ML Center, Tsinghua University（清华大学-博世联合机器学习中心，清华大学）；Shengshu Technology（盛舒科技）；Pazhou Lab (Huangpu), Guangzhou, China（黄埔实验室（广州），中国）
RAPiD: Reward-Guided Consistency Distillation of Diffusion Planners for Real-Time Autonomous Driving
School of Information Technology, Monash University, Malaysia（墨尔本大学信息科技学院）；Faculty of Information Technology, Monash University, Australia（墨尔本大学信息技术学院）；TCS Research, India（印度TCS研究）
Syndrome-Flow Consistency Model Achieves One-step Denoising Error Correction Codes
Department of Computer Science and Engineering, The Chinese University of Hong Kong（计算机科学与工程系，香港中文大学）；Huawei Technologies Co., Ltd.（华为技术有限公司）；Theory Lab（理论实验室）；Noah’s Ark Lab（诺亚实验室）
School of Electronic Information and Communications, Huazhong University of Science and Technology（华中科技大学电子信息学院）；School of Electronic Information（电子信息学院）；Communications, Huazhong University of Science（华中科技大学通信学院）；School of Science, Wuhan University of Technology（武汉理工大学理学院）
Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics
Technical University Berlin（柏林工业大学）；Free University of Berlin（柏林自由大学）；Zuse School ELIZA；Google DeepMind（谷歌DeepMind）；Rice University（莱斯大学）；MPI for Informatics, Saarbrücken（马克斯·普朗克信息学研究所，萨尔布吕肯）
