Trajectory Consistency for One-Step Generation on Euler Mean Flows

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.

Generative models can create realistic images, 3D shapes, and other complex data, but many high-quality methods require a long sequence of computational steps to turn random noise into a final sample. This makes generation slower and also makes training more expensive. Our work studies how to train such models so that they can generate high-quality samples in only one or a few steps. The main challenge is ensuring that these short generation paths remain consistent with the longer trajectories used by standard flow-based models. We propose Euler Mean Flows, a new training framework that replaces a difficult long-range consistency constraint with a simple local linear approximation inspired by Euler integration. This approximation allows the model to receive direct supervision from data while avoiding costly Jacobian-vector product computations. As a result, the method is easier to train, uses less memory, and is more stable than related one-step generation approaches. We evaluate the method on image generation, point-cloud generation, signed-distance-function generation, and functional image generation. Across these tasks, Euler Mean Flows improves training efficiency while maintaining or improving generation quality. These results suggest a practical path toward faster and more accessible generative modeling across visual and scientific domains.