Flow Map Learning via Nongradient Vector Flow

2026-07-28

AI Summary

Existing flow matching methods rely on model inversion or iterative backpropagation, incurring substantial computational costs and lacking theoretical guarantees that the target ODE flow corresponds to a stationary point of the loss function. This work proposes SGFlow, which introduces—for the first time—a non-conservative dynamics mechanism based on stop-gradient (stopgrad) to directly learn ODE trajectories and their implicit velocity fields from scratch, without requiring explicit invertibility constraints or expensive differential operations. The method is rigorously proven to ensure that the target flow constitutes a stationary point of the system. Trained end-to-end with a standard regression loss, SGFlow achieves state-of-the-art FID on CIFAR benchmarks with 10-step sampling and matches leading methods at other step counts, making it the only flow matching approach currently endowed with such a stationary-point theoretical guarantee.

Abstract

Diffusion and flow-based models benefit from simple regression losses, but inference incurs significant overhead because sampling requires integration. Consistency models address this by directly learning the flow maps along the ODE trajectory, opening a design space between one-step and many-step approaches. However, existing methods face computational challenges such as requiring model inverses or backpropagation through iterated model calls, and do not always prove that the desired ODE flow map is a solution to the loss. We introduce SGFlow, an approach for learning flow maps that bypasses explicit invertibility constraints and expensive differentiation through model iteration. SGFlow trains a model to compute both the ODE solutions and the implied velocity from scratch by following non-conservative dynamics with a stationary point at the desired flow map. On the CIFAR image benchmark, no single method attains the best FID at every step count: SGFlow attains the best FID at 10 sampling steps and remains competitive with flow matching, Meanflow, and Lagrangian map matching at other step counts, while being the only one with a proven stationary-point guarantee for its stopgrad-based dynamics.

Problem

• flow map learning
• consistency models
• ODE trajectory
• invertibility constraints
• differentiation overhead

Innovation

• flow map learning
• nongradient dynamics
• SGFlow
• stationary-point guarantee
• ODE-based generative modeling