Diffusion and flow-based models benefit from simple regression losses, but inference (i.e, producing samples) incurs significant computational overhead because it requires integration. Consistency models address this overhead by directly learning the flow maps along the ODE trajectory, revealing a design space for the learning problem between one-step and many-step approaches. However, existing consistency training methods feature 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 CurlFlow, an approach for learning flow maps that bypasses explicit invertibility constraints and expensive differentiation through model iteration. CurlFlow trains a model to compute both the ODE solutions and the implied velocity from scratch by following non-conservative dynamics (i.e., those with curl) with stationary point at the desired flow map. On the CIFAR image benchmark, CurlFlow attains a favorable relationship of FID to step count, relative to flow matching, MeanFlow, and several other flow map learning methods.
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