Learning Long-Horizon Flow Maps in High Dimensions: From a Regression Surrogate to Efficient Statistical Learning

Establishing statistically efficient neural flow-map learning for one-step generation requires a valid training objective in the absence of direct flow-map supervision. To this end, we develop (i) a flow-map regression surrogate perspective and (ii) a dimension-efficient statistical learning theory that exploits intrinsic mixture structure of real data. Specifically, we introduce transport loss, which identifies the true flow map and controls its regression error, providing a common framework for MeanFlow, CM, and sCM objectives. We then investigate statistical learnability under this objective. While generic learning bounds suffer from the curse of dimensionality, we establish learning guarantees with polynomial dependence on intrinsic and latent dimensions under a low-rank Gaussian mixture assumption, using a layered residual network construction to control approximation and estimation errors. These insights motivate Gaussian Mixture Residual Flow (GMRFlow), which combines analytic Gaussian component flow maps with a learned neural residual correction. Experiments on CelebA80 and FFHQ64 show that GMRFlow achieves improved one-step FID over MeanFlow baselines.