  We propose the characteristic generator, a novel one-step generative model
that combines the efficiency of sampling in Generative Adversarial Networks
(GANs) with the stable performance of flow-based models. Our model is driven by
characteristics, along which the probability density transport can be described
by ordinary differential equations (ODEs). Specifically, We estimate the
velocity field through nonparametric regression and utilize Euler method to
solve the probability flow ODE, generating a series of discrete approximations
to the characteristics. We then use a deep neural network to fit these
characteristics, ensuring a one-step mapping that effectively pushes the prior
distribution towards the target distribution. In the theoretical aspect, we
analyze the errors in velocity matching, Euler discretization, and
characteristic fitting to establish a non-asymptotic convergence rate for the
characteristic generator in 2-Wasserstein distance. To the best of our
knowledge, this is the first thorough analysis for simulation-free one step
generative models. Additionally, our analysis refines the error analysis of
flow-based generative models in prior works. We apply our method on both
synthetic and real datasets, and the results demonstrate that the
characteristic generator achieves high generation quality with just a single
evaluation of neural network.
