  We introduce a novel unit-time ordinary differential equation (ODE) flow
called the preconditioned F\"{o}llmer flow, which efficiently transforms a
Gaussian measure into a desired target measure at time 1. To discretize the
flow, we apply Euler's method, where the velocity field is calculated either
analytically or through Monte Carlo approximation using Gaussian samples. Under
reasonable conditions, we derive a non-asymptotic error bound in the
Wasserstein distance between the sampling distribution and the target
distribution. Through numerical experiments on mixture distributions in 1D, 2D,
and high-dimensional spaces, we demonstrate that the samples generated by our
proposed flow exhibit higher quality compared to those obtained by several
existing methods. Furthermore, we propose leveraging the F\"{o}llmer flow as a
warmstart strategy for existing Markov Chain Monte Carlo (MCMC) methods, aiming
to mitigate mode collapses and enhance their performance. Finally, thanks to
the deterministic nature of the F\"{o}llmer flow, we can leverage deep neural
networks to fit the trajectory of sample evaluations. This allows us to obtain
a generator for one-step sampling as a result.
