  A generative model based on a continuous-time normalizing flow between any
pair of base and target probability densities is proposed. The velocity field
of this flow is inferred from the probability current of a time-dependent
density that interpolates between the base and the target in finite time.
Unlike conventional normalizing flow inference methods based the maximum
likelihood principle, which require costly backpropagation through ODE solvers,
our interpolant approach leads to a simple quadratic loss for the velocity
itself which is expressed in terms of expectations that are readily amenable to
empirical estimation. The flow can be used to generate samples from either the
base or target, and to estimate the likelihood at any time along the
interpolant. In addition, the flow can be optimized to minimize the path length
of the interpolant density, thereby paving the way for building optimal
transport maps. In situations where the base is a Gaussian density, we also
show that the velocity of our normalizing flow can also be used to construct a
diffusion model to sample the target as well as estimate its score. However,
our approach shows that we can bypass this diffusion completely and work at the
level of the probability flow with greater simplicity, opening an avenue for
methods based solely on ordinary differential equations as an alternative to
those based on stochastic differential equations. Benchmarking on density
estimation tasks illustrates that the learned flow can match and surpass
conventional continuous flows at a fraction of the cost, and compares well with
diffusions on image generation on CIFAR-10 and ImageNet $32\times32$. The
method scales ab-initio ODE flows to previously unreachable image resolutions,
demonstrated up to $128\times128$.
