  We introduce a training objective for continuous normalizing flows that can
be used in the absence of samples but in the presence of an energy function.
Our method relies on either a prescribed or a learnt interpolation $f_t$ of
energy functions between the target energy $f_1$ and the energy function of a
generalized Gaussian $f_0(x) = ||x/\sigma||_p^p$. The interpolation of energy
functions induces an interpolation of Boltzmann densities $p_t \propto
e^{-f_t}$ and we aim to find a time-dependent vector field $V_t$ that
transports samples along the family $p_t$ of densities. The condition of
transporting samples along the family $p_t$ is equivalent to satisfying the
continuity equation with $V_t$ and $p_t = Z_t^{-1}e^{-f_t}$. Consequently, we
optimize $V_t$ and $f_t$ to satisfy this partial differential equation. We
experimentally compare the proposed training objective to the reverse
KL-divergence on Gaussian mixtures and on the Boltzmann density of a quantum
mechanical particle in a double-well potential.
