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 $f_0(x) = |x/\sigma|^p$. This, in turn, 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 this family of densities. Concretely, this condition can be translated to a PDE between $V_t$ and $f_t$ and we minimize the amount by which this PDE fails to hold. We compare this objective to the reverse KL-divergence on Gaussian mixtures and on the $\phi^4$ lattice field theory on a circle.

翻译：我们提出一种用于连续归一化流的训练目标，该目标可在无样本但存在能量函数的情况下使用。我们的方法依赖于目标能量$f_1$与广义高斯能量函数$f_0(x) = |x/\sigma|^p$之间能量函数的指定或学习插值$f_t$。这进而诱导出玻尔兹曼密度$p_t \propto e^{-f_t}$的插值，我们旨在寻找一个依赖于时间的向量场$V_t$，使样本沿这族密度进行传输。具体而言，该条件可转化为关于$V_t$和$f_t$的偏微分方程，我们最小化该偏微分方程不成立的偏差程度。我们将此目标与高斯混合模型及圆上$\phi^4$格点场论中的反向KL散度进行了比较。