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 $p_t$ can be translated to a PDE between $V_t$ and $f_t$ and we optimize $V_t$ and $f_t$ to satisfy this PDE. 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.
翻译:我们引入了一种连续归一化流的训练目标,可用于在没有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$f_1$和广义高斯能量函数$f_0(x) = ||x/\sigma||_p^p$之间的预设或学习插值$f_t$。能量函数的插值引出Boltzmann密度的插值$p_t \propto e^{-f_t}$,我们旨在寻找一个沿着密度族$p_t$运输样本的时变向量场$V_t$。沿着族$p_t$运输样本的条件可以转化为$V_t$和$f_t$之间的PDE,我们优化$V_t$和$f_t$以满足该PDE。我们在高斯混合和双阱势能量子力学粒子的Boltzmann密度上,实验比较了所提出的训练目标和KL散度的反转。