Learning Interpolations between Boltzmann Densities
This work addresses a challenge in generative modeling for physics and machine learning where samples are unavailable, offering a novel training objective that could benefit applications in computational chemistry and statistical mechanics.
The authors tackled the problem of training continuous normalizing flows without samples by using an energy function, introducing a method based on interpolations between Boltzmann densities. They experimentally validated their approach on Gaussian mixtures and a quantum mechanical double-well potential, showing competitive results compared to reverse KL-divergence.
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 $f_0(x) = ||x/σ||_p^p$. 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.