Learning Interpolations between Boltzmann Densities

Published: 30 May 2023, Last Modified: 30 May 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: 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.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: - rewrote the background section to connect it better to the continuity equation - this also means and updated and considerably simplified terminology
Code: https://github.com/balintmate/boltzmann-interpolations
Assigned Action Editor: ~Ruoyu_Sun1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 809