We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics, via solving the high-dimensional Fokker-Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. We show the proposed method can recover the meanfield approximation of an equilibrium density. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

翻译：我们提出了一种方法，通过基于低阶团簇矩或边际量求解高维Fokker-Planck方程，直接估计统计力学中高维平衡分布的矩或边际量。该方法规避了估计完整高维分布所需的指数级复杂度，直接求解针对低阶矩/边际量简化后的偏微分方程。此外，所提出的矩/边际松弛是完全凸的，可使用现成求解器进行求解。我们还提出了一种依赖时间的凸规划版本，用于研究非平衡动力学。研究表明，该方法能恢复平衡密度的平均场近似。数值结果验证了该算法在高维系统中的性能表现。