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：本文提出了一种直接估计统计力学中高维平衡分布矩或边缘分布的方法，通过从低阶簇矩或边缘分布的角度求解高维Fokker-Planck方程。该方法规避了估计完整高维分布所需的指数级复杂度，转而直接求解低阶矩/边缘分布的简化偏微分方程。此外，所提出的矩/边缘分布松弛方法具有完全凸性，可直接通过现有求解器进行计算。我们进一步提出了该凸与时间相关变量结合的时变版本，以研究非平衡动力学。研究表明，该方法能够恢复平衡密度的平均场近似。数值实验结果展示了所提算法在高维系统中的性能表现。