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. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

翻译：我们提出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，该方法通过低阶团簇矩或边缘分布求解高维Fokker-Planck方程。借助该方法，我们绕过了估计完整高维分布的指数复杂度，直接求解针对低阶矩/边缘分布简化后的偏微分方程。此外，所提出的矩/边缘松弛是完全凸的，可通过现成的求解器求解。我们进一步提出这些凸程序的时间依赖版本，用于研究非平衡动力学。数值结果展示了所提算法在高维系统中的性能。