One essential problem in quantifying the collective behaviors of molecular systems lies in the accurate construction of free energy surfaces (FESs). The main challenges arise from the prevalence of energy barriers and the high dimensionality. Existing approaches are often based on sophisticated enhanced sampling methods to establish efficient exploration of the full-phase space. On the other hand, the collection of optimal sample points for the numerical approximation of FESs remains largely under-explored, where the discretization error could become dominant for systems with a large number of collective variables (CVs). We propose a consensus sampling-based approach by reformulating the construction as a minimax problem which simultaneously optimizes the function representation and the training set. In particular, the maximization step establishes a stochastic interacting particle system to achieve the adaptive sampling of the max-residue regime by modulating the exploitation of the Laplace approximation of the current loss function and the exploration of the uncharted phase space; the minimization step updates the FES approximation with the new training set. By iteratively solving the minimax problem, the present method essentially achieves an adversarial learning of the FESs with unified tasks for both phase space exploration and posterior error-enhanced sampling. We demonstrate the method by constructing the FESs of molecular systems with a number of CVs up to 30.

翻译：量化分子系统集体行为的一个核心问题在于精确构建自由能曲面（FES）。主要挑战来自能量势垒的普遍存在以及高维特性。现有方法通常基于复杂的增强采样技术，以实现对整个相空间的高效探索。另一方面，用于数值逼近自由能曲面的最优样本点收集问题仍远未得到充分研究，对于具有大量集体变量（CV）的系统，离散化误差可能成为主导因素。我们提出一种基于共识采样的方法，将自由能曲面构建重新表述为一个极大极小问题，同时优化函数表示和训练集。具体而言，最大化步骤建立了一个随机相互作用粒子系统，通过调节当前损失函数拉普拉斯近似的利用与未知相空间的探索，实现对最大残差区域的自适应采样；最小化步骤则利用新训练集更新自由能曲面近似。通过迭代求解该极大极小问题，本方法实质上实现了自由能曲面的对抗学习，统一了相空间探索与后验误差增强采样的任务。我们通过构建多达30个集体变量的分子系统自由能曲面来演示该方法。