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：在量化分子系统集体行为的一个关键问题在于准确构建自由能曲面（FESs），主要挑战源于能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，以实现对全相空间的高效探索。另一方面，用于数值逼近FES的最优采样点收集问题仍未被充分研究，当系统具有大量集体变量（CVs）时，离散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，将曲面构建重新表述为一个同时优化函数表示和训练集的极小化极大问题。具体而言，最大化步骤通过结合对当前损失函数拉普拉斯近似的利用与对未知相空间的探索，建立随机相互作用粒子系统以实现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FES近似。通过迭代求解该极小化极大问题，本方法本质上实现了FES的对抗性学习，统一了相空间探索与后验误差增强采样两项任务。我们通过构建包含多达30个CV的分子系统FES验证了该方法的有效性。