Macroscopic observables of a system are of keen interest in real applications such as the design of novel materials. Current methods rely on microscopic trajectory simulations, where the forces on all microscopic coordinates need to be computed or measured. However, this can be computationally prohibitive for realistic systems. In this paper, we propose a method to learn macroscopic dynamics requiring only force computations on a subset of the microscopic coordinates. Our method relies on a sparsity assumption: the force on each microscopic coordinate relies only on a small number of other coordinates. The main idea of our approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters. We provide a theoretical justification of this under suitable conditions. We demonstrate the accuracy, force computation efficiency, and robustness of our method on learning macroscopic closure models from a variety of microscopic systems, including those modeled by partial differential equations or molecular dynamics simulations.

翻译：在现实应用中，如新材料设计，系统的宏观可观测量备受关注。现有方法依赖于微观轨迹模拟，需要计算或测量所有微观坐标上的作用力。然而，对于实际系统而言，这可能在计算上难以实现。本文提出一种仅需在部分微观坐标上计算作用力即可学习宏观动力学的方法。我们的方法基于一个稀疏性假设：每个微观坐标上的作用力仅依赖于少量其他坐标。该方法的核心思想是将宏观坐标上的训练过程映射回微观坐标，在微观坐标上可利用部分作用力计算作为随机估计来更新模型参数。我们在适当条件下为此提供了理论依据。通过在多种微观系统（包括偏微分方程或分子动力学模拟建模的系统）上学习宏观闭合模型，我们验证了本方法在准确性、作用力计算效率及鲁棒性方面的优势。