Complex spatiotemporal dynamics of physicochemical processes are often modeled at a microscopic level (through e.g. atomistic, agent-based or lattice models) based on first principles. Some of these processes can also be successfully modeled at the macroscopic level using e.g. partial differential equations (PDEs) describing the evolution of the right few macroscopic observables (e.g. concentration and momentum fields). Deriving good macroscopic descriptions (the so-called "closure problem") is often a time-consuming process requiring deep understanding/intuition about the system of interest. Recent developments in data science provide alternative ways to effectively extract/learn accurate macroscopic descriptions approximating the underlying microscopic observations. In this paper, we introduce a data-driven framework for the identification of unavailable coarse-scale PDEs from microscopic observations via machine learning algorithms. Specifically, using Gaussian Processes, Artificial Neural Networks, and/or Diffusion Maps, the proposed framework uncovers the relation between the relevant macroscopic space fields and their time evolution (the right-hand-side of the explicitly unavailable macroscopic PDE). Interestingly, several choices equally representative of the data can be discovered. The framework will be illustrated through the data-driven discovery of macroscopic, concentration-level PDEs resulting from a fine-scale, Lattice Boltzmann level model of a reaction/transport process. Once the coarse evolution law is identified, it can be simulated to produce long-term macroscopic predictions. Different features (pros as well as cons) of alternative machine learning algorithms for performing this task (Gaussian Processes and Artificial Neural Networks), are presented and discussed.

翻译：物理化学过程的复杂表面动态,通常在基于第一条原则的微观层面(例如,原子学、代理学或拉蒂基模型)建模,以显微镜层次(例如,原子学、代理学或拉蒂基模型)建模。其中一些过程也可以在宏观层次上建模,例如,部分差异方程式(PDEs),描述一些右大scocos(例如,集中和动力场)的演变情况。提出良好的宏观描述(所谓的“闭路反应问题”)往往是一个耗时的过程,需要深入了解/了解兴趣系统。数据科学的最新发展为有效提取/读取准确的宏观描述提供了替代方法。在本文中,我们引入了一个数据驱动框架,用于通过机器学习算法来识别无法使用的微镜级PDEs 。具体地说,使用高山进程、人工神经网络和(或者)数字级的货币级数据级进化地图,拟议的框架将明确揭示其长期数据级关系。