  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.
