(344a) Coarse-Scale PDEs from Microscopic Observations Via Machine Learning

Complex spatiotemporal dynamics of physicochemical processes are often modeled on a microscopic level (through molecular dynamic models, agent-based models, lattice-based models) based on first principles. Some of these processes can also be successfully modeled at the macroscopic level using partial differential equations (PDE) for the right macroscopic observables (not molecules and their velocities, say, but rather concentration and momentum fields). However, deriving the macroscopic PDE that effectively models the microscopic physics (the so-called "closure problem") requires deep understanding/intuition about the complex system of interest; the discovery of macroscopic governing equations is often a difficult and time-consuming process. Recent developments of data-driven approaches via machine learning algorithms provide an alternative way to effectively extract the macroscopic nonlinear PDEs approximating the underlying microscopic observations, without prior knowledge/hypotheses.

In this work, we introduce a novel framework to identify unavailable coarse-scale PDEs from microscopic observations through machine learning algorithms. Specifically, using Gaussian processes, 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). This framework will be illustrated through the data-driven discovery of the macroscopic, concentration-level PDE resulting from a fine-scale, Lattice Boltzmann model of a reaction/transport process. Long-term macroscopic prediction is facilitated by simulation of the coarse-grained PDE identified from data. The different features, as well as the pros and cons of our three different machine learning approaches for performing this task (Gaussian Processes, neural networks, and geometric harmonics based on diffusion maps) will also be discussed.