Coarse-scale PDEs from fine-scale observations via machine learning
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.
Code (0)
등록된 구현이 없습니다.
Tasks
BIG-bench Machine LearningGaussian ProcessesSimilar Papers 제목 키워드 기반
Particles to Partial Differential Equations Parsimoniously
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the…
Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data
We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -- and the closures that lead to them -- from high-fidelity, individual-based stochas…
BIG-bench Machine LearningGaussian ProcessesConstructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach
We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit ma…
feature selectionProbabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations
We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the coefficients (e.g. material parameters) exhibit random, fine scale variability. The direct solution to such problems requires grids that are sma…
Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning
Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-tem…
Inductive BiasMulti-agent Reinforcement Learning