Physics-Informed and Data-Driven Discovery of Governing Equations for Complex Phenomena in Heterogeneous Media

TL;DR

This paper reviews recent advances in physics-informed and data-driven methods for discovering governing equations of complex, heterogeneous phenomena, leveraging large datasets and machine learning to address unknown or partially known physical laws.

Contribution

It provides a comprehensive overview of emerging techniques combining physics-informed modeling with data-driven approaches for complex systems in heterogeneous media.

Findings

Various machine learning and symbolic regression methods are used to identify unknown governing equations.

Data-driven approaches enable modeling of complex phenomena where traditional equations are incomplete or unknown.

The paper discusses future directions for integrating physics-based and data-driven models.

Abstract

Rapid evolution of sensor technology, advances in instrumentation, and progress in devising data-acquisition softwares/hardwares are providing vast amounts of data for various complex phenomena, ranging from those in atomospheric environment, to large-scale porous formations, and biological systems. The tremendous increase in the speed of scientific computing has also made it possible to emulate diverse high-dimensional, multiscale and multiphysics phenomena that contain elements of stochasticity, and to generate large volumes of numerical data for them in heterogeneous systems. The difficulty is, however, that often the governing equations for such phenomena are not known. A prime example is flow, transport, and deformation processes in macroscopically-heterogeneous materials and geomedia. In other cases, the governing equations are only partially known, in the sense that they either contain various coefficients that must be evaluated based on data, or that they require constitutive relations, such as the relationship between the stress tensor and the velocity gradients for non-Newtonian fluids in the momentum conservation equation, in order for them to be useful to the modeling. Several classes of approaches are emerging to address such problems that are based on machine learning, symbolic regression, the Mori-Zwanzig projection operator formulation, sparse identification of nonlinear dynamics, data assimilation, and stochastic optimization and analysis, or a combination of two or more of such approaches. This Perspective describes the latest developments in this highly important area, and discusses possible future directions.