Using machine learning to extract coarse-scale PDEs from fine-scale data

Due to the complexity of the spatiotemporal evolution in microscopic complex systems, scientists often need to use simpler, macroscale approaches for generating models to cut back on computational cost. Even though recent efforts have opened the possibility of determining coarse-scale governing equations from fine-scale data, the determination of proper coarse-scale variables remains a major challenge for such approaches.

Lee et al. propose a framework for extracting coarse-scale partial differential equations (PDEs) from fine-scale data using machine learning. The approach generates alternative possible forms of governing equations by enhancing existing techniques such as Gaussian processes and feature selection methods for neural networks, and introducing additional manifold learning approaches, such as Diffusion Maps.

“The fact that one can encode the same phenomenon in many apparently different ways is something that we have become more and more interested in,” said author Ioannis Kevrekidis. “How do we understand that the law I found and the one you found, even though in terms of different observables and different formulas, are, in effect, the same law?”

The group focused on finding PDEs for reaction and transport processes, but also assessed how input feature selection can affect the framework’s ability to predict the behavior of a Lattice Boltzmann mesoscopic model.

“What was initially surprising to us is that if the spatiotemporal data live on low-dimensional manifolds, these manifolds can be parametrized in many different ways that can be transformed to each other,” Kevrekidis said. “This work is a set of simple numerical experiments demonstrating how different machine learning and data mining tools can be synthesized to learn macroscopic laws from microscopic simulations without explicit closures.”

Source: “Coarse-scale PDEs from fine-scale observations via machine learning,” by Seungjoon Lee, Mahdi Kooshkbaghi, Konstantinos Spiliotis, Constantinos I. Siettos, and Ioannis G. Kevrekidis, Chaos (2019). The article can be accessed at https://doi.org/10.1063/1.5126869 .