Enabling equation-free modeling via diffusion maps

TL;DR

This paper introduces an application-independent method using diffusion maps to construct lifting and restriction operators for equation-free modeling, enabling efficient analysis of complex systems without prior domain insight.

Contribution

The authors develop a novel diffusion map-based algorithm that automatically constructs lifting and restriction operators from simulation data, removing the need for a priori knowledge.

Findings

Successfully constructs operators from data without prior insight

Enables efficient low-dimensional modeling of complex systems

Code implementation is publicly available

Abstract

Equation-free modeling aims at extracting low-dimensional macroscopic dynamics from complex high-dimensional systems that govern the evolution of microscopic states. This algorithm relies on lifting and restriction operators that map macroscopic states to microscopic states and vice versa. Combined with simulations of the microscopic state, this algorithm can be used to apply Newton solvers to the implicitly defined low-dimensional macroscopic system or solve it more efficiently using direct numerical simulations. The key challenge is the construction of the lifting and restrictions operators that usually require a priori insight into the underlying application. In this paper, we design an application-independent algorithm that uses diffusion maps to construct these operators from simulation data. Code is available at https://doi.org/10.5281/zenodo.5793299.