Enhancing Computational Efficiency in Multiscale Systems Using Deep Learning of Coordinates and Flow Maps

TL;DR

This paper introduces a deep learning-based framework for efficient multiscale system simulation by jointly discovering coordinates and flow maps, significantly reducing computational costs while maintaining high predictive accuracy.

Contribution

It presents a novel deep learning approach that combines coordinate discovery and flow map estimation for multiscale systems, improving efficiency and accuracy.

Findings

Achieves state-of-the-art predictive accuracy

Reduces computational costs compared to traditional methods

Successfully applied to neuron and chaotic PDE models

Abstract

Complex systems often show macroscopic coherent behavior due to the interactions of microscopic agents like molecules, cells, or individuals in a population with their environment. However, simulating such systems poses several computational challenges during simulation as the underlying dynamics vary and span wide spatiotemporal scales of interest. To capture the fast-evolving features, finer time steps are required while ensuring that the simulation time is long enough to capture the slow-scale behavior, making the analyses computationally unmanageable. This paper showcases how deep learning techniques can be used to develop a precise time-stepping approach for multiscale systems using the joint discovery of coordinates and flow maps. While the former allows us to represent the multiscale dynamics on a representative basis, the latter enables the iterative time-stepping estimation of the reduced variables. The resulting framework achieves state-of-the-art predictive accuracy while incurring lesser computational costs. We demonstrate this ability of the proposed scheme on the large-scale Fitzhugh Nagumo neuron model and the 1D Kuramoto-Sivashinsky equation in the chaotic regime.