Multiscale and Nonlocal Learning for PDEs using Densely Connected RNNs

TL;DR

This paper introduces Densely Connected RNNs (DC-RNNs), a novel framework that effectively learns multiscale, nonlocal PDEs from data, capturing complex dynamics like transport and collision operators.

Contribution

The paper proposes a new DC-RNN architecture incorporating multiscale ansatz and IMEX schemes to identify complex PDEs from discrete data, advancing data-driven PDE modeling.

Findings

DC-RNN accurately captures multiscale PDE dynamics.

The method outperforms existing approaches in numerical experiments.

It effectively identifies transport and nonlocal operators.

Abstract

Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of numerically challenging PDEs arising in multiscale modeling and simulation -- kinetic equations. These equations are usually nonlocal and contain scales/parameters that vary by several orders of magnitude. We introduce an efficient framework, Densely Connected Recurrent Neural Networks (DC-RNNs), by incorporating a multiscale ansatz and high-order implicit-explicit (IMEX) schemes into RNN structure design to identify analytic representations of multiscale and nonlocal PDEs from discrete-time observations generated from heterogeneous experiments. If present in the observed data, our DC-RNN can capture transport operators, nonlocal projection or collision operators, macroscopic diffusion limit, and other dynamics. We provide numerical results to demonstrate the advantage of our proposed framework and compare it with existing methods.