Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrence

TL;DR

This paper introduces a neural network-based method that uses inner recurrence to infer detailed system dynamics from coarse, under-sampled observational data, enabling accurate long-term predictions of complex systems.

Contribution

The paper proposes a novel inner recurrence technique within deep neural networks to recover fine-scale dynamics from coarse observations, supported by mathematical proof and numerical experiments.

Findings

Successfully recovers fine-scale dynamics from coarse data

Demonstrates effectiveness on systems of ODEs and PDEs

Provides mathematical justification for the approach

Abstract

Recent work has focused on data-driven learning of the evolution of unknown systems via deep neural networks (DNNs), with the goal of conducting long term prediction of the dynamics of the unknown system. In many real-world applications, data from time-dependent systems are often collected on a time scale that is coarser than desired, due to various restrictions during the data acquisition process. Consequently, the observed dynamics can be severely under-sampled and do not reflect the true dynamics of the underlying system. This paper presents a computational technique to learn the fine-scale dynamics from such coarsely observed data. The method employs inner recurrence of a DNN to recover the fine-scale evolution operator of the underlying system. In addition to mathematical justification, several challenging numerical examples, including unknown systems of both ordinary and partial differential equations, are presented to demonstrate the effectiveness of the proposed method.