Error analysis based on inverse modified differential equations for discovery of dynamics using linear multistep methods and deep learning

TL;DR

This paper extends error analysis for discovering dynamical systems with deep learning and linear multistep methods by introducing inverse modified differential equations, providing bounds on discovery errors.

Contribution

It introduces the concept of inverse modified differential equations for linear multistep methods and derives a priori error estimates for deep learning-based system discovery.

Findings

Error bounds are established combining discretization and learning loss.

Numerical experiments verify the theoretical error estimates.

Learned models closely approximate inverse modified differential equations.

Abstract

Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates for the discovery of dynamics using linear multistep methods. Several numerical experiments are performed to verify the theoretical analysis.