An Error Analysis Framework for Neural Network Modeling of Dynamical Systems

TL;DR

This paper introduces a theoretical framework to analyze and quantify modeling errors caused by numerical integration when training neural networks to learn dynamical systems, using classical numerical analysis tools.

Contribution

It formulates the modeling error in neural network-based dynamical system learning using the Dahlquist test equation, bridging neural modeling and numerical analysis.

Findings

Error analysis using Dahlquist test equation

Application to Runge--Kutta methods

Insights into discretization errors in neural training

Abstract

We propose a theoretical framework for investigating a modeling error caused by numerical integration in the learning process of dynamics. Recently, learning equations of motion to describe dynamics from data using neural networks has been attracting attention. During such training, numerical integration is used to compare the data with the solution of the neural network model; however, discretization errors due to numerical integration prevent the model from being trained correctly. In this study, we formulate the modeling error using the Dahlquist test equation that is commonly used in the analysis of numerical methods and apply it to some of the Runge--Kutta methods.