Inverse modified differential equations for discovery of dynamics

TL;DR

This paper introduces inverse modified differential equations (IMDE) to analyze the behavior and errors in discovering dynamics with ODE-net, providing bounds on errors and insights into learning Hamiltonian systems.

Contribution

It presents a novel IMDE framework for error analysis in ODE-net based dynamics discovery, including bounds on errors and theoretical insights into Hamiltonian systems.

Findings

Difference between learned ODE and IMDE is bounded by learning loss and small discrepancy.

Total error in ODE-net is bounded by discrete error and learning loss.

Experimental results verify theoretical error bounds.

Abstract

The combination of numerical integration and deep learning, i.e., ODE-net, has been successfully employed in a variety of applications. In this work, we introduce inverse modified differential equations (IMDE) to contribute to the behaviour and error analysis of discovery of dynamics using ODE-net. It is shown that the difference between the learned ODE and the truncated IMDE is bounded by the sum of learning loss and a discrepancy which can be made sub exponentially small. In addition, we deduce that the total error of ODE-net is bounded by the sum of discrete error and learning loss. Furthermore, with the help of IMDE, theoretical results on learning Hamiltonian system are derived. Several experiments are performed to numerically verify our theoretical results.