Modeling Error and Nonuniqueness of the Continuous-Time Models Learned via Runge–Kutta Methods

Abstract

In the present study, we consider continuous-time modeling of dynamics using observed data and formulate the modeling error caused by the discretization method used in the process. In the formulation, a class of linearized dynamics called Dahlquist’s test equations is used as representative of the target dynamics, and the characteristics of each discretization method for various dynamics are taken into account. The family of explicit Runge–Kutta methods is analyzed as a specific discretization method using the proposed framework. As a result, equations for predicting the modeling error are derived, and it is found that there can be multiple possible models obtained when using these methods. Several learning experiments using a simple neural network exhibited consistent results with theoretical predictions, including the nonuniqueness of the resulting model.

Keywords

• Runge–Kutta method
• ODE modeling
• dynamics learning
• modeling error analysis
• stability analysis
• Dahlquist’s test equation