Artifacts of Numerical Integration in Learning Dynamical Systems

TL;DR

This paper investigates how the choice of numerical integrator affects the learning of dynamical systems from data, revealing artifacts and proposing the implicit midpoint method for better property preservation.

Contribution

It demonstrates the impact of numerical schemes on learned dynamics and advocates for the implicit midpoint method to preserve system properties.

Findings

Explicit integrators can distort system stability and dynamics.

Reducing step size or increasing order does not always fix artifacts.

Implicit midpoint method preserves conservative or dissipative properties.

Abstract

In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the optimization procedure, prediction data from generic dynamics requires a numerical integrator to assess the mismatch with the observed data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. Specifically, the analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, even though it adequately fits the given data points. This paper shows that the stability region of the selected integrator will distort the nature of the learned dynamics. Crucially, reducing the step size or raising the order of an explicit integrator does not, in general, remedy this artifact, because higher-order explicit methods have stability regions that extend further into the right half complex plane. Furthermore, it is shown that the implicit midpoint method can preserve either conservative or dissipative properties from discrete data, offering a principled integrator choice even when the only prior knowledge is that the system is autonomous.