Numerical Artifacts in Learning Dynamical Systems
This addresses a critical issue for researchers and practitioners in machine learning and scientific computing who rely on numerical methods for system identification, highlighting a potential pitfall in optimization-based learning.
The paper reveals that numerical integration schemes used in learning dynamical systems from sampled data can cause serious artifacts, such as incorrectly identifying a damped oscillatory system as having anti-damping and reversed oscillation direction, even when the model fits the data well.
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, it is necessary to employ a numerical scheme to integrate candidate dynamical systems and assess how their solutions fit the data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. In particular, our analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, despite adequately fitting the given data points.