🤖 AI Summary
This paper identifies a critical implicit bias arising from numerical integration scheme selection in learning dynamical systems from sparse temporal observations: even with perfect data fit, inappropriate schemes (e.g., explicit Euler) can fundamentally mischaracterize system dynamics—such as misidentifying a true damped oscillator as exhibiting “anti-damping” and reverse oscillation. Method: We formulate a unified optimization framework to systematically analyze modeling bias induced by diverse numerical integrators (explicit/implicit, low-/high-order) in system identification. Contribution/Results: Through rigorous theoretical analysis and empirical validation, we demonstrate that such numerical artifacts can yield physically contradictory conclusions. Crucially, this work is the first to explicitly identify, formalize, and quantify the threat posed by numerical pseudospectra to the reliability of dynamical system learning—thereby providing both a foundational warning and methodological grounding for trustworthy physics-informed machine learning.
📝 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, 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.