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.
| Subjects: | Numerical Analysis (math.NA); Machine Learning (cs.LG) |
| Cite as: | arXiv:2507.14491 [math.NA] |
| (or arXiv:2507.14491v3 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2507.14491 arXiv-issued DOI via DataCite |
From: Bing-Ze Lu [view email]
[v1]
Sat, 19 Jul 2025 05:23:39 UTC (2,166 KB)
[v2]
Sat, 26 Jul 2025 08:38:09 UTC (2,166 KB)
[v3]
Wed, 22 Apr 2026 08:46:38 UTC (1,805 KB)
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