数值积分方法会扭曲动态系统学习结果，导致错误识别系统性质。
Artifacts of Numerical Integration in Learning Dynamical Systems
- 用数值积分评估数据拟合时，积分器选择会影响学习结果
- 低阶显式积分器可能让阻尼系统误判为反向振荡
- 隐式中点法可保持系统能量守恒或耗散特性，适合无先验场景
许多应用需从有限时间点的解中学习动力系统。学习问题常被表述为在选定函数类上的优化问题。然而，在优化过程中，通用动力系统的预测数据需通过数值积分器来评估与观测数据的差异。本文揭示了所选数值方案对学习结果可能产生严重干扰：例如，一个阻尼振荡系统可能被错误识别为具有‘反阻尼’特性并呈现相反振荡方向，尽管其拟合观测数据良好。分析表明，所选积分器的稳定性区域会扭曲所学动力学的本质。关键的是，减小步长或提高显式积分器阶数通常无法消除此类伪影，因为高阶显式方法的稳定性区域更深入复平面右半部分。此外，本文证明隐式中点法能从离散数据中保留保守或耗散性质，即使仅知系统为自治系统，也提供了合理的积分器选择依据。
原文摘要 · Abstract (English)
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.
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