AI 中文总结
本研究揭示直接学习周期轨迹因相位随参数快速变化而失败，提出分别学习对齐波形与周期的方法，有效避免泛化误差。
AI 中文摘要
周期解的算子学习需要决定如何记录和表示模拟数据。一个自然的选择是积分足够长的时间以使瞬态衰减，并记录一个足够宽的窗口以包含所有轨迹的至少一个完整周期。我们发现，这些保守的选择可能使生成的轨迹难以学习，即使底层周期轨道随系统参数规则变化。未对齐的轨迹即使在训练分布内也泛化能力差。相位对齐显著改善了分布内泛化，但在固定物理时间窗口上训练的模型对周期超出训练范围的轨迹仍有较大误差。我们通过一个共同的机制解释这两种失败：频率差异随时间累积，因此目标相位随参数快速变化。无法跟踪这种变化的预测器在两种设置下都会产生总体均方误差下限；对于固定窗口预测，我们还推导出每个样本的下界。然后，我们研究了一种能避开这些下限的最简单表示之一：分别学习对齐的归一化波形及其周期。我们在常微分方程假设下建立了解耦目标的规律性，并通过实验表明，这种方法在常微分方程系统和偏微分方程案例研究中均避免了这两种失败。
英文摘要
Operator learning of periodic solutions requires deciding how simulation data should be recorded and represented. A natural choice is to integrate long enough for transients to decay and record a window wide enough to contain at least one full period of all trajectories. We find that these conservative choices can make the resulting trajectories difficult to learn, even when the underlying periodic orbits vary regularly with system parameters. Unaligned trajectories generalize poorly even within the training distribution. Phase alignment substantially improves in-distribution generalization, but models trained on a fixed physical-time window still have large errors on trajectories with periods outside the training range. We explain both failures through a common mechanism: frequency differences accumulate over time, so the target phase varies rapidly with the parameters. Predictors that cannot track this variation incur a population MSE floor in both settings; for fixed window prediction, we also derive a per-sample lower bound. We then study one of the simplest representations that escape these floors: learning an aligned, normalized waveform and its period separately. We establish regularity of the decoupled targets under ODE assumptions and show experimentally that this approach avoids both failures in ODE systems and a PDE case study.
