Why Directly Learning Periodic Trajectories Can Fail

2026-09-26

AI Summary

This study addresses the generalization failure in directly learning periodic trajectories, which arises from phase misalignment and the accumulation of frequency discrepancies. We elucidate the intrinsic mechanism by which frequency differences induce a lower bound on prediction error and identify the inherent limitations of recording data within fixed time windows. To overcome these issues, this work proposes a learning framework that decouples waveform and period representations. By integrating operator learning with phase alignment techniques, we establish regularity theory for the decoupled objectives and validate the approach through numerical simulations of ordinary and partial differential equations. The proposed method effectively prevents predictive divergence and significantly enhances cross-domain generalization capabilities for periodic systems.

Abstract

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.

Problem

• operator learning
• periodic trajectories
• phase alignment
• generalization failure
• frequency variation

Innovation

• Operator learning
• Periodic trajectories
• Phase alignment
• Decoupled representation
• Frequency accumulation