In-Span Learning Adapts Models Using Their Own Predictions

Reduced-order models (ROMs) are valuable for compressing high-dimensional dynamics into faster, lower-dimensional representations. However, their accuracy degrades when real-world dynamics deviate from the initial training data. Existing adaptive methods typically rely on external information, such as full-order corrections or sensor data, to update the model's underlying subspace. This research identifies a new, complementary adaptation channel: "in-span learning." In-span learning allows a ROM to adapt using only its own predictions. By continuously streaming the model's generated trajectories through an incremental singular-value decomposition with forgetting, the model's basis is reweighted and realigned towards the modes most visited by the dynamics. Crucially, the underlying subspace itself remains unchanged. This internal adaptation mechanism enables the model to more effectively incorporate future external corrections, enhancing its robustness to drifting online dynamics. The concept is demonstrated on various dynamic systems, suggesting a new principle for computational science where model-generated data holds more usable information than previously recognized.