In-span learning: adapting reduced-order models using their own predictions

2026-07-03

AI Summary

This work addresses the degradation in accuracy of conventional reduced-order models when online dynamics deviate from the training distribution, a limitation stemming from their reliance on external information to update the reduced subspace. The authors propose an intrinsic span-learning mechanism that, for the first time, reveals endogenous signals embedded within the model’s own trajectory, which can be leveraged for adaptation. By employing incremental singular value decomposition with forgetting, the method dynamically reweights and aligns the subspace basis, recasting basis reconstruction as a dynamic preconditioner from the perspective of dynamical systems. This enables in-context learning without external supervision. The approach demonstrates significantly enhanced adaptability and predictive accuracy in out-of-distribution scenarios across three benchmark problems: three-dimensional helical flow, the viscous Burgers equation, and Fisher–KPP dynamics.

Abstract

Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data. Adaptive methods address this by updating the subspace online with external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. By streaming the model's own predictions through an incremental singular-value decomposition with forgetting, we obtain a trajectory-informed spectral preconditioner, in which the subspace is unchanged but the basis is reweighted and realigned toward the modes visited by the dynamics. This enables the model to absorb future out-of-span corrections more effectively. We expose aspects of this mechanism on a three-dimensional spiral and confirm it on viscous Burgers and Fisher-KPP dynamics. We also discuss how in-span learning can be viewed as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.

Problem

• reduced-order models
• online adaptation
• in-span learning
• dynamical systems
• model drift

Innovation

• in-span learning
• reduced-order models
• adaptive modeling
• incremental SVD
• trajectory-informed preconditioning