In-span learning: adapting reduced-order models using their own predictions
2026-07-03 • Machine Learning
Machine LearningComputational Engineering, Finance, and Science
AI summaryⓘ
The authors explain that models designed to simplify complex systems struggle when faced with new situations outside their original training. They found a new way to improve these models by learning from their own ongoing predictions without changing the overall structure, but by adjusting how important each internal pattern is. This method helps the model better handle unexpected changes in the system it’s modeling. They tested their idea on several example systems and suggest it’s similar to how some AI models learn from context within a task.
reduced-order modelssingular-value decompositionadaptive methodssubspacespectral preconditionerBurgers equationFisher-KPP equationin-span learningdynamical systemsincremental learning
Authors
Amirpasha Hedayat, Laura Balzano, Karthik Duraisamy
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.