Scientific Ml

Koopman Operator Learning

The Koopman operator recasts nonlinear dynamics as a linear evolution of observables, enabling linear tools on nonlinear systems.

Linearizing without approximation

Nonlinear dynamics are hard to analyze, but the Koopman viewpoint offers a striking alternative. Instead of tracking the state, it tracks functions of the state, called observables. The Koopman operator advances these observables in time and is exactly linear, even when the underlying dynamics are strongly nonlinear. The price is that the space of observables is infinite-dimensional.

Finding good observables

Kronos motion — operator learning

The practical challenge is choosing a finite set of observables whose evolution is approximately closed under the Koopman operator, so that the operator restricted to them stays linear. Classical dictionaries of candidate functions can work, but learning the observables with a neural network, an encoder into a coordinate system where the dynamics are linear, is often far more effective.

Why linearity is valuable

Linear systems have a complete spectral theory: eigenvalues set growth and oscillation
Prediction, control, and stability analysis all have mature linear tools
Modes reveal coherent structures and their time scales
A linear latent evolution is stable and interpretable over long horizons

Relation to dynamic mode decomposition

Dynamic mode decomposition is the standard data-driven method for approximating the Koopman operator: it fits the best linear operator relating snapshots one step apart. Extended and deep variants enrich the observables, with neural network encoders learning the coordinates in which the dynamics become linear. These methods share the goal of a linear surrogate for nonlinear behavior.

Limits

Systems with multiple stable states or chaos cannot be captured globally by a finite linear model, because no finite set of observables closes the dynamics everywhere. Within a single regime, however, Koopman-based models give compact, stable, and interpretable surrogates, which is why they are widely used for reduced-order modeling and control.