Koopman operators

Gold definitionUpdated Apr 2, 2026

Definition

Koopman operators linearize nonlinear dynamical systems by lifting them to an infinite-dimensional observable space, enabling the analysis and prediction of complex behaviors using linear system theory. In machine learning, they are used to learn parsimonious, generalizable representations of dynamics, improving extrapolation and long-term prediction.

At a glance

Executive summary

Koopman operators are a mathematical tool that simplifies complex, nonlinear systems by finding a way to describe their behavior using simpler, linear equations in a different mathematical space. When combined with AI models like Physics-Informed Neural Networks, they help these models better understand and predict how systems change over time and space, even for situations they haven't seen before.

TL;DR

Koopman operators turn complicated system behaviors into simpler linear ones, helping AI models predict complex dynamics more accurately and generalize better.

Key points

Lifts nonlinear dynamics to an observable space where their evolution becomes linear.
Improves generalization, temporal extrapolation, and long-term prediction for complex dynamical systems, especially when integrated with PINNs.
Used by researchers in scientific machine learning, fluid dynamics, control theory, and applied mathematics.
Unlike direct nonlinear modeling, Koopman operators provide a linear framework for analysis and prediction, offering unconditional stability for stiff systems.
Growing trend of integration with neural networks for data-driven discovery of dynamics and enhancing the robustness of physics-informed models.

Use cases

Predicting long-term behavior of chaotic systems like the Lorenz attractor.
Modeling and forecasting fluid dynamics, such as Navier-Stokes equations.
Improving the generalization of Physics-Informed Neural Networks for various PDEs (parabolic, hyperbolic, dispersive, stiff).
Developing stable numerical integration schemes for stiff differential equations.
Reduced-order modeling for complex engineering systems.

Also known as

Koopman operator theory, Koopman analysis, Koopman spectral analysis, Koopman eigenfunctions