Koopman Operator Theory

Koopman Operator Theory is a mathematical framework that represents nonlinear dynamical systems as linear operators in an infinite-dimensional function space — enabling the use of powerful linear analysis tools (eigenvalues, modes, spectral decomposition) on inherently nonlinear systems.

What Is the Koopman Operator?

• Concept: Instead of tracking the state $x(t)$ directly, track "observables" $g(x(t))$ — functions of the state.
• Linearity: The Koopman operator $\mathcal{K}$ advances observables linearly: $\mathcal{K}g(x) = g(f(x))$, even if $f$ is nonlinear.
• Approximation: In practice, approximate the infinite-dimensional operator using data-driven methods (Dynamic Mode Decomposition, deep learning).

Why It Matters

• Linear Control: Once the nonlinear system is "linearized" via Koopman, standard linear control methods apply.
• Process Modeling: Used in semiconductor manufacturing for modeling plasma etch dynamics and other nonlinear processes.
• Interpretability: Koopman modes provide physical insight into the dominant dynamics of complex systems.

Koopman Operator Theory is seeing nonlinear systems through a linear lens — a mathematical transformation that makes the intractable tractable.