Koopman Generator Skill

Core Idea

The Koopman operator K linearizes nonlinear dynamics by lifting to infinite-dimensional observable space:

State space (nonlinear)     Observable space (linear)
      x_{t+1} = f(x_t)   →   (Kg)(x) = g(f(x))

Key property: K is linear even when f is nonlinear.

Connection to DMD

DMD finds finite-rank approximation of K:

K ≈ Φ Λ Φ†

• Φ = DMD modes (approximate Koopman eigenfunctions)
• Λ = eigenvalues

As ACSet Morphism

Koopman = natural transformation on observable presheaves:

# Observable functor
F: StateSpace → ObservableSpace

# Koopman as pushforward
K = f_*: Sh(X) → Sh(X)

GF(3) Triads

dmd-spectral (-1) ⊗ structured-decomp (0) ⊗ koopman-generator (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ acsets (0) ⊗ koopman-generator (+1) = 0 ✓

References

• Brunton et al. "Modern Koopman Theory" (2021)
• Mezić "Spectral Properties of Dynamical Systems" (2005)
• PyDMD: https://github.com/mathLab/PyDMD