majiayu000 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