Koopman operator methods

Koopman theory offers a linear-but-infinite-dimensional view of a nonlinear dynamical system: instead of tracking the nonlinear evolution of the state x, one tracks the linear evolution of observables g(x) under the Koopman operator, which advances any function of the state one step forward. If one can find a set of observables (a Koopman-invariant subspace) that the operator maps linearly among themselves, a nonlinear system becomes exactly linear in those coordinates, with eigenvalues and eigenfunctions (Koopman modes) describing intrinsic frequencies, growth/decay rates and coherent structures.

For neural data the promise is a principled nonlinear-to-linear lift: learn observables — classically via extended-DMD dictionaries, now often via autoencoders that jointly learn the lifting and a linear latent operator — so that population dynamics become linear, interpretable, and amenable to the whole linear-systems toolbox (spectra, controllability, stability). The honest caveats are that finite-dimensional Koopman-invariant subspaces exist only approximately for generic nonlinear systems, that spectra from short, noisy neural recordings are hard to estimate reliably, and that a learned lift can overfit; Koopman is a powerful lens, not a guaranteed exact linearization.