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This paper introduces a method to accelerate the computation of Koopman operator eigenspaces for continuous-time dynamical systems with reversible trajectories by exploiting the multiplicative group property of nowhere-vanishing eigenfunctions. The approach constructs a larger set of eigenfunctions by generating polynomials from a small set of conventionally approximated "principal" eigenfunctions, enriching the representation of observables. The paper also addresses the challenge of localized and extended singularities in eigenfunctions, enabling eigenfunction matching/continuation across these singularities.
Unlock exponentially faster computation of dynamical system representations by exploiting the algebraic structure of Koopman eigenfunctions.
For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.
