Tensor network approximation of Koopman operators

We propose a tensor network framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator by a diagonalizable, skew-adjoint operator W_τ that acts on a reproducing kernel Hilbert space H_τ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators et W_τ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space F( H_τ) generated by H_τ that acts multiplicatively with respect to the tensor product. Our scheme also employs a representation of classical observables (L^∞ functions of the state) by quantum observables (self-adjoint operators) acting on the Fock space, and a representation of probability densities in L¹ by quantum states. Combining these constructions leads to an approximation of the Koopman evolution of observables that is representable as evaluation of a tree tensor network built on a tensor product subspace H_τ⊗ n ⊂ F( H_τ) of arbitrarily high grading n ∈ N. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension (2d+1)ⁿ, generated from a collection of $2d + 1$ eigenfunctions of W_τ. Furthermore, the approximation is positivity preserving. The paper contains a theoretical convergence analysis of the method and numerical applications to two dynamical systems on the 2-torus: an ergodic torus rotation as an example with pure point Koopman spectrum and a Stepanoff flow as an example with topological weak mixing.