Tensor network approximation of Koopman operators

• URL: http://arxiv.org/abs/2407.07242v1
• Date: Tue, 9 Jul 2024 21:40:14 GMT
• Title: Tensor network approximation of Koopman operators
• Authors: Dimitrios Giannakis, Mohammad Javad Latifi Jebelli, Michael Montgomery, Philipp Pfeffer, Jörg Schumacher, Joanna Slawinska,
• Abstract summary: We propose a 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. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)n$.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $W_\tau$ that acts on a reproducing kernel Hilbert space $\mathcal H_\tau$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_\tau}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_\tau)$ generated by $\mathcal H_\tau$ that acts multiplicatively with respect to the tensor product. Our scheme also employs a representation of classical observables ($L^\infty$ functions of the state) by quantum observables (self-adjoint operators) acting on the Fock space, and a representation of probability densities in $L^1$ 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 $\mathcal H_\tau^{\otimes n} \subset F(\mathcal H_\tau)$ of arbitrarily high grading $n \in \mathbb N$. 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 $W_\tau$. 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.

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