Tensor network approximation of Koopman operators Article Swipe
Dimitrios Giannakis
,
Mohammad Javad Latifi Jebelli
,
Michael Montgomery
,
Philipp Pfeffer
,
Jörg Schumacher
,
Joanna Sławińska
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2407.07242
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2407.07242
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 $\mathcal H_τ$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_τ}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_τ)$ generated by $\mathcal H_τ$ 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 $\mathcal H_τ^{\otimes n} \subset F(\mathcal H_τ)$ 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 $W_τ$. 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.
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- en
- Landing Page
- http://arxiv.org/abs/2407.07242
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- OA Status
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- Related Works
- 10
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https://openalex.org/W4400600694Canonical identifier for this work in OpenAlex
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https://doi.org/10.48550/arxiv.2407.07242Digital Object Identifier
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Tensor network approximation of Koopman operatorsWork title
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preprintOpenAlex work type
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enPrimary language
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2024Year of publication
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2024-07-09Full publication date if available
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Dimitrios Giannakis, Mohammad Javad Latifi Jebelli, Michael Montgomery, Philipp Pfeffer, Jörg Schumacher, Joanna SławińskaList of authors in order
- Landing page
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https://arxiv.org/abs/2407.07242Publisher landing page
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https://arxiv.org/pdf/2407.07242Direct link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2407.07242Direct OA link when available
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Tensor (intrinsic definition), Mathematics, Computer science, Algebra over a field, Pure mathematicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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