Numerical approximation of the Koopman-von Neumann equation: Operator learning and quantum computing

TL;DR

This paper develops numerical methods to approximate the Koopman-von Neumann operator using operator learning and quantum computing concepts, enabling analysis of classical dynamical systems with quantum-inspired tools.

Contribution

It introduces a framework for approximating the Koopman-von Neumann operator and its eigenfunctions from data, leveraging quantum circuit representations and basis function choices.

Findings

The operator can be represented as a unitary matrix suitable for quantum circuits.

Choice of basis functions and domain critically affects the operator's well-definedness.

Illustrations include oscillators and the Lotka-Volterra model.

Abstract

The Koopman-von Neumann equation describes the evolution of wavefunctions associated with autonomous ordinary differential equations and can be regarded as a quantum physics-inspired formulation of classical mechanics. The main advantage compared to conventional transfer operators such as Koopman and Perron-Frobenius operators is that the Koopman-von Neumann operator is unitary even if the dynamics are non-Hamiltonian. Projecting this operator onto a finite-dimensional subspace allows us to represent it by a unitary matrix, which in turn can be expressed as a quantum circuit. We will exploit relationships between the Koopman-von Neumann framework and classical transfer operators in order to derive numerical methods to approximate the Koopman-von Neumann operator and its eigenvalues and eigenfunctions from data. Furthermore, we will show that the choice of basis functions and domain are crucial to ensure that the operator is well-defined. We will illustrate the results with the aid of guiding examples, including simple undamped and damped oscillators and the Lotka-Volterra model.