Numerical approximation of the Koopman-von Neumann equation: Operator learning and quantum computing
This work addresses the challenge of numerical approximation in operator learning for classical mechanics, with potential applications in quantum computing, but it appears incremental as it builds on existing Koopman and transfer operator frameworks.
The paper tackled the problem of approximating the Koopman-von Neumann operator from data, leveraging its unitary property for non-Hamiltonian dynamics, and demonstrated results on examples like oscillators and the Lotka-Volterra model.
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.