# Quantum mechanics and data assimilation

## TL;DR

This paper introduces a novel data assimilation framework inspired by quantum mechanics, utilizing operator theory and machine learning to improve filtering in complex dynamical systems.

## Contribution

It develops a quantum-inspired data assimilation method combining operator-theoretic ergodic theory with data-driven techniques, enabling effective filtering of chaotic and non-Gaussian systems.

## Key findings

- Successfully applied to Lorenz 63 system demonstrating robustness.
- Handles highly non-Gaussian statistics and complex geometries.
- Converges with large data under mild assumptions.

## Abstract

A framework for data assimilation combining aspects of operator-theoretic ergodic theory and quantum mechanics is developed. This framework adapts the Dirac--von Neumann formalism of quantum dynamics and measurement to perform sequential data assimilation (filtering) of a partially observed, measure-preserving dynamical system, using the Koopman operator on the $L^2$ space associated with the invariant measure as an analog of the Heisenberg evolution operator in quantum mechanics. In addition, the state of the data assimilation system is represented by a trace-class operator analogous to the density operator in quantum mechanics, and the assimilated observables by self-adjoint multiplication operators. An averaging approach is also introduced, rendering the spectrum of the assimilated observables discrete, and thus amenable to numerical approximation. We present a data-driven formulation of the quantum mechanical data assimilation approach, utilizing kernel methods from machine learning and delay-coordinate maps of dynamical systems to represent the evolution and measurement operators via matrices in a data-driven basis. The data-driven formulation is structurally similar to its infinite-dimensional counterpart, and shown to converge in a limit of large data under mild assumptions. Applications to periodic oscillators and the Lorenz 63 system demonstrate that the framework is able to naturally handle highly non-Gaussian statistics, complex state space geometries, and chaotic dynamics.