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Quantum mechanics and data assimilation.

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Summary
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This study introduces a novel data assimilation framework inspired by quantum mechanics and ergodic theory. It effectively processes complex data from dynamical systems, handling non-Gaussian statistics and chaos.

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Area of Science:

  • * Physics and Applied Mathematics
  • * Quantum Mechanics and Dynamical Systems

Background:

  • * Traditional data assimilation methods struggle with complex, chaotic, and non-Gaussian systems.
  • * Operator-theoretic ergodic theory offers tools for analyzing measure-preserving dynamical systems.
  • * Quantum mechanics provides a formalism for dynamics, measurement, and state representation.

Purpose of the Study:

  • * To develop a novel data assimilation framework by integrating quantum mechanics and ergodic theory.
  • * To adapt quantum formalisms for sequential data assimilation (filtering) of dynamical systems.
  • * To create a data-driven formulation for practical implementation.

Main Methods:

  • * Adaptation of the Dirac-von Neumann formalism for quantum dynamics and measurement.
  • * Utilizing the Koopman operator on L^{2} space as an analog to the Heisenberg evolution operator.
  • * Representing system states with trace-class operators and observables with self-adjoint multiplication operators.
  • * Employing kernel methods from machine learning and delay-coordinate maps for a data-driven approach.

Main Results:

  • * The framework successfully performs sequential data assimilation on partially observed, measure-preserving dynamical systems.
  • * A data-driven formulation converges under mild assumptions, mirroring the infinite-dimensional theory.
  • * Applications to periodic oscillators and the Lorenz 63 system demonstrate robustness with non-Gaussian statistics, complex geometries, and chaotic dynamics.

Conclusions:

  • * The developed quantum-inspired framework provides a powerful new approach to data assimilation.
  • * It offers a robust method for analyzing complex dynamical systems previously intractable for standard techniques.
  • * The data-driven formulation ensures practical applicability and scalability.