Quantum mechanics and data assimilation

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.

arXiv comment20 pages, 6 figures

References in corpus (15)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Applied Koopmanism
• Consistency of spectral clustering
• Data-driven model reduction and transfer operator approximation
• Data Assimilation: A Mathematical Introduction
• Evaluating Data Assimilation Algorithms
• Nonparametric forecasting of low-dimensional dynamical systems
• The Lorenz attractor is mixing
• Delay-coordinate maps and the spectra of Koopman operators
• Koopman analysis of the long-term evolution in a turbulent convection cell
• A Priori Estimation Of Memory Effects In Coarse-Grained Nonlinear Systems Using The Mori-Zwanzig Formalism
• A Quantum Extended Kalman Filter
• Conditions for successful data assimilation
• Bi-stochastic kernels via asymmetric affinity functions
• Semiparametric forecasting and filtering: correcting low-dimensional model error in parametric models

Cited by in corpus (5)

• Quantum Cognitive Triad. Semantic geometry of context representation
• Quantum dynamics of the classical harmonic oscillator
• Data Assimilation in Operator Algebras
• Koopman and transfer operator techniques from the perspective of quantum theory
• Covariance Linkage Assimilation method for Unobserved Data Exploration