From Classical to Quantum-Mechanical Data Assimilation: A Comparison between DATO and QMDA

TL;DR

This paper compares two operator-based data assimilation methods, DATO and QMDA, analyzing their mathematical structures, computational costs, and empirical performance on benchmark systems.

Contribution

It provides a unified operator-theoretic comparison of DATO and QMDA, highlighting their differences, advantages, and limitations in various observational regimes.

Findings

DATO and QMDA have distinct assimilation paradigms.

Both methods perform differently depending on observational noise and sparsity.

The study clarifies regimes where each method is most effective.

Abstract

Data assimilation provides a systematic framework for combining dynamical models with partial and noisy observations to infer the evolving state of a system. In this work, we undertake a comparative study of Data Assimilation with Transfer Operators (DATO) and Quantum Mechanical Data Assimilation (QMDA), focusing on their mathematical formulation, algorithmic structure, and empirical performance. Both methods are first cast within a common operator-theoretic framework, which makes it possible to compare, on a unified basis, their representations of uncertainty, forecast propagation, and assimilation updates. We then analyse their principal similarities and differences with respect to state-space structure, update mechanisms, structural preservation properties, and computational cost. To complement the theoretical analysis, we assess both approaches on benchmark dynamical systems across a range of observational settings, including noisy, sparse, and partially observed regimes. Our results show that, despite their shared operator-theoretic motivation, DATO and QMDA embody substantially different assimilation paradigms, leading to distinct advantages and limitations in terms of interpretability, robustness, and scalability. The present study helps delineate the regimes in which each framework is most effective and offers broader insight into the design of operator-based methodologies for data assimilation.