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We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification. 

Abstract Introduction Digital twins, a form of artificial intelligence, are virtual representations of the physical world. In the past 20 years, digital twins have been utilized to track wind turbines' operations, monitor spacecraft's status, and even create a model of the Earth for climate research. While digital twins hold much promise for the neurocritical care unit, the question remains on how to best establish the rules that govern these models. This model will expand on our group’s existing digital twin model for the treatment of sepsis. Methods The authors of this project collaborated to create a Direct Acyclic Graph (DAG) and an initial series of 20 DELPHI statements, each with six accompanying sub-statements that captured the pathophysiology surrounding the management of acute ischemic strokes in the practice of Neurocritical Care (NCC). Agreement from a panel of 18 experts in the field of NCC was collected through a 7-point Likert scale with consensus defined a-priori by ≥ 80% selection of a 6 (“agree”) or 7 (“strongly agree”). The endpoint of the study was defined as the completion of three separate rounds of DELPHI consensus. DELPHI statements that had met consensus would not be included in subsequent rounds of DELPHI consensus. The authors refined DELPHI statements that did not reach consensus with the guidance of de-identified expert comments for subsequent rounds of DELPHI. All DELPHI statements that reached consensus by the end of three rounds of DELPHI consensus would go on to be used to inform the construction of the digital twin model. Results After the completion of three rounds of DELPHI, 93 (77.5%) statements reached consensus, 11 (9.2%) statements were excluded, and 16 (13.3%) statements did not reach a consensus of the original 120 DELPHI statements. Conclusion This descriptive study demonstrates the use of the DELPHI process to generate consensus among experts and establish a set of rules for the development of a digital twin model for use in the neurologic ICU. Compared to associative models of AI, which develop rules based on finding associations in datasets, digital twin AI created by the DELPHI process are easily interpretable models based on a current understanding of underlying physiology.