Quantum Theory for Spectral Analysis, Prediction, and Control of Classical Systems
Abstract
Modern science is undergoing what may arguably be called a data revolution, whereby an exponentialrate of increase of computing and data acquisition capabilities, coupled with advances inmachine learning, has enabled the creation of data-driven models for complex physical processesthat match or exceed the skill of first-principles models. Realizing the full spectrum of benefits ofthis new para digm hinges upon the development of physical and mathematical frameworks that canintrinsically handle probabilistic descriptions of complex systems in the presence of highly nonlineardynamics. Quantum mechanics, together with its mathematical underpinnings in oper atoralgebras and gauge theory, provides such a framework which has historically shown tremendoussuccess in describing atomic- and su batomic-scale phenomena, but whose applicability and potentialbenefits in the context of classical complex systems, such as turbulen t fluid flows, remainlargely unexplored. In response, we propose a comprehensive research program to devise novelquantum mechanical formulations of key problems in modeling of classical dynamical systems,and create data-driven implementations of these techniques t hrough machine learning approaches.Specifically, the proposed research will focus on (i) Learning dynamical laws from data throughsp ectral analysis of Koopman operators on non-commutative algebras of observables; (ii) Analysisof observables with spatial structure and dynamical symmetries using gauge theory; (iii) Quantummechanical approaches for stochastic subgrid-scale modeling and control; a nd (iv) Developmentof compilation strategies for simulation of classical dynamics on quantum computers.Collectively, these technique s will create a new paradigm for statistical modeling of classicalsystems based on state-of-art approaches from fundamental physics and mathematics, realizedthrough machine learning. A core element of this paradigm is that formulating finite-dimensionalapproximati on schemes in non-abelian operator spaces enables preservation of intrinsic algebraicstructures of classical dynamical systems, such as the Leibniz rule for vector fields, in ways whichare not possible through conventional discretizations. Moreover, for systems wi th spatial structure,the proposed gauge-covariant methodologies will enable seamless fusion of sensor data acquiredfrom different fr ames of reference, providing representations of physical configurations and lawsin their intrinsic geometrical form. Looking ahead t o the quantum computing era, the project willlay out the foundations for scalable, consistent quantum simulation of classical system s throughencodings of Koopman operators in quantum circuits.The proposed research program also has a strong applied component, addre ssing challengingcurrent problems in climate dynamics and fluid dynamics. Areas of focus include uncertaintyquantifiedforecasting of the El Nino Southern Oscillation, subgrid-scale modeling of convectiveprocesses, and spatiotemporal interpolation of high-resoluti on, along-track satellite altimetry foroceanic turbulence.The mathematical and computati will find wideapplicability in many DoD-relevant contexts including artificial intelligence, simulation of naturaland engineered sys tems, and quantum information. In addition, results from domain-scientificapplications will impact DoD strategic and operational cap abilities through improved analysis andforecasting of environmental data. The project will contribute to workforce development throu ghinterdisciplinary training of postdoctoral researchers and PhD students.Dimitrios Giannakis, Dartmouth College Approved for Public Release
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Oct 22, 2021
- Source ID
- N000142112946
Entities
People
- Dimitrios Giannakis
Organizations
- Board of Trustees of Dartmouth College
- Office of Naval Research
- United States Navy