Edriss Titi
Texas A&M University, Mathematics, Faculty Member
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Generating high‐resolution flow fields is of paramount importance for various applications in engineering and climate sciences. This is typically achieved by solving the governing dynamical equations on high‐resolution meshes, suitably... more
Generating high‐resolution flow fields is of paramount importance for various applications in engineering and climate sciences. This is typically achieved by solving the governing dynamical equations on high‐resolution meshes, suitably nudged toward available coarse‐scale data. To alleviate the computational cost of such downscaling process, we develop a physics‐informed deep neural network (PI‐DNN) that mimics the mapping of coarse‐scale information into their fine‐scale counterparts of continuous data assimilation (CDA). Specifically, the PI‐DNN is trained within the theoretical framework described by Foias et al. (2014, https://doi.org/10.1070/rm2014v069n02abeh004891) to generate a surrogate of the theorized determining form map from the coarse‐resolution data to the fine‐resolution solution. We demonstrate the PI‐DNN methodology through application to 2D Rayleigh‐Bénard convection, and assess its performance by contrasting its predictions against those obtained by dynamical downscaling using CDA. The analysis suggests that the surrogate is constrained by similar conditions, in terms of spatio‐temporal resolution of the input, as the ones required by the theoretical determining form map. The numerical results also suggest that the surrogate's downscaled fields are of comparable accuracy to those obtained by dynamically downscaling using CDA. Consistent with the analysis of Farhat, Jolly, and Titi (2015, https://doi.org/10.48550/arxiv.1410.176), temperature observations are not needed for the PI‐DNN to predict the fine‐scale velocity, pressure and temperature fields.
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Research Interests: Mathematics, Applied Mathematics, Turbulence, Navier-Stokes Equations, Grid, and 6 moreBo, Turbulence Models, D, F, L, and Q
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Research Interests: Mathematics, Geology, Geophysics, Physics, Geophysical Fluid Dynamics, and 6 moreUniqueness, D, B, A, Q, and Springer Ebooks
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Research Interests: Mathematics, Applied Mathematics, Physics, Turbulence, Large Eddy Simulation, and 15 morePure Mathematics, Computation Fluid Dynamics, Global Attractor, Navier Stokes, Uniqueness, Length scale, Three Dimensional, Upper Bound, Turbulence Model, Global existence, Initial Condition, Degree of Freedom, periodic boundary condition, Euler Equation, and attractor
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Research Interests: Applied Mathematics, Mechanics, Convection, Pure Mathematics, Dissipation, and 4 moreUniqueness, K, A, and Q
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One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters -... more
One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters - finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is being determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.