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Explore advanced computational mechanics through this seminar presentation that introduces information geometric regularization (IGR) as a novel approach to solving shock wave problems in high-speed gas dynamics. Learn how classical finite element solvers and scientific machine learning face severe numerical difficulties when dealing with shock waves, and discover how modifying the geometry of the diffeomorphism manifold can provide smooth solutions without excessive dissipation. Examine the theoretical foundation showing that shock formation occurs when the flow map reaches the boundary of the manifold of diffeomorphisms, and understand how IGR modifies this geometry so geodesics approach but never reach the boundary. Study the mathematical proof of global strong IGR solutions in the unidimensional pressureless case and observe practical applications through multidimensional examples featuring complex shock interactions. Discover how this approach enabled the first compressible flow simulation exceeding a quadrillion degrees of freedom, representing a significant computational achievement. Delve into the connection between the modified diffeomorphism manifold geometry and the information geometry of mass density, which motivates a broader framework of information geometric mechanics. Understand how this perspective views solutions of continuum mechanical partial differential equations as parameters of probability distributions derived from statistical physics, and explore how replacing Euclidean geometry of individual particles with information geometry of statistical families can lead to more performant numerical methods that preserve positivity of densities and energies while integrating seamlessly with scientific machine learning approaches.
