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Explore a mathematical lecture that addresses the numerical challenges of shock waves in high-speed gas dynamics through innovative geometric approaches. Learn how shock formation occurs when flow maps reach the boundary of diffeomorphism manifolds and discover a novel solution using modified geometry where geodesics approach but never reach this boundary. Understand information geometric regularization (IGR) as a method that maintains smooth solutions while avoiding excessive dissipation typical of viscous regularizations, thereby accelerating shock flow simulations. Examine the theoretical foundations including proof of global strong IGR solutions in unidimensional pressureless cases 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 in collaboration with researchers. Delve into the connection between the modified diffeomorphism manifold geometry and information geometry of mass density, leading to the broader concept of information geometric mechanics that treats continuum mechanical PDE solutions as parameters of probability distributions from statistical physics. Gain insights into how replacing Euclidean geometry of individual particles with information geometry of statistical families can create more performant numerical methods that preserve density and energy positivity while integrating seamlessly with scientific machine learning approaches.
