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## Modeling and Computation Seminar

#### An Invitation to Machine Learning for Pure and Applied Mathematicians

### Abstract

Machine learning is rapidly evolving as a powerful tool in enhancing computational physics, pattern recognition, artificial intelligence, etc. In this talk we will first describe some of the basic elements of the theory of statistical learning by V. N. Vapnik and A. Chervonenkis such as support vector machine, reproducing kernel methods etc. We will then indicate possible future directions in connecting these developments to some of the latest results in statistical and stochastic Navier-Stokes equations such as invariant-ergodic measures, martingale solutions, nonlinear stochastic filtering and large deviations (all deal with the probability laws of fluid velocity), with the long term vision of developing machine learning to turbulence.## Applied Math Colloquium

#### Rigorous Questions in Compressible Euler and Navier-Stokes Equations

### Abstract

Two famous developments in nonlinear partial differential equations in the past century continue to inspire modern mathematicians: (1) Solvability theory of incompressible three dimensional fluid dynamics initiated by. J. Leray in the 1930’s and further crystallized by E. Hopf and O. A. Ladyzhenskaya in the 1950’s which is also one of the six remaining Clay Institute Millennium Prize Problems; (2) Solvability theory of compressible three dimensional fluid dynamics initiated by James Serrin (1959) and John Nash (1962), with the most decisive results coming from P. L. Lions in the 1990’s. Both of these subjects are important in engineering applications as they address mathematical aspects of low-speed and high-speed aerodynamics. In particular, stochastic analysis, control theory, filtering/estimation, turbulence closure modeling, and numerical analysis of viscous aerodynamics essentially depend on our understanding of the solvability theory of these physical problems which add to the importance of such analysis. In this talk we will discuss the recent progress and open problems in the rigorous theory of compressible viscous and inviscid fluid dynamics in multi-dimensions. Both weak and strong solutions have famous developments and open problems. We will also discuss the stochastic counterpart and control theoretic aspects.