Abstract：For the evolutionary nonlinear differential equations, a significant phenomenon that happens frequently is the instability of the solution on the initial values, especially for the fluid dynamics. Perhaps the simplest one is a set of chaotic solutions of the Lorenz system, a simplified mathematical model for atmospheric convection. Due to imperfect initial data, the "butterfly effect" stemming from the real-world implications of the Lorenz attractor makes the long-time prediction very challenging, e.g., weather forecast. Generally speaking, the short-term behavior of a predictive model with imperfect initial data is a critical issue for weather and climate predictability. Furthermore, understanding the model's sensitivity to errors in the initial data to assess subsequent errors in forecasts is important. In this talk, I will present some results on the sensitivity of solutions for the burgers equation and the forced nonlinear Schrodinger equation by use of the methods of nonlinear optimization and statistical machine learning, as well as the applications in atmospheric blocking and extreme weather, which are based on the joint work with Dehai Luo, Guodong Sun and Wenqi Zhang.
