Solving Differential Equations with Machine Learning

Topic: Study basics of applying of machine learning technics for solving ordinarily differential equations and partial differential equations
Description:
In the last years machine learning becomes a major tool in a lot of practical disciplines. In the course we consider applications of such type of techniques for numerical simulation of ordinarily partial differential equations. The main difference with the classical learning is that instead of trainining dataset we use differential equations. In the course we would consider 3 main techniques Residual Neural Networks (ResNets), Physically inspired neural networks (PINNS), and Gaussian-process regression. We would consider theoretical basis as well as positive (and negative) examples of applications. 
The topics includes

-- Basics of neural networks

-- Basics of numerical methods for ODE

-- Residual neural networks

-- Solving ODE with residual neural networks

-- Neural networks as universal approximator

-- Physically inspired neural networks

-- elliptic equations

-- parabolic equations

-- Bayesian method

-- Basics of Gaussian Fields

-- Gaussian-process regression