Deep Neural Networks and Adaptive Quadrature for Solving Variational Problems

The great success of deep neural networks (DNNs) in such areas as image processing, natural language processing has motivated also their usage in many other areas. It has been shown that in particular cases they provide very good approximation to different classes of functions. The aim of this work is to explore the usage of deep learning methods for approximation of functions, which are solutions of boundary value problems for particular differential equations. More specific, the class of methods known as physics-informed neural network will be explored. Components of the DNN algorithms, such as the definition of loss function and the choice of the minimization method will be discussed while presenting results from the computational experiments.