Physics-Informed Neural Network for Numerical PDE Subtitle: with examples of shockwave solution to Burger's equation

We use fully-connected neural networks as smooth functions to approximate the solution of a 1D Burger's equation with discontinuous initial condition, and propose a loss functional to evaluate the approximation which gives an update direction with respect to the parameters of the neural network. The ideas from preceding papers don't fully work and a minor fix needs to be posed. The well-trained neural network approximates the exact shockwave solution with small error, which outperforms traditional finite-difference methods. This method can be easily extended to high dimensions, high orders, and a large class of PDEs with more general boundaries.