PINNS (physics informed neural nets) are becoming increasingly popular methods for using deep learning techniques to solve a wide variety of differential equations. They have been advertised as ‘mesh free methods’ which can out perform traditional methods. But how good are they in practice? In this talk I will look at how they compare with traditional techniques such as the finite element method on different types of PDE , linking their performance to that of general nonlinear approximation methods such as Free Knot Splines. I will show that a combination of ‘traditional’ numerical analysis and deep learning can yield good results. But there is still a lot to be learned about the performance and reliability of a PINN based method.
Joint work with Simone Appela, Teo Deveney, and Lisa Kreusser.