Alex Bihlo (Memorial University)
Title:Deep neural networks for solving differential equations on general orientable surfaces.
Abstract:We present a general method for solving partial differential equations on orientable surfaces using deep neural networks. The method rests on embedding the given differential equation on a surface in a higher-dimensional Cartesian space and solving the differential equations in extrinsic coordinates that are then restricted in a suitable way to the surface itself. The solution is approximated with a neural network, hence allowing for derivatives being computed using automatic differentiation. We illustrate the method by solving the shallow-water equations on the sphere, and various reaction-diffusion equations on general surfaces such as the bumpy sphere, Boy's surface and the Stanford bunny. This is joint work with Roman O. Popovych.