Solving PDEs with Neural Networks

Partial differential equations (PDEs) are used to model a multitude of phenomena encountered in science and engineering. However, a closed-form expression of the solution is rarely available for most practical problems. Thus, numerical algorithms are employed to approximately solve PDEs. Some commonly used methods include finite difference/volume methods, finite element methods, spectral Galerkin methods, and also deep neural networks! To better appreciate some of these methods, especially deep neural networks, let us consider a simple model problem describing the scalar advection-diffusion problem in one-dimension.