Rapid machine learning-based solutions of partial differential equations on complex domains.

Conventional methods like FEM (finite element method) and FVM (finite volume method) are mesh-based. However, if the computational domain consists of a complicated polygon with very short line segments, then it puts serious restrictions on mesh generation methods like triangulation. Recently, Berg et. al [1] have developed a deep unified ANN algorithm to
solve PDEs on complex computational domains. Their method is based on an ansatz for the solution which requires deep neural networks and an unconstrained gradient-based optimization method such as gradient descent or a quasi-Newton method. In this paper, we present physics informed extreme learning machine (PIELM), a new machine-learning algorithm,
which solves this problem with a simpler neural network architecture and an extremely fast learning routine. We demonstrate the efficacy of our method by solving the Poisson and biharmonic equation on complex 2D and 3D geometries such as the gyroid, which has important engineering applications.
Reference
[1] Berg, Jens, and Kaj Nyström. ”A unified deep artificial neural network approach to partial differential equations in complex geometries.” Neurocomputing 317 (2018): 28-41.