Least squares with equality constraints extreme learning machines for the resolution of PDEs

In this paper, we investigate the use of single hidden-layer neural networks as a family of ansatz functions for the resolution of partial differential equations (PDEs). In particular, we train the network via Extreme Learning Machines (ELMs) on the residual of the equation collocated on -eventually randomly chosen- points. Because the approximation is done directly in the formulation, such a method falls into the framework of Physically Informed Neural Networks (PINNs) and has been named PIELM. Since its first introduction, the method has been refined variously, and one successful variant is the Extreme Theory of Functional Connections (XTFC). However, XTFC strongly takes advantage of the description of the domain as a tensor product. Our aim is to extend XTFC to domains with general shapes. The novelty of the procedure proposed in the present paper is related to the treatment of boundary conditions via constrained imposition, so that our method is named Least Squares with Equality constraints ELM (LSE-ELM). An in-depth analysis and comparison with the cited methods is performed, again with the analysis of the convergence of the method in various scenarios. We show the efficiency of the procedure both in terms of computational cost and in terms of overall accuracy.