  We address a new numerical scheme based on a class of machine learning
methods, the so-called Extreme Learning Machines with both sigmoidal and
radial-basis functions, for the computation of steady-state solutions and the
construction of (one dimensional) bifurcation diagrams of nonlinear partial
differential equations (PDEs). For our illustrations, we considered two
benchmark problems, namely (a) the one-dimensional viscous Burgers with both
homogeneous (Dirichlet) and non-homogeneous mixed boundary conditions, and, (b)
the one and two-dimensional Liouville-Bratu-Gelfand PDEs with homogeneous
Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs,
exact analytical solutions are available and used for comparison purposes
against the numerical derived solutions. Furthermore, the numerical efficiency
(in terms of accuracy and size of the grid) of the proposed numerical machine
learning scheme is compared against central finite differences (FD) and
Galerkin weighted-residuals finite-element methods (FEM). We show that the
proposed ELM numerical method outperforms both FD and FEM methods for medium to
large sized grids, while provides equivalent results with the FEM for low to
medium sized grids.
