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For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville\u2013Bratu\u2013Gelfand 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 numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.<\/jats:p>","DOI":"10.1007\/s10915-021-01650-5","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T11:04:07Z","timestamp":1633950247000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":77,"title":["Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines"],"prefix":"10.1007","volume":"89","author":[{"given":"Gianluca","family":"Fabiani","sequence":"first","affiliation":[]},{"given":"Francesco","family":"Calabr\u00f2","sequence":"additional","affiliation":[]},{"given":"Lucia","family":"Russo","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9568-3355","authenticated-orcid":false,"given":"Constantinos","family":"Siettos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,10,11]]},"reference":[{"key":"1650_CR1","doi-asserted-by":"crossref","unstructured":"Allen, E.J., Burns, J.A., Gilliam, D.S.: Numerical approximations of the dynamical system generated by burgers\u2019 equation with Neumann\u2013Dirichlet boundary conditions. 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