On computing the hyperparameter of extreme learning machines: Algorithm and application to computational PDEs, and comparison with classical and high-order finite elements

We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM hidden-layer coefficients in neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed, where $R_m$ is a user-provided constant, output-layer trained by linear or nonlinear least squares computation. present method computing optimal value based evolution algorithm. The presented enables us illuminate characteristics two types configurations: (i) Single-Rm-ELM, which single used generating all hidden layers, (ii) Multi-Rm-ELM, multiple constants involved with each different layer. adopt from this also incorporate other improvements into implementation. particular, here we compute operators involving output fields last layer forward-mode auto-differentiation, as opposed reverse-mode auto-differentiation previous work. These significantly reduce training time enhance performance. systematically compare performance current improved that finite element (FEM), both classical second-order FEM high-order Lagrange elements higher degrees, solving number PDEs. It shown far outperforms FEM. Its comparable smaller problem sizes, larger sizes markedly