  Obtaining the solutions of partial differential equations based on various
machine learning methods has drawn more and more attention in the fields of
scientific computation and engineering applications. In this work, we first
propose a coupled Extreme Learning Machine (called CELM) method incorporated
with the physical laws to solve a class of fourth-order biharmonic equations by
reformulating it into two well-posed Poisson problems. In addition, some
activation functions including tangent, gauss, sine, and trigonometric
(sin+cos) functions are introduced to assess our CELM method. Notably, the sine
and trigonometric functions demonstrate a remarkable ability to effectively
minimize the approximation error of the CELM model. In the end, several
numerical experiments are performed to study the initializing approaches for
both the weights and biases of the hidden units in our CELM model and explore
the required number of hidden units. Numerical results show the proposed CELM
algorithm is high-precision and efficient to address the biharmonic equation in
both regular and irregular domains.
