Alternating method for solving a biharmonic inverse problem in detection of Robin coefficients

In this paper, we are interested in an inverse problem of detection of corrosion for a biharmonic equation which consists to determine the Robin coefficients in the inaccessible part of the boundary from the Riquiert-Neumann data on the accessible one. For this end, we consider the factorisation of the biharmonic problem which gives rise to two Cauchy problems for Laplace and Poisson equations. An algorithm based on an alternative iterative method is proposed allowing to complete the missing Cauchy data and then recover the Robin coefficients. We show the feasibility of this approach by numerical reconstructions.Publisher's Versio

Extreme Learning Machine-Assisted Solution of Biharmonic Equations via Its Coupled Schemes

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