Augmented physics informed extreme learning machine to solve the biharmonic equations via Fourier expansions

To address the sensitivity of parameter and dissatisfactory precision for physics informed extreme machine learning (called PIELM) with common sigmoid, tangent and gaussian activation functions in solving high order partial differential equations (PDEs) arised from the fields of scientific computation and engineering applications. In this work, a Fourier induced PIELM (dubbed FPIELM) is proposed to approximate the solution for a class of fourth-order biharmonic equations with two types boundary conditions on unitized and non-unitized domains. By calculating carefully the differential operator and boundary operator of biharmonic equation on discretized collections, the solution of this high-order equation is casted into a linear least squares minimization problem. Additionally, we assess our FPIELM method with various hidden nodes and scale factor of uniform distribution initialization, then determine the optimal value range of the two hyperparameters required. Numerical experiments and comparsions for the FPIELM method and the other PIELM methods demonstrate that the proposed FPIELM method is more stable and robust, more high-precious and efficient in addressing biharmonic equations in both regular and irregular domains.