Fourier-feature induced physics informed randomized neural network method to solve the biharmonic equation

To address the sensitivity of parameters and limited precision for randomized neural network method (RNN) with common activation functions, such as sigmoid, tangent, and Gaussian, in solving high-order partial differential equations (PDEs) relevant to scientific computation and engineering applications, this work develops a Fourier-feature induced physics informed single layer RNN (FPISRNN) method. This approach aims to approximate the solution for a class of fourth-order biharmonic equation with two boundary conditions on both unitized and non-unitized domains. By carefully calculating the differential and boundary operators of the biharmonic equation on discretized collections, the solution for this high-order equation is reformulated as a linear least squares minimization problem. We further evaluate the FPISRNN method with varying hidden nodes and scaling factors for uniform distribution initialization, and then determine the optimal range for these two hyperparameters. Numerical experiments and comparative analyses demonstrate that the proposed FPISRNN method is more stable, robust, precise, and efficient than other FPISRNN approaches in solving biharmonic equations for both regular and irregular domains.