Fourier heuristic PINNs to solve the biharmonic equations based on its coupled scheme

Abstract: Physics-informed neural networks (PINNs) have been widely utilized for solving a range of partial differential equations (PDEs) in various scientific and engineering disciplines. This paper presents a Fourier heuristic-enhanced PINN (termed FCPINN) designed to address a specific class of biharmonic equations with Dirichlet and Navier boundary conditions. The method achieves this by decomposing the high-order equations into two Poisson equations. FCPINN integrates Fourier spectral theory with a reduced-order formulation for high-order PDEs, significantly improving approximation accuracy and reducing computational complexity. This approach is especially beneficial for problems with intricate boundary constraints and high-dimensional inputs. To assess the effectiveness and robustness of the FCPINN algorithm, we conducted several numerical experiments on both linear and nonlinear biharmonic problems across different Euclidean spaces. The results show that FCPINN provides an optimal trade-off between speed and accuracy for high-order PDEs, surpassing the performance of conventional PINN and deep mixed residual method (MIM) approaches, while also maintaining stability and robustness with varying numbers of hidden layer nodes.