Advanced machine learning technique for solving elliptic partial differential equations using Legendre spectral neural networks

In this work, a novel approach based on a single-layer machine learning Legendre spectral neural network (LSNN) method is used to solve an elliptic partial differential equation. A Legendre polynomial based approach is utilized to generate neurons that fulfill the boundary conditions. The loss function is computed by using the error back-propagation principles and a feed-forward neural network model combined with automatic differentiation. The main advantage of using this methodology is that it does not need to solve a system of nonlinear and nonsparse equations compared with other traditional numerical schemes, which makes this algorithm more convenient for solving higher-dimensional equations. Further, the hidden layer is eliminated with the help of a Legendre polynomial to enlarge the input pattern. The neural network's training accuracy and efficiency were significantly enhanced by the innovative sampling technique and neuron architecture. Moreover, the Legendre spectral approach can handle equations on more complex domains because of numerous networks. Several test problems were used to validate the proposed scheme, and a comparison was made with other neural network schemes consisting of the physics-informed neural network (PINN) scheme. We found that our proposed scheme has a very good agreement with PINN, which further enhances the reliability and efficiency of our proposed method. The absolute and relative error in both $L_2$ and $L_{\mathrm{\infty }}$ between exact and numerical solutions are provided, which shows that our numerical method converges exponentially.