On the use of exponential basis functions in the PINN architecture: An enhanced solution approach for the Laplace, Helmholtz, and elasto-static equations

Abstract

This paper presents a novel physics-informed neural network (PINN) architecture that employs exponential basis functions (EBFs) to solve many boundary value problems. The EBFs are organized so that the PINN architecture may employ their simple differentiation features to solve partial differential equations (PDEs). The proposed approach has been meticulously investigated and compared to the conventional PINN in the solution of many instances involving Laplace and Helmholtz differential equations, as well as linear and nonlinear elasto-static problems. By utilizing EBFs, fewer interpolation functions are needed and the derivation rule can be explicitly applied, significantly reducing computation time. In addition, in some problems, PINN embedded with EBFs shows better precision near the boundaries of the problem, which is one of the disadvantages of conventional PINNs. Two specific applications of the presented method have been investigated, namely the inverse identification of material properties and the solution of complex-valued partial differential equations. The proposed method was found to be accurate and converging when applied to inverse problems. As shown, it is also suitable for the solution of PDEs containing both imaginary and real parts.