PHYSICS-INFORMED NEURAL NETWORK FOR SOLVING HAUSDORFF DERIVATIVE POISSON EQUATIONS

Abstract

This paper proposed a new physics-informed neural network (PINN) for solving the Hausdorff derivative Poisson equations (HDPEs) on irregular domains by using the concept of Hausdorff fractal derivative. The present scheme transforms the numerical solution of partial differential equation into an optimization problem including governing equation and boundary conditions. Like the meshless method, the developed PINN does not require grid generation and numerical integration. Moreover, it can freely address irregular domains and non-uniformly distributed nodes. The present study investigated different activation functions, and given an optimal choice in solving the HDPEs. Compared to other existing approaches, the PINN is simple, straightforward, and easy-to-program. Numerical experiments indicate that the new methodology is accurate and effective in solving the HDPEs on arbitrary domains, which provides a new idea for solving fractal differential equations.