Solving partial differential equations in real-time using artificial neural network signal processing as an alternative to finite-element analysis

摘要

Finite element methods (FEM) have been widely utilized for evaluating partial differential equations (PDEs). Although these methods have been highly successful, they require time-consuming procedures to build numerous volumetric elements and solve large-size linear systems of equations. In this paper, a new signal processing method is utilized to solve PDEs numerically by using an artificial neural network. We investigate the theoretical aspects of this approach and show that the numerical computation can be formulated as a machining learning problem and implemented by a supervised function approximation neural network. We also show that, for the case of the Poisson equation, the solution is unique and continuous with respect to the boundary surface. We apply this method to bio-potential computation where the solution of a standard volume conductor is mapped to the solutions of a set of volume conductors in different shapes.