ARTIFICIAL NEURAL NETWORKS FOR SOLVING ORDINARY AND PARTIAL-DIFFERENTIAL EQUATIONS

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equa tion is written as a sum of two parts. The first part satisfies the in itial/boundary conditions and contains no adjustable parameters. The s econd part is constructed so as not to affect the initial/boundary con ditions. This part involves a feedforward neural net work containing a djustable parameters (the weights). Hence by construction the initial/ boundary conditions are satisfied and the network is trained to satisf y the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of co upled ODE's and also to partial differential equations (PDE's). In thi s article, we illustrate the method by solving a variety of model prob lems and present comparisons with solutions obtained using the Galekrk in finite element method for several cases of partial differential equ ations. With the advent of neuroprocessors and digital signal processo rs the method becomes particularly interesting due to the expected ess ential gains in the execution speed.