A Constrained Backpropogation Approach to Solving Partial Differential Equations in Non-stationary Environment

A constrained-backpropagation (CPROP) training technique is presented to solve Partial Differential Equations (PDEs). The technique is based on constrained optimization and minimizes an error function subject to a set of equality constraints provided by the boundary conditions of the differential problem. As a result, sigmoidal neural networks can be trained to approximate the solution of PDEs while avoiding the discontinuity in the derivative of the solution, which may affect the stability of classical methods. Also, the memory provided to the network through the constrained approach may be used to solve PDEs on line when the forcing term changes over time, learning different solutions of the differential problem through a continuous nonlinear mapping. The effectiveness of this method is demonstrated by solving a nonlinear PDE on a circular domain. When the underlying process changes but is subjected to the same boundary conditions, the CPROP network is able to adapt online and approximate the new solution, while memory of the boundary conditions is maintained virtually intact at all times.