System Modeling with Partial Differential Equations

This chapter develops a model using neural networks for solving the strongly degenerate parabolic and Burgers–Fisher equations. A trial solution of this system is subdivided into two parts. The initial and boundary conditions comprise the first part, which contains no adjustable parameters. The involvement of a neural network containing adjustable parameters (weights and biases) comprise the second part. A training technique is implemented for the training of the network, which requires the calculation of the error gradient in consideration of the network parameters. In order to maintain a continuation, a significantly modified methodology is illustrated for solving a wave equation that is modeled on the basis of two patterned Bernstein neural networks: static and dynamic models. The theoretical analysis and simulation results show that these new methods, neural networks and Bernstein neural networks, are effective in estimating solutions to partial differential equations.