Multi-Layer Solution of Heat Equation

A new approach to the construction of multilayer neural network approximate solutions for evolutionary partial differential equations is considered. The approach is based on the application of the recurrence relations of the Euler, Runge-Kutta, etc. methods to variable length intervals. The resulting neural-like structure can be considered as a generalization of a feedforward multilayer network or a recurrent Hop-field network. This analogy makes it possible to apply known methods to the refinement of the obtained solution, for example, the backpropagation algorithm. Earlier, a similar approach has been successfully used by the authors in the case of ordinary differential equations. Computational experiments are performed on one test problem for the one-dimensional (in terms of spatial variables) heat equation. Explicit formulas are derived for the dependence of the resulting neural network output on the number of layers. It was found that the error tends to zero with an increasing number of layers, even without the use of the network learning.