Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

A systematic approach is developed for constructing proper trial solutions to Partial Differential Equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. The spatial domain considered is of the rectangular hyper-box type. On each face either Dirichlet or Neumann conditions may apply. Robin conditions may be accommodated as well. Interface conditions that induce discontinuities, have not been treated to date in the relevant neural network literature. As an illustration a common problem of heat conduction through a system of two rods in thermal contact is considered.