Augmented neural forms with parametric condition-matching operators for solving ordinary differential equations

The approximation of solutions to ordinary and partial differential equations is an important task. Complementing the classical numerical analysis methods of solution, neural forms offer a valuable alternative approach. Neural forms, which are closed-form expressions involving neural networks, are specifically designed to exactly satisfy prescribed initial or boundary conditions. Starting from the important class of ordinary differential equations, the present work aims to refine and extend the methodology of neural forms, paving the way for its application to the highly challenging field of partial differential equations. First, a formalism is developed for the systematic construction of proper neural forms with parametric condition-matches, amenable to optimization. Second, a novel technique is described for converting Neumann or Robin conditions into equivalent parametric Dirichlet conditions. Third, a methodology is introduced for determining an upper bound on the absolute deviation from the exact solution. The proposed approach was applied on a set of diverse test problems, including first and second order ordinary differential equations, as well as first order systems. Stiff differential equations have been considered as well. The obtained solutions were evaluated against known exact solutions, solutions derived by the physics-informed neural networks method, and solutions obtained via a contemporary finite difference technique. The reported results demonstrate that the augmented neural forms provide closed-form solutions featuring high-quality interpolation and controllable overall accuracy. These attributes are essential for expanding the approach to treat challenging partial differential equation problems.