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

TL;DR

This paper advances neural forms for solving ODEs by introducing boundary-matching operators, converting complex boundary conditions, and providing bounds on solution accuracy, validated across diverse differential equations.

Contribution

It presents a systematic formalism for neural forms with adaptable boundary matches, a technique to convert complex boundary conditions, and a method to estimate solution deviation bounds.

Findings

Successfully solved diverse ODEs including stiff equations.

Exact boundary condition satisfaction and high-quality solutions.

Outperformed existing neural and numerical methods.

Abstract

Approximating solutions of ordinary and partial differential equations constitutes a significant challenge. Based on functional expressions that inherently depend on neural networks, neural forms are specifically designed to precisely satisfy the prescribed initial or boundary conditions of the problem, while providing the approximate solutions in closed form. Departing from the important class of ordinary differential equations, the present work aims to refine and validate the neural forms methodology, paving the ground for further developments in more challenging fields. The main contributions are as follows. First, it introduces a formalism for systematically crafting proper neural forms with adaptable boundary matches that are amenable to optimization. Second, it describes a novel technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. Third, it outlines a method for determining an upper bound on the absolute deviation from the exact solution. The proposed augmented neural forms approach was tested on a set of diverse problems, encompassing first- and second-order ordinary differential equations, as well as first-order systems. Stiff differential equations have been considered as well. The resulting solutions were subjected to assessment against existing exact solutions, solutions derived through the common penalized neural method, and solutions obtained via contemporary numerical analysis methods. The reported results demonstrate that the augmented neural forms not only satisfy the boundary and initial conditions exactly, but also provide closed-form solutions that facilitate high-quality interpolation and controllable overall precision. These attributes are essential for expanding the application field of neural forms to more challenging problems that are described by partial differential equations.