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.

翻译：近似求解常微分方程和偏微分方程的解构成了一项重大挑战。基于本质上依赖于神经网络的功能表达式，神经形式被专门设计为精确满足问题规定的初始或边界条件，同时以封闭形式提供近似解。本文从重要的常微分方程类别出发，旨在完善和验证神经形式方法学，为更具挑战性领域的进一步发展奠定基础。主要贡献如下：首先，引入了一种形式化方法，用于系统构建具有可优化自适应边界匹配的适当神经形式。其次，描述了一种将诺伊曼或罗宾条件问题转化为等效参数化狄利克雷条件问题的新技术。第三，概述了一种确定与精确解绝对偏差上界的方法。所提出的增强型神经形式方法在一系列多样化问题上进行了测试，涵盖一阶和二阶常微分方程以及一阶方程组。刚性微分方程也被纳入考量。所得解通过以下三类结果进行评估：现有精确解、通过常见惩罚神经方法推导的解，以及通过现代数值分析方法获得的解。报告结果表明，增强型神经形式不仅能精确满足边界和初始条件，还能提供便于高质量插值且具有可控整体精度的封闭形式解。这些特性对于将神经形式的应用领域扩展到由偏微分方程描述的更具挑战性问题至关重要。