A critical issue in approximating solutions of ordinary differential equations using neural networks is the exact satisfaction of the boundary or initial conditions. For this purpose, neural forms have been introduced, i.e., functional expressions that depend on neural networks which, by design, satisfy the prescribed conditions exactly. Expanding upon prior progress, the present work contributes in three distinct aspects. First, it presents a novel formalism for crafting optimized neural forms. Second, it outlines a method for establishing an upper bound on the absolute deviation from the exact solution. Third, it introduces a technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed optimized neural forms were numerically tested on a set of diverse problems, encompassing first-order and second-order ordinary differential equations, as well as first-order systems. Stiff and delay differential equations were also considered. The obtained solutions were compared against solutions obtained via Runge-Kutta methods and exact solutions wherever available. The reported results and analysis verify that in addition to the exact satisfaction of the boundary or initial conditions, optimized neural forms provide closed-form solutions of superior interpolation capability and controllable overall accuracy.

翻译：利用神经网络逼近常微分方程解的一个关键问题在于精确满足边界条件或初始条件。为此，研究者引入了神经形式，即依赖于神经网络的功能表达式，这类表达式通过设计能够完全满足预设条件。本文在先前研究基础上进行了三个方面的创新：首先，提出了一种构建优化神经形式的新形式体系；其次，概述了确定解与精确解绝对偏差上界的方法；第三，引入将纽曼条件或罗宾条件问题转化为等价参数化狄利克雷条件问题的技术。所提出的优化神经形式在一系列不同问题上进行了数值测试，涵盖一阶和二阶常微分方程以及一阶方程组，同时考虑了刚性微分方程和延迟微分方程。将所得解与龙格-库塔方法解得的结果及精确解（如有）进行了对比。报告的结果与分析证实，优化神经形式不仅能够精确满足边界条件或初始条件，还能提供具有出色插值能力和可控整体精度的闭式解。