In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for solving different classes of singular ODEs, namely the well known second-order Lane-Emden equations, third order-order Emden-Fowler equations, and fourth-order Lane-Emden-Fowler equations. Two variants of PINNs technique are considered and compared. First, a minimization procedure is used to constrain the total loss function of the neural network, in which the equation residual is considered with some weight to form a physics-based loss and added to the training data loss that contains the initial/boundary conditions. Second, a specific choice of trial solutions ensuring these conditions as hard constraints is done in order to satisfy the differential equation, contrary to the first variant based on training data where the constraints appear as soft ones. Advantages and drawbacks of PINNs variants are highlighted.

翻译：本文提出基于物理信息的神经网络（PINNs）数值方法，旨在求解高阶常微分方程（ODEs）。该深度学习技术成功应用于求解不同类别的奇异常微分方程，包括著名的二阶Lane-Emden方程、三阶Emden-Fowler方程以及四阶Lane-Emden-Fowler方程。研究考虑并比较了两种PINNs技术变体：第一种采用最小化过程约束神经网络的总损失函数，其中方程残差以特定权重构成物理损失项，并与包含初始/边界条件的训练数据损失项相加；第二种通过特定试解的选择，将初始/边界条件作为硬约束确保微分方程满足，这与基于训练数据的第二种变体（其中约束表现为软约束）形成对比。文中重点阐述了两种PINNs变体的优势与局限性。