As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems.~Recently, the physics-informed neural network (PINN) has gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with a neural network structure via the attention mechanisms, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders of the Radau IIA method affect the accuracy of neural network solutions. The experimental results demonstrate that the PINN based on a 5th-order Radau IIA method achieves the highest level of system accuracy. Specifically, the absolute errors for all differential variables remains as low as $10^{-6}$, and the absolute errors for algebraic variables is maintained at $10^{-5}$, surpassing the results found in existing literature. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

翻译：众所周知，微分代数方程（DAE）能够描述动态变化与潜在约束，已广泛应用于流体动力学、多体动力学、机械系统及控制理论等工程领域。在这些领域的实际物理建模中，系统常生成高指标DAE。经典隐式数值方法在求解高指标系统时，通常会导致数值精度出现不同程度的阶降。近年来，物理信息神经网络（PINN）在求解DAE系统方面受到关注，但面临无法直接求解高指标系统、预测精度较低及泛化能力较弱等挑战。本文提出一种PINN计算框架，通过注意力机制将Radau IIA数值方法与神经网络结构相结合，直接求解高指标DAE。此外，我们采用区域分解策略以提高解精度。以两个经典高指标系统为例进行数值实验，探究不同阶次Radau IIA方法对神经网络解精度的影响。实验结果表明，基于5阶Radau IIA方法的PINN实现了系统最高精度。具体而言，所有微分变量的绝对误差低至$10^{-6}$，代数变量的绝对误差维持在$10^{-5}$，超越了现有文献的结果。因此，该方法展现出优异的计算精度与强泛化能力，为更大规模、更高指标DAE或高维偏微分代数方程系统的高精度求解提供了可行途径。