Abstract:

To address the issues of high computational and storage costs， poor stability of stiff systems， and difficulties in handling algebraic constraints in the traditional numerical methods for Integro-Differential-Algebraic Equations （IDAEs）， an Auxiliary Physics-Informed Neural Network （APINN）-based method for solving IDAEs， named APINN-IDAE， was proposed. The core idea of this method is to use the multi-task learning mechanism of APINN， introducing auxiliary network outputs to specifically approximate the integral history terms， so that the intractable integral constraints in the original IDAE system were transformed into differential relations that are easier to optimize， thereby avoiding the computationally expensive numerical integration in the traditional methods. The differential variables were handled by the main network output， while the integral terms and algebraic constraint residuals were processed by the auxiliary outputs， respectively. In addition， a unified weighted loss function was constructed， incorporating the differential equations， algebraic equations， auxiliary variable differential relations， and initial conditions. Experimental results on several typical IDAE examples （including nonlinear， variable-coefficient， and strongly coupled systems） show that the predicted solutions of the proposed method highly coincide with the exact solutions within the solution domain. And the relative L₂ errors for both differential and algebraic variables reach the order of 10?3 to 10??， without observable error accumulation over time. It can be seen that APINN-IDAE retains the mesh-free and automatic differentiation advantages while using auxiliary variables to avoid explicit differentiation of integral terms， thereby reducing the computational complexity of integral operators and improving gradient stability， offering potential robustness for stiff or high-index IDAE systems， and providing a new perspective for solving multi-physics coupling problems.

Key words: Integro-Differential-Algebraic Equation (IDAE), Physics-Informed Neural Network (PINN), auxiliary variable, deep learning, meshless method

摘要：

针对积分微分代数方程（IDAE）的传统数值方法存在计算存储成本高、刚性系统稳定性差及代数约束处理困难等问题，提出一种基于辅助物理信息神经网络（APINN）的IDAE求解方法——APINN-IDAE。该方法的核心思想是：利用APINN的多任务学习机制，通过引入辅助网络输出专门逼近积分历史项，将原IDAE系统中难以处理的积分约束转化为易于优化的微分关系，从而避免传统方法中昂贵的数值积分计算。网络主输出负责微分变量，辅助输出分别处理积分项与代数约束残差，并构建融合微分方程、代数方程、辅助变量微分关系及初值条件的统一加权损失函数。在多个典型IDAE算例（包括非线性、变系数及强耦合系统）上的实验结果表明，所提方法在求解域内的预测解与精确解高度吻合，而微分变量与代数变量的相对L₂误差均达到10?3至10??量级，且误差无时间积累效应。可见，APINN-IDAE在保持无网格和自动微分优势的同时，以辅助变量避免了对积分项的显式微分，从而降低了积分算子的计算复杂度并改善了梯度稳定性，在处理刚性或高指标IDAE系统时具有潜在的鲁棒性，为多物理耦合问题的求解提供了新的思路。

关键词: 积分微分代数方程, 物理信息神经网络, 辅助变量, 深度学习, 无网格方法