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

Abstract: To address the issues of high computational and storage costs, poor stability for stiff systems, and difficulties in handling algebraic constraints associated with traditional numerical methods for Integro-Differential-Algebraic Equations (IDAEs), an Auxiliary Physics-Informed Neural Network (APINN) method named APINN-IDAE was proposed. The core idea of this method leverages the multi-task learning mechanism of APINN by introducing auxiliary network outputs to specifically approximate the integral history terms. This transforms the intractable integral constraints in the original IDAE system into differential relations that are easier to optimize, thereby avoiding the expensive numerical integration required in traditional methods. The main network output handles the differential variables, while the auxiliary outputs process the integral terms and algebraic constraint residuals. A unified weighted loss function is constructed, incorporating the differential equations, algebraic equations, differential relations of the auxiliary variables, 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 throughout the domain. The relative L² errors for both differential and algebraic variables reach the order of 10⁻³ to 10⁻⁵, with no observable error accumulation over time. 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. This structure offers potential robustness for stiff or high-index IDAE systems and provides a new perspective for solving multiphysics coupling problems.

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