Physics-informed neural networks utilizing the Legendre-Gauss-Lobatto collocation method for solving differential-algebraic equation with discrete event

Canyi Che; Qingli Zhao; Funing Yang; Xintong Zhang; Canyi Che; Qingli Zhao; Funing Yang; Xintong Zhang

doi:10.3934/era.2026183

Electronic Research Archive

2026, Volume 34, Issue 6: 4080-4106. doi: 10.3934/era.2026183

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Research article

Physics-informed neural networks utilizing the Legendre-Gauss-Lobatto collocation method for solving differential-algebraic equation with discrete event

School of Science, Shandong Jianzhu University, Jinan 250101, China

Received: 25 December 2025 Revised: 16 April 2026 Accepted: 07 May 2026 Published: 15 May 2026

Differential-algebraic equation (DAE) is widely used in engineering domains, such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory, owing to their ability to effectively characterize dynamic variations and inherent constraints. In recent years, physics-informed neural networks (PINNs) have manifested remarkable advantages in solving both the forward and inverse problems of DAE by integrating physical prior knowledge into neural network models. Presently, PINNs-based approaches still encounter challenges of inadequate solution accuracy and limited generalization performance when dealing with DAE involving discrete events. This paper presented a physics-informed neural network that integrates the Legendre-Gauss-Lobatto (LGL) collocation method from spectral methods to solve the aforementioned DAE with discrete event. To further augment the accuracy and continuity of the solution, the model employed a time-domain decomposition strategy to construct the network architecture, thereby enabling high-precision continuous-time prediction of DAE. Numerical examples illustrated that the LGL-PINN can attain high-precision solutions of DAE. In comparison with the PINNs, the error between the predicted solution and the exact solution of the LGL-PINN was substantially reduced, with the accuracy improved by one to two orders of magnitude. Therefore, the proposed solution model demonstrated excellent computational accuracy for solving DAE problems involving discrete event.
- differential-algebraic equation,
- discrete event,
- physics-informed neural network,
- LGL collocation method,
- time-domain decomposition
Citation: Canyi Che, Qingli Zhao, Funing Yang, Xintong Zhang. Physics-informed neural networks utilizing the Legendre-Gauss-Lobatto collocation method for solving differential-algebraic equation with discrete event[J]. Electronic Research Archive, 2026, 34(6): 4080-4106. doi: 10.3934/era.2026183

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Abstract

Differential-algebraic equation (DAE) is widely used in engineering domains, such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory, owing to their ability to effectively characterize dynamic variations and inherent constraints. In recent years, physics-informed neural networks (PINNs) have manifested remarkable advantages in solving both the forward and inverse problems of DAE by integrating physical prior knowledge into neural network models. Presently, PINNs-based approaches still encounter challenges of inadequate solution accuracy and limited generalization performance when dealing with DAE involving discrete events. This paper presented a physics-informed neural network that integrates the Legendre-Gauss-Lobatto (LGL) collocation method from spectral methods to solve the aforementioned DAE with discrete event. To further augment the accuracy and continuity of the solution, the model employed a time-domain decomposition strategy to construct the network architecture, thereby enabling high-precision continuous-time prediction of DAE. Numerical examples illustrated that the LGL-PINN can attain high-precision solutions of DAE. In comparison with the PINNs, the error between the predicted solution and the exact solution of the LGL-PINN was substantially reduced, with the accuracy improved by one to two orders of magnitude. Therefore, the proposed solution model demonstrated excellent computational accuracy for solving DAE problems involving discrete event.

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