Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods

• URL: http://arxiv.org/abs/2310.12846v1
• Date: Thu, 19 Oct 2023 15:57:10 GMT
• Title: Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods
• Authors: Jiasheng Chen and Juan Tang and Ming Yan and Shuai Lai and Kun Liang and Jianguang Lu and Wenqiang Yang
• Abstract summary: 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. Our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs.
• Score: 10.974537885042613
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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