基于Lobatto方法和Legendre多项式的PINN求解微分代数方程

doi:10.11772/j.issn.1001-9081.2024030313

《计算机应用》唯一官方网站 ›› 2025, Vol. 45 ›› Issue (3): 911-919.DOI: 10.11772/j.issn.1001-9081.2024030313

基于Lobatto方法和Legendre多项式的PINN求解微分代数方程

赖帅¹, 唐卷¹^,²(), 梁锟², 陈佳盛²

^1.广州大学计算机科学与网络工程学院，广州 510006
^2.广州大学计算科技研究院，广州 510006

收稿日期:2024-03-21 修回日期:2024-05-12 接受日期:2024-05-15 发布日期:2024-06-06 出版日期:2025-03-10
通讯作者: 唐卷
作者简介:赖帅（1997—），男，广东茂名人，硕士研究生，主要研究方向：科学机器学习、人工智能
梁锟（1999—），男，广东茂名人，硕士研究生，主要研究方向：科学机器学习、人工智能
陈佳盛（1998—），男，广东揭阳人，硕士研究生，主要研究方向：科学机器学习、人工智能。
基金资助:
国家自然科学基金资助项目(12201144)

Physics-informed neural network based on Lobatto method and Legendre polynomials for solving differential-algebraic equations

Shuai LAI¹, Juan TANG¹^,²(), Kun LIANG², Jiasheng CHEN²

^1.School of Computer Science and Cyber Engineering，Guangzhou University，Guangzhou Guangdong 510006，China
^2.Institute of Computing Science and Technology，Guangzhou University，Guangzhou Guangdong 510006，China

Received:2024-03-21 Revised:2024-05-12 Accepted:2024-05-15 Online:2024-06-06 Published:2025-03-10
Contact: Juan TANG
About author:LAI Shuai， born in 1997， M. S. candidate. His research interests include scientific machine learning， artificial intelligence.
LIANG Kun， born in 1999， M. S. candidate. His research interests include scientific machine learning， artificial intelligence.
CHEN Jiasheng， born in 1998， M. S. candidate. His research interests include scientific machine learning， artificial intelligence.
Supported by:
National Natural Science Foundation of China(12201144)

摘要/Abstract

摘要：

当前求解微分代数方程（DAE）的神经网络方法基本都采用数据驱动策略，需要大量的数据集，因此存在对神经网络的结构和参数选择敏感、求解结果精度低、稳定性差等问题。针对这些问题，提出一种基于Lobatto方法和Legendre多项式的物理信息神经网络（LL-PINN）。首先，基于离散型物理信息神经网络（PINN）的计算框架，结合Lobatto IIIA方法求解DAE高精度和高稳定性的优点，将DAE的物理信息嵌入Lobatto IIIA时间迭代格式中，并使用PINN对该时间迭代进行近似数值求解；其次，采用单隐藏层的神经网络结构，利用勒让德多项式展开项的逼近能力，应用这些多项式作为激活函数来简化网络模型调整的过程；最后，采用时间区域分解方案构建网络模型，即对每个等分的子时间区域依次使用一个微分神经网络和一个代数神经网络，从而实现DAE的高精度连续时间预测。数值算例结果表明，基于勒让德多项式和4阶的Lobatto方法的LL-PINN实现了对DAE的高精度求解。与函数连接理论（TFC）试验解模型和PINN模型相比，LL-PINN的微分变量和代数变量的预测解与精确解的绝对误差显著降低，精度提高了一个或两个量级。因此，所提求解模型对求解DAE问题具有较好的计算精度，可为解决具有挑战性的偏DAE提供可行的解决方案。

关键词: 微分代数方程, 物理信息神经网络, Lobatto IIIA方法, 勒让德多项式, 时间区域分解

Abstract:

Current neural network methods solving Differential-Algebraic Equations （DAEs） basically adopt data-driven strategies， and require a large number of datasets. So that， there are problems such as sensitive structure and parameter selection of neural networks， low accuracy of solution， and poor stability. In response to these issues， a Physics-Informed Neural Network based on Lobatto method and Legendre polynomials （LL-PINN） was proposed. Firstly， based on the discrete Physics-Informed Neural Network （PINN） computing framework， combined with the advantages of high accuracy and high stability of Lobatto IIIA method solving DAEs， the physical information of DAEs was embedded in the Lobatto IIIA time iteration format， and PINN was used to solve the approximate numerical value of this time iteration. Secondly， a neural network structure with single hidden layer was utilized， by using the approximation capability of Legendre polynomials， these polynomials were applied as activation functions to simplify the process of adjusting the network model. Finally， a time domain decomposition scheme was employed to construct the network model， which a differential neural network and an algebraic neural network were used for each equally divided sub-time domain one by one， enabling high-precision continuous-time prediction of DAEs. Results of numerical examples demonstrate that the LL-PINN based on Legendre polynomials and the 4th-order Lobatto method achieves high-precision solutions for DAEs. Compared to the Theory of Functional Connections （TFC） trial solution method and PINN model， LL-PINN significantly reduces the absolute error between the predicted and exact solutions of differential variables and algebraic variables， and improves accuracy by one or two orders of magnitude. Therefore， the proposed solution model exhibits good computational accuracy for solving DAE problems， providing a feasible solution for challenging partial DAEs.

Key words: Differential-Algebraic Equation (DAE), Physics-Informed Neural Network (PINN), Lobatto IIIA method, Legendre polynomial, time domain decomposition

中图分类号:

O241

赖帅, 唐卷, 梁锟, 陈佳盛. 基于Lobatto方法和Legendre多项式的PINN求解微分代数方程[J]. 计算机应用, 2025, 45(3): 911-919.

Shuai LAI, Juan TANG, Kun LIANG, Jiasheng CHEN. Physics-informed neural network based on Lobatto method and Legendre polynomials for solving differential-algebraic equations[J]. Journal of Computer Applications, 2025, 45(3): 911-919.

图/表 11

表1 Lobatto IIIA方法系数表

Tab. 1 Coefficient table of Lobatto IIIA method

$c j$	$b 1$	$b 2$	$…$	$b s - 1$	$b s$
$c 1 = 0$	$a 11$	$a 12$	$…$	$a 1, s - 1$	$a 1 s$
$c 2$	$a 21$	$a 12$	$…$	$a 2, s - 1$	$a 2 s$
$⋮$	$⋮$	$⋮$		$⋮$	$⋮$
$c s - 1$	$a s - 1,1$	$a s - 1,2$	$…$	$a s - 1, s - 1$	$a s - 1, s$
$c s = 1$	$a s 1$	$a s 2$	$…$	$a s, s - 1$	$a s s$

表1 Lobatto IIIA方法系数表

Tab. 1 Coefficient table of Lobatto IIIA method

$c j$	$b 1$	$b 2$	$…$	$b s - 1$	$b s$
$c 1 = 0$	$a 11$	$a 12$	$…$	$a 1, s - 1$	$a 1 s$
$c 2$	$a 21$	$a 12$	$…$	$a 2, s - 1$	$a 2 s$
$⋮$	$⋮$	$⋮$		$⋮$	$⋮$
$c s - 1$	$a s - 1,1$	$a s - 1,2$	$…$	$a s - 1, s - 1$	$a s - 1, s$
$c s = 1$	$a s 1$	$a s 2$	$…$	$a s, s - 1$	$a s s$

图1 勒让德多项式激活函数的结构

Fig. 1 Structure of Legendre polynomial activation function

图2 并行的微分网络和代数网络的结构

Fig. 2 Structures of parallel differential network and algebraic network

图3 LL-PINN的框架

Fig. 3 Framework of LL-PINN

图4 时间域分解方案的结构

Fig. 4 Structure of time domain decomposition scheme

图5 高精度连续时间预测方案

Fig. 5 High-precision continuous-time prediction scheme

图6 3种模型的预测解与精确解的绝对误差对比（数值算例1）

Fig. 6 Comparison of absolute errors among predicted and exact solutions of three models （numerical example 1）

表2 3种模型的预测解与精确解的L2误差对比（数值算例1）

Tab. 2 Comparison of L2 errors among predicted and exact solutions of three models （numerical example 1）

模型

x 1

x 2

x 3

x 4

x 5

TFC

试验解

8.94E-07

4.09E-06

1.02E-06

4.03E-06

1.76E-06

表2 3种模型的预测解与精确解的L2误差对比（数值算例1）

Tab. 2 Comparison of L2 errors among predicted and exact solutions of three models （numerical example 1）

模型

x 1

x 2

x 3

x 4

x 5

TFC

试验解

7.32E-04

1.22E-04

5.40E-04

2.07E-04

2.51E-03

PINN

8.04E-04

2.36E-03

7.17E-04

2.31E-03

1.61E-03

LL-PINN

8.94E-07

4.09E-06

1.02E-06

4.03E-06

1.76E-06

图7 单摆模型

Fig. 7 Single pendulum model

图8 3种模型的预测解与精确解的绝对误差比较（数值算例2）

Fig. 8 Comparison of absolute errors among predicted and exact solutions of three models （numerical example 2）

表3 3种模型的预测解与精确解的L2误差对比（数值算例2）

Tab. 3 Comparison of L2 errors among predicted and exact solutions of three models （numerical example 2）

模型

x

u

y

v

λ

TFC

试验解

2.51E-05

2.64E-04

6.09E-05

9.49E-05

1.90E-04

表3 3种模型的预测解与精确解的L2误差对比（数值算例2）

Tab. 3 Comparison of L2 errors among predicted and exact solutions of three models （numerical example 2）

模型

x

u

y

v

λ

TFC

试验解

3.46E-03

3.61E-02

1.72E-02

1.43E-02

2.25E-02

PINN

2.15E-03

4.68E-03

5.48E-03

8.24E-03

1.96E-02

LL-PINN

2.51E-05

2.64E-04

6.09E-05

9.49E-05

1.90E-04

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基于Lobatto方法和Legendre多项式的PINN求解微分代数方程

Physics-informed neural network based on Lobatto method and Legendre polynomials for solving differential-algebraic equations

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