残差分裂自适应物理信息神经网络求解偏微分方程

范昆昆; 张皓然; 岳煜铖; 袁冬芳

doi:10.21656/1000-0887.460018

残差分裂自适应物理信息神经网络求解偏微分方程

doi: 10.21656/1000-0887.460018

1.
内蒙古科技大学自动化与电气工程学院，内蒙古包头 014010
2.
内蒙古科技大学理学院，内蒙古包头 014010

基金项目:

国家自然科学基金地区科学基金 12261067

国家自然科学基金地区科学基金 12361088

内蒙古自然科学基金 2022MS01008

内蒙古科技大学基本研究业务费专项资金 2024QNJS052

详细信息

作者简介:
范昆昆(1998—)，男，硕士生(E-mail: fankunkun914@163.com)

通讯作者:
袁冬芳(1985—)，女，副教授，硕士(通信作者. E-mail: yuandf@imust.edu.cn)

中图分类号: O241.82
计量
- 文章访问数: 63
- HTML全文浏览量: 16
- PDF下载量: 21
- 被引次数: 0
出版历程
- 收稿日期: 2025-01-24
- 修回日期: 2025-03-24
- 刊出日期: 2026-05-01

Residual Splitting Adaptive Physics-Informed Neural Networks for Solving Partial Differential Equations

1.
School of Automation and Electrical Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, P.R. China
2.
School of science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, P.R. China

摘要

摘要: 物理信息神经网络(PINN)损失函数之间的量级差异，导致训练过程收敛缓慢，有时甚至会在某些区域训练失败.为解决这一挑战，本文提出了一种融合残差分裂和权重自适应的PINN模型.该方法通过将主导PINN训练过程的偏微分方程(PDE)残差项，按照区域分解的方式分裂为多个独立子项，并采用权重自适应加权策略，自动调节各个子项之间的权重，从而改善了PINN的收敛性.该方法弥补了全局残差策略忽略和抹平局部特征的缺陷，通过分裂子项的方式增加了对局部特征的关注，改善了优化过程的效率，从而提升了求解精度.数值实验结果表明，本文方法不仅在精度上超越了现有几种模型，且达到了2~3个数量级的提升，计算效率也表现出优越性能.
- 物理信息神经网络 /
- 自适应权重 /
- 残差分裂 /
- 区域分解
Abstract: The magnitude difference between the loss functions of physical-informed neural networks (PINN) leads to a slow convergence of the training process and sometimes even training failure in some regions. To address this challenge, a physical-informed neural network model incorporating residual splitting and weight self-adaptation was proposed. The method improves the convergence of PINN by splitting the residual terms of PDE dominating the training process of PINN, into multiple independent components according to the domain decomposition, and adopts a self-adaptive weighting strategy to automatically adjust the weights among the components, thus promoting the convergence of PINN. This method makes up for the defects of the global residual strategy ignoring and smoothing out the local features, and increases the attention to the local features by splitting the subterms, which improves the efficiency of the optimization process, and thus enhances the solution accuracy. Through numerical experiments, the results show that, the proposed method not only surpasses the existing models in terms of accuracy, but also achieves an improvement of 2-3 orders of magnitude with superior computational efficiency.
- physics-informed neural network /
- adaptive weighting /
- residual splitting /
- domain decomposition

HTML全文

图 1 区域分裂后各区域表示情况

Figure 1. The situation expressed by each region after regional division

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图 2 本文方法求解PDE的模型框架图

Figure 2. The model frame diagram of solving PDE with this method

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图 3 不同方法的损失函数收敛曲线

Figure 3. Loss function convergence curves for different methods

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图 4 四种不同方法的预测解和误差分布

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 4. Predicted solutions and error distributions for 4 different methods

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图 5 不同方法损失函数收敛曲线

Figure 5. Convergence curves of loss functions for different methods

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图 6 四种不同方法的预测解和误差分布

Figure 6. Predicted solutions and error distributions for 4 different methods

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图 7 不同方法的损失函数误差收敛图

Figure 7. Error convergence curves of the loss functions for different methods

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图 8 四种方法的预测解和误差分布

Figure 8. Predicted solutions and error distributions for the 4 methods

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图 9 不同区域分裂策略下预测解u(x, y, 0, 2)的误差对比

Figure 9. Comparison of prediction errors of solution u(x, y, 0, 2) under different regional division strategies

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表 1 算例1在不同频率下的误差、训练时间

Table 1. Errors and training times at different frequencies for example 1

method	ω	subdomain	e_loss	e₂	e_∞	training time/s
PINN	15	-	6.92E-2	7.64E-2	8.14E-2	190.63
	20	-	7.40E-1	3.00E-1	1.86E-1	189.02
	25	-	3.26	8.75E-1	8.13E-1	195.78
lbPINN	15	-	1.58E-2	1.55E-2	1.57E-2	180.67
	20	-	9.34E-3	1.82E-2	1.83E-2	199.56
	25	-	2.63E-2	9.41E-2	5.20E-2	210.10
SAPINN	15	-	6.12	3.76	1.78	1 554.46
	20	-	2.85E-1	4.20	2.89	1 499.22
	25	-	6.84E-2	6.59E-1	2.10E-1	1 514.71
our approach	15	(3, 1)	7.34E-7	8.03E-4	7.84E-4	196.52
	20	(3, 1)	2.49E-6	1.20E-3	8.41E-4	180.71
	25	(3, 1)	1.17E-5	3.45E-3	2.29E-3	196.85

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表 2 不同残差分裂策略的误差

Table 2. Errors of different residual splitting strategies

method	ω	subdomain	e_loss	e₂	e_∞	training time/s
our approach	25	(1, 1)	1.57E-4	3.59E-2	2.02E-2	202.11
		(3, 1)	1.17E-5	3.45E-3	2.29E-3	196.85
		(5, 1)	8.84E-5	4.87E-3	5.34E-3	212.77
		(3, 2)	1.07E-4	3.55E-3	1.19E-3	199.49
		(3, 3)	4.72E-4	2.10E-2	1.90E-2	217.94

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表 3 在不同波速情况下四种方法误差、时间对比

Table 3. Comparison of errors and times of 4 methods under different wave speeds

method	λ	subdomain	e_loss	e₂	e_∞	training time/s
PINN	0.2	-	2.07E-8	3.78E-2	5.81E-3	109.72
	0.4	-	1.12E-6	3.90E-1	1.14E-1	111.06
	0.6	-	2.25E-8	5.34E-2	3.20E-2	114.39
lbPINN	0.2	-	1.20E-11	3.01E-2	4.08E-3	117.08
	0.4	-	2.19E-11	5.35E-2	2.04E-2	123.77
	0.6	-	6.66E-11	2.72E-1	1.36E-1	146.07
SAPINN	0.2	-	9.39E-6	1.80E-2	3.37E-3	981.95
	0.4	-	1.60E-6	8.82E-3	2.76E-3	962.94
	0.6	-	1.02E-5	8.93E-1	2.96E-1	1019.10
our approach	0.2	(5, 1)	5.74E-16	1.52E-5	1.54E-6	103.90
	0.4	(5, 1)	1.34E-13	9.01E-5	2.94E-5	101.15
	0.6	(5, 1)	8.18E-14	9.10E-5	4.28E-5	103.82

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表 4 四种方法在不同扩散速率下的各种误差与训练时间

Table 4. Various errors vs. training times for the 4 methods at different diffusion rates

method	λ	subdomain	e_loss	e₂	e_∞	training time/s
PINN	8	-	3.78E-5	1.02E-2	1.86E-2	109.72
	10	-	9.84E-3	7.80E-2	1.38E-1	111.06
	12	-	1.63E-3	3.11E-2	5.27E-2	114.39
lbPINN	8	-	7.39E-3	1.12E-2	2.22E-2	117.08
	10	-	1.76E-3	4.61E-2	8.38E-2	123.77
	12	-	5.50E-3	1.16E-1	1.83E-1	146.07
SAPINN	8	-	4.51E-1	1.36	1.99	778.91
	10	-	4.51E-1	1.36	1.99	773.56
	12	-	4.51E-1	1.36	1.99	775.83
our approach	8	(2, 2)	4.67E-10	7.75E-4	1.61E-3	103.90
	10	(2, 2)	9.29E-9	3.76E-3	5.50E-3	101.15
	12	(2, 2)	5.09E-10	9.31E-4	1.58E-3	103.82

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表 5 不同时间区域的不同分裂策略误差对比

Table 5. Comparison of different splitting strategies' errors in different time domain

method	T=1 s				T=2 s
method	subdomain	e₂	e_∞	training time/s	subdomain	e₂	e_∞	training time/s
PINN	-	7.92E-4	8.24E-3	223.56	-	6.92E-4	3.49E-3	205.89
lbPINN	-	6.54E-4	6.85E-3	210.66	-	5.86E-4	3.79E-3	189.19
our approach	1	0.67E-5	9.66E-4	198.67	1	9.18E-4	1.10E-3	188.98
	2	8.86E-6	7.29E-5	158.82	2	8.64E-5	5.44E-4	191.56
	4	1.15E-5	1.38E-4	165.73	4	8.18E-6	4.44E-5	229.71
	8	1.23E-5	6.60E-5	248.29	8	1.22E-5	8.09E-5	252.92

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