基于AGA-PINNs方法求解具有间断解的方程

刘博宇; 江林峰; 杨凤莲

doi:10.21656/1000-0887.460129

基于AGA-PINNs方法求解具有间断解的方程

doi: 10.21656/1000-0887.460129

河海大学数学学院，南京 210000

基金项目:

国家自然科学基金（12271140）；中央高校基本科研业务费(B220202081)

详细信息

作者简介:
刘博宇（2001—），男，硕士生(E-mail:231362010004@hhu.edu.cn);江林峰（2000—），男，博士生(E-mail: mathlfjiang@hhu.edu.cn);杨凤莲（1982—），女，副教授，博士(通信作者. E-mail: yangfenglian@hhu.edu.cn).

中图分类号: O241
计量
- 文章访问数: 11
- HTML全文浏览量: 1
- PDF下载量: 5
- 被引次数: 0
出版历程
- 收稿日期: 2025-06-25
- 修回日期: 2025-07-30
- 网络出版日期: 2026-07-23

Solving Equations With Discontinuous Solutions Based on the AGA-PINNs Method

School of Mathematics, Hohai University, Nanjing 210000, P.R. China

Funds:

The National Science Foundation of China(12271140)

摘要

摘要: 物理信息神经网络（physics-informed neural networks,PINNs）是求解偏微分方程的重要工具，在偏微分方程的数值求解中，具有间断解的方程是目前的研究难题，PINNs及其现有的改进算法通常无法捕捉到间断的特性，然而在流体力学领域需要考虑许多间断问题.针对PINNs处理具有间断解的方程的不足，本文提出了自适应梯度消灭的物理信息神经网络（adaptive gradient-annihilated PINNs,AGA-PINNs）方法来求解具有间断解的Burgers方程与Allen-Cahn方程.该方法利用与梯度相关的权重函数来优化损失函数，并在训练过程中根据当前的残差分布动态调整训练点，以逐步强化模型对间断区域的学习能力.算例结果表明，AGA-PINNs方法比传统的PINNs方法、gPINNs方法、GA-PINNs方法、高阶DG方法等对于物理量的预测精度有显著提升，在求解Burgers方程与Allen-Cahn方程时均方误差降低了大约一个数量级，准确地再现了Burgers方程中冲击波的特征与Allen-Cahn方程中相分离现象.
- AGA-PINNs /
- Burgers方程 /
- Allen-Cahn方程 /
- 自适应方法
Abstract: The physical information neural networks (PINNs) are an important tool for solving partial differential equations. In the numerical solution of partial differential equations, equations with discontinuous solutions are currently a research challenge. The PINNs and their existing improved algorithms often cannot capture the characteristics of discontinuities. However, many discontinuous problems need to be considered in the field of fluid mechanics. In response to the shortcomings of the PINNs in handling equations with discontinuous solutions, an adaptive gradient-annihilated PINNs (AGA-PINNs) method was proposed to solve the Burgers equations and the Allen-Cahn equations with discontinuous solutions. The gradient related weight functions were utilized to optimize the loss function, and the training points were dynamically adjusted based on the current residual distribution during the training process to gradually enhance the model’s learning ability in discontinuous regions. The experimental results show that, the AGA-PINNs method significantly improves the accuracy of predicted physical quantities compared to the traditional PINNs, the gradient-enhanced physics-informed neural networks (gPINNs), the gradient-annihilated physics-informed neural networks (GA-PINNs) method, and the high-order deep Galerkin （DG）method. In solving the Burgers equation and the Allen-Cahn equation, the mean square errors decrease by approximately 1 order of magnitude, accurately reproducing the characteristics of shock waves in the Burgers equation and the phase separation phenomena in the Allen-Cahn equation.
- AGA-PINNs /
- Burgers equation /
- Allen-Cahn equation /
- adaptive method

HTML全文

参考文献(30)

[2]Wang Y, Cheung S W, Chung E T, et al. Deep multiscale model learning[J].Journal of Computational Physics,2020,406: 109071.

Zhang K, Zhao X G, Zhang L M, et al. Current status and prospect for the research and application of big data and intelligent optimization methods in oilfield development[J].Journal of China University of Petroleum (Edition of Natural Science),2020,44(4): 28-38.

[3]邱天威, 魏光美, 宋禹欣, 等. 基于PINN方法的KdV类方程新孤子解的研究[J]. 应用数学和力学, 2025,46(1): 105-113.(Qiu Tianwei, Wei Guangmei, Song Yuxin, et al. Novel soliton solutions to KdV-type equations based on physics-informed neural networks[J].Applied Mathematics and Mechanics,2025,46(1): 105-113. (in Chinese))

[4]Zha W, Li X, Li D, et al. Shale digital core image generation based on generative adversarial networks[J].Journal of Energy Resources Technology,2021,143(3): 033003.

[5]肖争光, 张春利, 陈伟球. 基于PINNs的压电半导体梁的非线性多场耦合力学分析[J]. 应用数学和力学, 2024,45(10): 1288-1299.(Xiao Zhengguang, Zhang Chunli, Chen Weiqiu. Analysis of nonlinear multi-field coupling mechanics of piezoelectric semiconductor beams via PINNs[J].Applied Mathematics and Mechanics,2024,45(10): 1288-1299. (in Chinese))

[6]Pang G, Lu L, Karniadakis G E. fPINNs: fractional physics-informed neural networks[J].SIAM Journal on Scientific Computing,2019,41(4): A2603-A2626.

[7]闵建, 傅卓佳, 郭远. 课程-迁移学习物理信息神经网络用于曲面长时间对流扩散行为模拟[J]. 应用数学和力学, 2024,45(9): 1212-1223.(Min Jian, Fu Zhuojia, Guo Yuan. Curriculum-transfer-learning-based physics-informed neural networks for simulating long-term-evolution convection-diffusion behaviors on curved surfaces[J].Applied Mathematics and Mechanics,2024,45(9): 1212-1223. (in Chinese))

[8]Li X, Chen H, Liu Z, et al. Identifying varying thermal diffusivity of inhomogeneous materials based on a hybrid physics-informed neural network[J].International Journal of Applied Mechanics,2022,14(7): 2250027.〖JP〗

[9]Hornik K, Stinchcombe M, White H. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks[J].Neural Networks,1990,3(5): 551-560.

[10]Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks:a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J].Journal of Computational Physics,2019,378: 686-707.

[11]Lu L, Meng X, Mao Z, et al. DeepXDE: a deep learning library for solving differential equations[J].SIAM Review,2021,63(1): 208-228.

[12]Yu J, Lu L, Meng X, et al. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems[J].Computer Methods in Applied Mechanics and Engineering,2022,393: 114823.

[13]Guo H, Zhuang X, Fu X, et al. Physics-informed deep learning for three-dimensional transient heat transfer analysis of functionally graded materials[J].Computational Mechanics,2023,72(3): 513-524.

[14]Raissi M, Yazdani A, Karniadakis G E. Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations[J].Science,2020,367(6481): 1026-1030.

[15]Bar-Kohany T, Jain A. Dissipation of boundary effects in multilayer heat conduction problems[J].International Journal of Heat and Mass Transfer,2024,223: 125207.

[16]Ferrer-Sánchez A, Martín-Guerrero J D, de Austri-Bazan R R, et al. Gradient-annihilated PINNs for solving Riemann problems: application to relativistic hydrodynamics[J].Computer Methods in Applied Mechanics and Engineering,2024,424: 116906.

[17]Mao Z, Jagtap A D, Karniadakis G E. Physics-informed neural networks for high-speed flows[J].Computer Methods in Applied Mechanics and Engineering,2020,360: 112789.

[18]Xu H, Chang H, Zhang D. DL-PDE: deep-learning based data-driven discovery of partial differential equations from discrete and noisy data[J].Communications in Computational Physics,2025,29(3): 698-728.

[19]Xu J, Wei H, Bao H. Physics-informed neural networks for studying heat transfer in porous media[J].International Journal of Heat and Mass Transfer,2023,217: 124671.

[20]Lu L, Pestourie R, Yao W, et al. Physics-informed neural networks with hard constraints for inverse design[J].SIAM Journal on Scientific Computing,2021,43(6): B1105-B1132.

[21]Mahmud M S, Huang J Z, Salloum S, et al. A survey of data partitioning and sampling methods to support big data analysis[J].Big Data Mining and Analytics,2020,3(2): 85-101.

[22]Nabian M A, Gladstone R J, Meidani H. Efficient training of physics-informed neural networks via importance sampling[J].Computer-Aided Civil and Infrastructure Engineering,2021,36(8): 962-977.

[23]Renardy M, Joslyn L R, Millar J A, et al. To Sobol or not to Sobol? The effects of sampling schemes in systems biology applications[J].Mathematical Biosciences,2021,337: 108593.

[24]Wu C, Zhu M, Tan Q, et al. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks[J].Computer Methods in Applied Mechanics and Engineering,2023,403: 115671.

[25]Mishra S, Molinaro R. Estimates on the generalization error of physics-informed neural networks for approximating PDEs[J].IMA Journal of Numerical Analysis,2023,43(1): 1-43.

[26]de Lara F M, Ferrer E. Accelerating high order discontinuous Galerkin solvers using neural networks: 1D Burgers’ equation[J].Computers & Fluids,2022,235: 105274.

[27]Du Q, Yang J, Zhou Z. Time-fractional Allen-Cahn equations: analysis and numerical methods[J].Journal of Scientific Computing,2020,85(2): 42.

[28]Toro E F.Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction[M]. Berlin: Springer, 2009.

[29]Zeng S, Zhang Z, Zou Q. Adaptive deep neural networks methods for high-dimensional partial differential equations[J].Journal of Computational Physics,2022,463: 111232.

[30]Zhang H, Xu Y, Liu Q, et al. Solving Fokker-Planck equations using deep KD-tree with a small amount of data[J].Nonlinear Dynamics,2022,108(4): 4029-4043.

施引文献

资源附件(0)

访问统计