|
|
Modern Physics 2025
基于热传导方程的自适应损失物理信息神经网络算法研究
|
Abstract:
在热传导方程的研究中，物理信息神经网络(PINN)的应用已初显成效，其损失函数由多个损失项的加权和组成，这些损失项的加权组合对PINN的有效训练具有关键作用。为此，我们引入了一个基于高斯概率模型的损失项定义，通过噪声参数来描述每个损失项的权重，并提出了一种基于极大似然估计原理的自适应损失函数方法，该方法通过不断更新每个训练周期中的噪声参数，实现损失权重的自动分配。采用自适应物理信息神经网络(SalPINN)对一维瞬态热传导方程进行求解，并与传统PINN方法对比，结果显示SalPINN在模拟热传导方程方面表现出更高的精确性和有效性。
In the field of research into heat transfer equations, the application of physical information neural network (PINN) has achieved some results. The loss function of PINN consists of a weighted sum of multiple loss terms, and the weighted combination of these loss terms plays an important role in PINN’s effective training. Therefore, we construct a loss term definition based on a Gaussian probability model, where the introduction of noise parameters is used to describe the weight of each loss term. We propose a self-adaptive loss function method based on the maximum likelihood estimation principle to automatically assign loss weights by constantly updating noise parameters in each training cycle. Then, we use self-adaptive loss physical information neural network (SalPINN) to solve the one-dimensional transient heat transfer equation, and compare it with the traditional PINN method, and the results show that SalPINN is more accurate and effective in simulating the heat transfer equation.
| [1] | Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019) Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. https://doi.org/10.1016/j.jcp.2018.10.045 |
| [2] | Jagtap, A.D., Kharazmi, E. and Karniadakis, G.E. (2020) Conservative Physics-Informed Neural Networks on Discrete Domains for Conservation Laws: Applications to Forward and Inverse Problems. Computer Methods in Applied Mechanics and Engineering, 365, Article ID: 113028. https://doi.org/10.1016/j.cma.2020.113028 |
| [3] | Pu, J., Li, J. and Chen, Y. (2021) Solving Localized Wave Solutions of the Derivative Nonlinear Schrödinger Equation Using an Improved PINN Method. Nonlinear Dynamics, 105, 1723-1739. https://doi.org/10.1007/s11071-021-06554-5 |
| [4] | Kadeethum, T., Jørgensen, T.M. and Nick, H.M. (2020) Physics-Informed Neural Networks for Solving Nonlinear Diffusivity and Biot’s Equations. PLOS ONE, 15, e0232683. https://doi.org/10.1371/journal.pone.0232683 |
| [5] | Lagergren, J.H., Nardini, J.T., Baker, R.E., Simpson, M.J. and Flores, K.B. (2020) Biologically-Informed Neural Networks Guide Mechanistic Modeling from Sparse Experimental Data. PLOS Computational Biology, 16, e1008462. https://doi.org/10.1371/journal.pcbi.1008462 |
| [6] | Cipolla, R., Gal, Y. and Kendall, A. (2018) Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics. 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Salt Lake City, 18-23 June 2018, 7482-7491. https://doi.org/10.1109/cvpr.2018.00781 |
| [7] | 张焕, 张庆, 于纪言. 激活函数的发展综述及其性质分析[J]. 西华大学学报(自然科学版), 2021, 40(4): 1-10. |
| [8] | Sharma, S., Sharma, S. and Athaiya, A. (2020) Activation Functions in Neural Networks. International Journal of Engineering Applied Sciences and Technology, 4, 310-316. https://doi.org/10.33564/ijeast.2020.v04i12.054 |
| [9] | 张海斌, 薛毅. 自动微分的基本思想与实现[J]. 北京工业大学学报, 2005, 31(3): 332-336. |
| [10] | Givoli, D. (1991) Non-Reflecting Boundary Conditions. Journal of Computational Physics, 94, 1-29. https://doi.org/10.1016/0021-9991(91)90135-8 |
| [11] | Abdolrasol, M.G.M., Hussain, S.M.S., Ustun, T.S., Sarker, M.R., Hannan, M.A., Mohamed, R., et al. (2021) Artificial Neural Networks Based Optimization Techniques: A Review. Electronics, 10, Article 2689. https://doi.org/10.3390/electronics10212689 |
