Article Details

Abstract

To address the problem of limited accuracy in solving differential equations with a Physics-Informed Neural Network (PINN), a readout-PINN architecture with integrated outputs of multiple hidden layers is proposed, where the final solution is obtained by fusing the outputs of each hidden layer. Taking one-dimensional and two-dimensional Poisson's equations as numerical examples, Limited-memory BFGS (L-BFGS) is adopted for optimization training, and comparisons are made with PINN of the same scale, Gradient-enhanced PINN (gPINN) and hp-Variational PINN (hp-VPINN). The results show that compared with PINN, the maximum absolute error and mean squared error are reduced by 93.1% and 99.1% for the one-dimensional case, and 72.6% and 93.6% for two-dimensional case, respectively. While maintaining high accuracy, the proposed model achieves a lower computational cost than that of gPINN and hp-VPINN.

Keywords

Poisson's equation; PINN; Neural Network; high-frequency differential equation

References

1. [1] Cuomo S，Cola V S D，Giampaolo F，et al. Scientific Machine Learning Through Physics-Informed Neural Networks: Where We are and What's Next [J].Journal of Scientific Computing，2022，92（3）：1-62.
2. [2] Arqub O A. Numerical Solutions for the Robin Time-fractional Partial Differential Equations of Heat and Fluid Flows Based on the Reproducing Kernel Algorithm [J].International Journal of Numerical Methods for Heat & Fluid Flow，2018，28（4）：828-856.
3. [3] Gomez L J，Dannhauera M，Koponena L M，et al. Conditions for Numerically Accurate TMS Electric Field Simulation [J].Brain Stimulation，2020，13（1）：157-166.
4. [4] Bonito A，Canuto C，Nochetto R H，et al. Adaptive Finite Element Methods [J].Acta Numerica，2024，33（4）：163-485.
5. [5] Wang Yizheng, Zhuang Xiaoying, Rabczcuk T， Wait Research progress of AI for PDEs in the field of solid mechanics [J]. Advances in Mechanics, 2025, 55 (2): 231-287
6. [6] Cybenko G. Approximation by Superpositions of a Sigmoidal Function [J].Mathematics of Control, Signals and Systems，1989，2（4）：303-314.
7. [7] Hornik K. Approximation Capabilities of Multilayer Feedforward Networks [J].Neural Networks，1991，4（2）：251-257.
8. [8] Telgarsky M. Benefits of Depth in Neural Networks [C]//Proceedings of the 29th Annual Conference on Learning Theory，2016：1517-1539.
9. [9] 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.
10. [10] Ko S，Park S H. VS-PINN: A Fast and Efficient Training of Physics-informed Neural Networks Using Variable-scaling Methods for Solving PDEs with Stiff Behavior [J/OL].arXiv:2406.06287 [math.NA].https://arxiv.org/abs/2406.06287.
11. [11] Song Y J，Wang H，Yang H，et al. Loss-attentional Physics-Informed Neural Networks [J/OL].Journal of Computational Physics，2024，501：112781.https://www.sciencedirect.com/science/article/pii/S0021999124000305?via%3Dihub.
12. [12] He K，Zhang X，Ren S，et al. Deep Residual Learning for Image Recognition [C]//Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition，2016：770-778.
13. [13] Huang G，Liu Z，Van Der Maaten L，et al. Densely Connected Convolutional Networks [C]//Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition，2017：2261-2269.
14. [14] Szegedy C，Liu W，Jia Y，et al. Going Deeper with Convolutions [C]//Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition，2015：1-9.
15. [15] Yu J，Lu L，Meng X H，et al. Gradient-enhanced Physics-informed Neural Networks for Forward and Inverse PDE Problems [J/OL].Computer Methods in Applied Mechanics and Engineering，2022，393：114823.https://www.sciencedirect.com/science/article/abs/pii/S0045782522001438.
16. [16] Kharazmi E，Zhang Z Q，Karniadakis G E，et al. hp-VPINNs: Variational Physics-informed Neural Networks with Domain Decomposition [J/OL].arXiv:2003.05385 [cs.NE].https://arxiv.org/abs/2003.05385.
17. [17] Rumelhart D E，Hinton G E，Williams R J. Learning Representations by Back-Propagating Errors [J].Nature，1986，323（6088）：533-536.
18. [18] Liu D C，Nocedal J. On the Limited Memory BFGS Method for Large Scale Optimization [J].Mathematical Programming，1989，45（1-3）：503-528.

About the authors

Lin Yuxuan (1999-), female, Han ethnicity, from Shangqiu, Henan Province, is currently pursuing a master's degree in neural network solutions for differential equations.