用梯度自学习提升PINN求解非线性PDE的精度与收敛性。
Gradient Enhanced Self-Training Physics-Informed Neural Network (gST-PINN) for Solving Nonlinear Partial Differential Equations
- 引入基于梯度的伪点自学习机制，增强模型对物理规律的捕捉能力。
- 在18500次迭代后，误差降至10⁻⁵，显著优于传统PINN的10⁻³~10⁻⁴。
- 无需真实标签数据，在无监督场景下仍表现优异，适合低精度难题。
偏微分方程（PDEs）为物理科学与工程中复杂行为的模拟与理解提供了数学基础。随着深度学习的发展，数据驱动的物理信息神经网络（PINNs）应运而生，以无网格、解析型框架高效求解各类PDE。然而传统PINNs常面临精度有限、训练缓慢、标签数据稀缺及多物理场交互处理不足等问题。为此，本文提出梯度增强自学习物理信息神经网络（gST-PINN），引入基于梯度的伪点自学习算法。在三个不同领域的典型PDE问题上验证：求解Burgers'方程时，标准PINN达到约10⁻³的均方误差（MSE），扩散-吸附方程则达10⁻⁴，且12500次迭代后不再改善；而gST-PINN的误差持续下降，18500次迭代后稳定在10⁻⁵，展现更强泛化能力。结果表明，该纯半监督方法在无真实解情况下仍优于标准PINN，可推广至精度与收敛性差的场景，尤其适用于缺乏标注数据的情形。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data$-$driven approaches like Physics$-$Informed Neural Networks (PINNs) have been developed, offering a mesh$-$free, analytic type framework for efficiently solving PDEs across a wide range of applications. However, traditional PINNs often struggle with challenges such as limited precision, slow training dynamics, lack of labeled data availability, and inadequate handling of multi$-$physics interactions. To overcome these challenging issues of PINNs, we proposed a Gradient Enhanced Self$-$Training PINN (gST$-$PINN) method that specifically introduces a gradient based pseudo point self$-$learning algorithm for solving PDEs. We tested the proposed method on three different types of PDE problems from various fields, each representing distinct scenarios. The effectiveness of the proposed method is evident, as the PINN approach for solving the Burgers$'$ equation attains a mean square error (MSE) on the order of $10^{-3}$, while the diffusion$-$sorption equation achieves an MSE on the order of $10^{-4}$ after 12,500 iterations, with no further improvement as the iterations increase. In contrast, the MSE for both PDEs in the gST$-$PINN model continues to decrease, demonstrating better generalization and reaching an MSE on the order of $10^{-5}$ after 18,500 iterations. Furthermore, the results show that the proposed purely semi$-$supervised gST$-$PINN consistently outperforms the standard PINN method in all cases, even when solution of the PDEs are unavailable. It generalizes both PINN and Gradient$-$enhanced PINN (gPINN), and can be effectively applied in scenarios prone to low accuracy and convergence issues, particularly in the absence of labeled data.
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