Physics-Informed Neural Networks (PINNs) have become a prominent application of deep learning in scientific computation, as they are powerful approximators of solutions to nonlinear partial differential equations (PDEs). There have been numerous attempts to facilitate the training process of PINNs by adjusting the weight of each component of the loss function, called adaptive loss-balancing algorithms. In this paper, we propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs). We treat the initial and boundary conditions as constraints for the optimization problem of the PDE residual. By employing Augmented Lagrangian relaxation, the constrained optimization problem becomes a sequential max-min problem so that the learnable parameters $\lambda$ adaptively balance each loss component. Our theoretical analysis reveals that the sequence of minimizers of the proposed loss functions converges to an actual solution for the Helmholtz, viscous Burgers, and Klein--Gordon equations. We demonstrate through various numerical experiments that AL-PINNs yield a much smaller relative error compared with that of state-of-the-art adaptive loss-balancing algorithms.

翻译：物理信息神经网络已成为深度学习在科学计算中的重要应用，因其能够有效逼近非线性偏微分方程的解。为促进PINNs的训练过程，已有多项研究通过调整损失函数各分量的权重（即自适应损失平衡算法）进行尝试。本文提出一种针对PINNs的增广拉格朗日松弛方法。我们将初始条件与边界条件视为偏微分方程残差优化问题的约束条件。通过采用增广拉格朗日松弛，该约束优化问题转化为序列最大-最小问题，使得可学习参数$\lambda$能够自适应地平衡各损失分量。理论分析表明，所提损失函数的最小化序列收敛至亥姆霍兹方程、黏性伯格斯方程和克莱因-戈登方程的真实解。多种数值实验证实，相比现有最先进的自适应损失平衡算法，AL-PINNs的相对误差显著更小。