Physics-Informed Neural Networks (PINNs) have been recognized as a mesh-free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or shock-dominated dynamics, traditional PINNs have been found to have limitations, including unbalanced training and inaccuracy in solution, even with small physics residuals. In this research, we seek to address these limitations using the viscous Burgers' equation with low viscosity and the Allen-Cahn equation as test problems. In addressing unbalanced training, we have developed a new adaptive loss balancing scheme using smoothed gradient norms to ensure satisfaction of initial and boundary conditions. Further, to address inaccuracy in the solution, we have developed an adaptive residual-based collocation scheme to improve the accuracy of solutions in the regions with high physics residuals. The proposed new approach significantly improves solution accuracy with consistent satisfaction of physics residuals. For instance, in the case of Burgers' equation, the relative L2 error is reduced by about 44 percent compared to traditional PINNs, while for the Allen-Cahn equation, the relative L2 error is reduced by approximately 70 percent. Additionally, we show the trustworthy solution comparison of the proposed method using a robust finite difference solver.

翻译：物理信息神经网络（PINNs）已被视为一种无需网格的偏微分方程求解方法，其特点在于融入了物理信息。然而，在处理具有高刚性或激波主导动力学特性的问题时，传统PINNs被发现存在局限，包括训练不平衡以及即使物理残差较小时解的不准确性。在本研究中，我们以低黏性黏性Burgers方程和Allen-Cahn方程作为测试问题，旨在解决这些局限。针对训练不平衡问题，我们开发了一种新的自适应损失平衡方案，利用平滑梯度范数来确保初始条件和边界条件的满足。此外，为解决解的不准确性，我们提出了一种自适应残差驱动配置方案，以提升物理残差较大区域解的精度。所提出的新方法在确保物理残差一致满足的同时，显著提高了解的计算精度。例如，在Burgers方程案例中，与传统PINNs相比，相对L2误差降低了约44%；而在Allen-Cahn方程案例中，相对L2误差降低了约70%。此外，我们通过稳健的有限差分求解器展示了所提方法的可信解对比结果。