In this study, we propose a new data-free framework, Feature Enforcing Physics Informed Neural Network (FE-PINN), to overcome the challenge of an imbalanced loss function in vanilla PINNs. The imbalance is caused by the presence of two terms in the loss function: the partial differential loss and the boundary condition mean squared error. A standard solution is to use loss weighting, but it requires hyperparameter tuning. To address this challenge, we introduce a process called smart initialization to force the neural network to learn only the boundary conditions before the final training in a designed process. In this method, clustered domain points are used to train a neural network with designed weights, resulting in the creation of a neural network called Foundation network. This results in a network with unique weights that understand boundary conditions. Then, additional layers are used to improve the accuracy. This solves the problem of an imbalanced loss function without further need for hyperparameter tuning. For 2D flow over a cylinder as a benchmark, smart initialization in FE-PINN is 574 times faster than hyperparameter tuning in vanilla PINN. Even with the optimal loss weight value, FE-PINN outperforms vanilla PINN by speeding up the average training time by 1.98. Also, the ability of the proposed approach is shown for an inverse problem. To find the inlet velocity for a 2D flow over a cylinder, FE-PINN is twice faster than vanilla PINN with the knowledge of optimal weight loss value for vanilla PINN. Our results show that FE-PINN not only eliminates the time-consuming process of loss weighting but also improves convergence speed compared to vanilla PINN, even when the optimal weight value is used in its loss function. In conclusion, this framework can be used as a fast and accurate tool for solving a wide range of Partial Differential Equations across various fields.

翻译：本研究提出了一种无数据框架——特征强制物理信息神经网络（FE-PINN），以克服经典PINN中损失函数不平衡的挑战。该不平衡源于损失函数中两项的存在：偏微分损失项与边界条件均方误差项。标准解决方案采用损失加权法，但需要超参数调优。为解决此问题，我们引入名为"智能初始化"的过程，通过设计流程强制神经网络在最终训练前仅学习边界条件。在该方法中，使用聚类域点结合设计权重训练神经网络，生成名为"基础网络"的神经网络。由此产生一个具有独特权重且理解边界条件的网络。随后通过附加层提升精度。该方法在无需进一步超参数调优的情况下解决了损失函数不平衡问题。以二维圆柱绕流基准测试为例，FE-PINN中的智能初始化比经典PINN的超参数调优快574倍。即便采用最优损失权重值，FE-PINN仍以平均训练时间加速1.98倍的优势优于经典PINN。此外，该方法的逆问题求解能力也得到验证：在二维圆柱绕流入口速度反演中，FE-PINN比已知最优损失权重的经典PINN快两倍。结果表明，FE-PINN不仅消除了耗时损失加权过程，即使在损失函数中使用最优权重值时，其收敛速度仍优于经典PINN。综上所述，该框架可作为快速精准的工具，广泛应用于各领域的偏微分方程求解。