论文标题
Rang：一种基于残留的物理信息神经网络的自适应节点生成方法
RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks
论文作者
论文摘要
具有物理知识神经网络（PINN）的部分微分方程（PDE）的学习解决方案是传统求解器的一种有吸引力的替代方法，因为它的灵活性和易于合并观察到的数据。尽管PINN在准确求解各种PDE方面取得了成功，但该方法仍需要改善计算效率。一种可能的改进想法是优化训练点集的产生。基于残留的自适应抽样和准均匀抽样方法分别应用于改善PINN的训练效应。为了从两种方法中受益，我们提出了有效训练PINN的基于残留的自适应节点产生（RANG）方法，该方法基于RBF-FD的可变密度淋巴结分布方法。记忆机制还可以增强该方法，以进一步提高训练稳定性。我们在三个线性PDE和三个具有各种节点生成方法的非线性PDE上进行实验，通过该方法，该方法的准确性和效率与主要均匀的采样方法相比得到了数值验证。
Learning solutions of partial differential equations (PDEs) with Physics-Informed Neural Networks (PINNs) is an attractive alternative approach to traditional solvers due to its flexibility and ease of incorporating observed data. Despite the success of PINNs in accurately solving a wide variety of PDEs, the method still requires improvements in terms of computational efficiency. One possible improvement idea is to optimize the generation of training point sets. Residual-based adaptive sampling and quasi-uniform sampling approaches have been each applied to improve the training effects of PINNs, respectively. To benefit from both methods, we propose the Residual-based Adaptive Node Generation (RANG) approach for efficient training of PINNs, which is based on a variable density nodal distribution method for RBF-FD. The method is also enhanced by a memory mechanism to further improve training stability. We conduct experiments on three linear PDEs and three nonlinear PDEs with various node generation methods, through which the accuracy and efficiency of the proposed method compared to the predominant uniform sampling approach is verified numerically.
