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.
翻译:与物理进化神经网络的局部差异方程式(PDEs)的学习解决方案(PDEs)与物理进化神经网络(PINNs)是传统解决方案的一种具有吸引力的替代方法,因为它具有灵活性,而且容易纳入观察到的数据。尽管PINNs成功地准确地解决了各种各样的PDEs,但该方法仍然需要在计算效率方面加以改进。一个可能的改进想法是优化产生训练点成套材料。分别采用了残余的适应性抽样和准统一的抽样方法来提高PINNs的培训效果。为了从这两种方法中受益,我们建议采用基于残余的适应性节点生成(RANG)的方法,对PINNs进行有效的培训,该方法以RFF-FD的可变密度节点分布法为基础。该方法还因一个记忆机制而得到加强,以进一步提高培训稳定性。我们用各种节点生成方法对三个线性PDEs和三个非线性PDEs进行了实验,通过这些方法对拟议方法与主要统一取样方法的准确度和效率进行数字核查。