We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by the radial basis function (RBF), particular solution neural network, and general neural network. Loss term is the potential energy, optimized based on the principle of minimum potential energy. The loss term at the interfaces has the lower order derivative compared to the strong form PINN with subdomains. The advantage of the proposed method is higher efficiency, more accurate, and less hyperparameters than the strong form PINN with subdomains. Another advantage of the proposed method is that it can apply to complex geometries based on the special construction of the admissible function. To analyze its performance, the proposed method CENN is used to model representative PDEs, the examples include strong discontinuity, singularity, complex boundary, non-linear, and heterogeneous problems. Furthermore, it outperforms other methods when dealing with heterogeneous problems.
翻译:我们建议一种基于神经网络的保守能源方法,该方法的特点是:无边界惩罚满足基本边界条件的可受理功能是由辐射基函数(RBF)、特定溶液神经网络和一般神经网络构建的。损失术语是潜在的能源,根据最低潜在能量原则加以优化。接口的损失术语是低排序衍生物,而不是具有子域的强势PINN。拟议方法的优点是效率更高、更准确、高超参数,比有子域的强型PINN的强型PINN要低。拟议方法的另一个优点是,该方法可以适用于基于特殊构建可受理功能的复杂地貌。为分析其性能,拟议的CENN方法用于模拟具有代表性的PDEs,其例子包括强烈的不连续性、独一性、复杂边界、非线性和多式问题。此外,该方法比处理多种问题的其他方法要好。