论文标题
用于解决自由边界系统的神经网络的收敛分析
Convergence analysis of neural networks for solving a free boundary system
论文作者
论文摘要
自由边界问题涉及部分微分方程的系统，其中域边界是未知的。由于这种特殊的特征，在理论上或数字上解决自由边界问题是具有挑战性的。在本文中，我们开发了一种新的方法，用于解决基于神经网络离散化的修改后的Hele-Shaw问题。理论上建立了使用这种离散化的数值解决方案的存在。我们还通过计算对称性的溶液来验证这种方法，这些溶液以径向对称分支附近的分叉分析为指导。此外，我们通过计算某些非特征于任何定理特征的非亚基对称解，进一步验证了这种方法的能力。
Free boundary problems deal with systems of partial differential equations, where the domain boundaries are apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problems either theoretically or numerically. In this paper, we develop a novel approach for solving a modified Hele-Shaw problem based on neural network discretization. The existence of the numerical solution with this discretization is established theoretically. We also numerically verify this approach by computing the symmetry-breaking solutions which are guided by the bifurcation analysis near the radially-symmetric branch. Moreover, we further verify the capability of this approach by computing some non-radially symmetric solutions which are not characterized by any theorems.
