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 the 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.

自由边界问题涉及偏微分方程组，其区域边界是先验未知的。由于这种特殊的特性，从理论上或在数值上解决自由边界问题都具有挑战性。在本文中，我们开发了一种基于神经网络离散化方法来解决改进的Hele-Shaw问题的新颖方法。理论上证明了这种离散化的数值解的存在性。我们还通过计算在径向对称分支附近的分叉分析指导的对称破坏解，对这种方法进行了数值验证。此外，我们通过计算一些没有任何定理表征的非径向对称解，进一步验证了这种方法的能力。