论文标题
一个统一的硬构框架，用于求解几何复杂的PDE
A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs
论文作者
论文摘要
我们提出了一个统一的硬构框架，用于与神经网络求解几何复杂的PDE，其中考虑了最常用的Dirichlet，Neumann和Robin边界条件（BCS）。具体而言，我们首先将“额外字段”从混合有限元方法进行重新重新调整PDE，以等效地将三种类型的BC转换为线性方程。基于重新制定，我们在分析上得出了BCS的一般解决方案，该解决方案被用于构建自动满足BCS的ANSATZ。通过这样的框架，我们可以训练神经网络而无需添加额外的损失项，从而有效处理几何复杂的PDE，从而减轻了与BCS和PDES相对应的损失项之间的不平衡竞争。从理论上讲，我们证明“额外的领域”可以稳定训练过程。现实世界中几何复杂的PDE的实验结果与最先进的基线相比，展示了我们方法的有效性。
We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first introduce the "extra fields" from the mixed finite element method to reformulate the PDEs so as to equivalently transform the three types of BCs into linear equations. Based on the reformulation, we derive the general solutions of the BCs analytically, which are employed to construct an ansatz that automatically satisfies the BCs. With such a framework, we can train the neural networks without adding extra loss terms and thus efficiently handle geometrically complex PDEs, alleviating the unbalanced competition between the loss terms corresponding to the BCs and PDEs. We theoretically demonstrate that the "extra fields" can stabilize the training process. Experimental results on real-world geometrically complex PDEs showcase the effectiveness of our method compared with state-of-the-art baselines.
