A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs
This work addresses computational challenges in solving PDEs for domains like engineering and physics, but it appears incremental as it builds on existing methods like mixed finite elements.
The authors tackled the problem of solving geometrically complex PDEs with neural networks by developing a unified hard-constraint framework that automatically satisfies boundary conditions, eliminating the need for extra loss terms and improving training stability. Experimental results demonstrated effectiveness compared to state-of-the-art baselines, though no concrete numbers were provided.
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.