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A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs

Songming LiuZhongkai HaoChengyang YingHang SuJun ZhuZe Cheng
Oct 2022
摘要
We present a unified hard-constraint framework for solving geometricallycomplex PDEs with neural networks, where the most commonly used Dirichlet,Neumann, and Robin boundary conditions (BCs) are considered. Specifically, wefirst introduce the "extra fields" from the mixed finite element method toreformulate the PDEs so as to equivalently transform the three types of BCsinto linear forms. Based on the reformulation, we derive the general solutionsof the BCs analytically, which are employed to construct an ansatz thatautomatically satisfies the BCs. With such a framework, we can train the neuralnetworks without adding extra loss terms and thus efficiently handlegeometrically complex PDEs, alleviating the unbalanced competition between theloss terms corresponding to the BCs and PDEs. We theoretically demonstrate thatthe "extra fields" can stabilize the training process. Experimental results onreal-world geometrically complex PDEs showcase the effectiveness of our methodcompared with state-of-the-art baselines.
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