A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs

Abstract

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 forms. 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.

A solution for finite journal bearings by using physics-informed neural networks with both soft and hard constrains

“…Mathematically, the hard constraint method can satisfy the corresponding pressure boundary conditions exactly. Moreover, the converge of the loss function for L PDE is normally faster than the one for L BC (Liu et al, 2022;Wang et al, 2021;Wang et al, 2022), and for the hard constraint method, it is unnecessary to consider the boundary conditions in the loss functions; therefore, the commutating cost can also be reduced.…”