We present PFNN, a penalty-free neural network method, to efficiently solve a class of second-order boundary-value problems on complex geometries. To reduce the smoothness requirement, the original problem is reformulated to a weak form so that the evaluations of high-order derivatives are avoided. Two neural networks, rather than just one, are employed to construct the approximate solution, with one network satisfying the essential boundary conditions and the other handling the rest part of the domain. In this way, an unconstrained optimization problem, instead of a constrained one, is solved without adding any penalty terms. The entanglement of the two networks is eliminated with the help of a length factor function that is scale invariant and can adapt with complex geometries. We prove the convergence of the PFNN method and conduct numerical experiments on a series of linear and nonlinear second-order boundary-value problems to demonstrate that PFNN is superior to several existing approaches in terms of accuracy, flexibility and robustness.

我们提出了PFNN，一种无惩罚的神经网络方法，以有效解决复杂几何上的一类二阶边值问题。为了降低平滑度要求，将原始问题重新构造为弱形式，从而避免了对高阶导数的评估。使用两个神经网络（而不只是一个神经网络）来构造近似解，其中一个网络满足基本边界条件，另一个网络处理域的其余部分。以此方式，解决了无约束的优化问题，而不是无约束的优化问题，而无需添加任何惩罚项。借助长度不变的长度因子函数可以消除两个网络的纠缠，并且可以适应复杂的几何形状。