论文标题
PFNN：一种无惩罚的神经网络方法，用于解决复杂几何形状上的一类二阶边界问题问题
PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
论文作者
论文摘要
我们提出PFNN是一种无惩罚的神经网络方法，可有效解决复杂几何形状上的一类二阶边界值问题。为了降低平滑性要求，原始问题被重新构成弱形式，以便避免对高阶导数的评估。两个神经网络，而不仅仅是一个神经网络来构建近似解决方案，一个网络满足基本边界条件，另一个网络满足域的其余部分。通过这种方式，无限制的优化问题而不是受约束的问题可以解决，而无需添加任何惩罚条款。借助比例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。我们证明了PFNN方法的收敛性，并在一系列线性和非线性二阶边界值问题上进行数值实验，以证明PFNN在准确性，灵活性和鲁棒性方面优于几种现有方法。
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.
