PFNN：求解一类复杂几何二阶边值问题的无惩罚神经网络方法
PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
摘要：
提出了一种无惩罚的神经网络方法PFNN，用于求解一类复杂几何条件下的二阶边值问题。为了降低光滑性要求，将原问题转化为弱形式，从而避免了高阶导数的计算。两个神经网络，而不是只有一个，被用来构造近似解，与一个网络满足本质边界条件和其他处理域的其余部分。用这种方式，一个无约束的最优化问题，而不是一个有约束的，解决了没有添加任何惩罚条款。两个网络的纠缠被消除的长度因子函数的帮助下，是规模不变的，可以适应复杂的几何形状。我们证明了PFNN方法的收敛性，并对一系列线性和非线性二阶边值问题进行了数值实验，以证明PFNN在精度、灵活性和鲁棒性方面优于现有的几种方法。
英文摘要：
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.
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