In this paper we propose a {\it discontinuous} plane wave neural network (DPWNN) method with $hp-$refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with $h-$refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving the minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.

翻译：摘要：本文提出了一种采用$hp-$精化的非连续平面波神经网络（DPWNN）方法，用于近似求解Helmholtz方程和时间谐波Maxwell方程组。该方法中，我们定义了与$h-$精化平面波最小二乘法（PWLS）中类似的二次泛函，并引入由逐元单隐层神经网络函数张成的新离散集，其中每个单元上的激活函数被选为类似平面波函数的复值指数函数。通过迭代求解与上述泛函及离散集相关联的极小化问题，递归生成所需近似解；该过程由底层残差泛函的一系列近似极小化序列定义，其中平面波方向角与激活系数通过迭代算法交替计算。与标准PWLS方法（需预先指定平面波方向）不同，所提DPWNN方法可在迭代过程中自适应确定平面波方向。数值实验表明，相较于PWLS方法，DPWNN方法能生成精度更高的近似解。