In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.

翻译：本文提出了一种新颖的迭代深度Ritz方法（IDRM），用于求解一类广泛的椭圆问题。该方法受到神经网络训练过程中最小化损失函数的迭代过程的启发，但在每一步中编码了底层函数空间的几何结构，并引入凸惩罚项以提升算法性能。该算法适用于涉及单调算子（不一定具有变分形式）的椭圆问题，且不对解施加任何严格的正则性假设。对于所关注的椭圆问题类别，该方法在精度上改进了多种现有的神经偏微分方程求解器，例如物理信息神经网络和深度Ritz方法。此外，我们利用Banach空间几何理论和单调算子理论工具，建立了该方法的收敛速度，并分析了学习误差。为说明方法的有效性，我们给出了若干具有挑战性的数值算例，包括与现有技术的对比研究。