For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the solution in the spatial variables. We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is a power of the popular ReLU activation function.

翻译：对于几类神经PDE求解器（Deep Ritz、PINNs、DeepONets），其逼近偏微分方程解或解算子的能力，关键在于神经网络在空间变量上逼近解的能力。本文在假设边界条件可被高效逼近的前提下，分析了神经网络逼近椭圆型PDE解的能力。我们重点关注半空间上带Dirichlet边界条件的拉普拉斯算子，以及使用单隐藏层、激活函数为流行ReLU激活函数幂次的神经网络。