Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective

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.