Before revision:  The generalization error of the gradient descent solution does not suffer from the curse of dimensonality if the solution is in a Barron-type space . The theories developed here could form a theoretical foundation of deep learning-based PDE solvers .
After revision:  The generalization error of the gradient descent solution does not suffer from the curse of dimensonality if the solution is in a Barron-type space and the least-squares optimization is regularized with a Barron-type norm, without the over-parametrization assumption .