Performance and architecture of neural network for increased dimensions
Some things to take into account:


Try to apply appropriate input space transformations, e.g. convert to polar coordinates.
Despite the fact that a single hidden layer feedforward network can be a universal approximator, there is no guarantees about the number of neurons needed to approximate an arbitrarily complex function. Instead of having a single hidden layer and making it increasingly wide, try stacking more layers. This enables the network to model non linearities in an easier way.
Do not verify the performance of your model based on visual inspection (at least not only), but based on error measurements (e.g. MSE).