Regularization term moving the hyperbolic plane
The point of the regularization term is usually to keep the network from overfitting by keeping the weights vector with low absolute values or sparse. Therefore I would say a sum of all weights is not a good regularization term, because even extreme positive and negative weights can sum to zero. In the regularization term, the $L_2$, $L_1$, $L_0$ norms are commonly used (read about it here). 

In a side note, if I understood you correctly, I would suggest to avoid imagining the target function (or loss function) as a hyperbolic plane, since usually DNNs contain non linear activation functions like ReLU.