  We propose a novel regularization method, called \textit{volumization}, for
neural networks. Inspired by physics, we define a physical volume for the
weight parameters in neural networks, and we show that this method is an
effective way of regularizing neural networks. Intuitively, this method
interpolates between an $L_2$ and $L_\infty$ regularization. Therefore, weight
decay and weight clipping become special cases of the proposed algorithm. We
prove, on a toy example, that the essence of this method is a regularization
technique to control bias-variance tradeoff. The method is shown to do well in
the categories where the standard weight decay method is shown to work well,
including improving the generalization of networks and preventing memorization.
Moreover, we show that the volumization might lead to a simple method for
training a neural network whose weight is binary or ternary.
