  Recent theoretical work has demonstrated that deep neural networks have
superior performance over shallow networks, but their training is more
difficult, e.g., they suffer from the vanishing gradient problem. This problem
can be typically resolved by the rectified linear unit (ReLU) activation.
However, here we show that even for such activation, deep and narrow neural
networks (NNs) will converge to erroneous mean or median states of the target
function depending on the loss with high probability. Deep and narrow NNs are
encountered in solving partial differential equations with high-order
derivatives. We demonstrate this collapse of such NNs both numerically and
theoretically, and provide estimates of the probability of collapse. We also
construct a diagram of a safe region for designing NNs that avoid the collapse
to erroneous states. Finally, we examine different ways of initialization and
normalization that may avoid the collapse problem. Asymmetric initializations
may reduce the probability of collapse but do not totally eliminate it.
