  The training of neural networks requires tedious and often manual tuning of
the network architecture. We propose a systematic method to insert new layers
during the training process, which eliminates the need to choose a fixed
network size before training. Our technique borrows techniques from constrained
optimization and is based on first-order sensitivity information of the
objective with respect to the virtual parameters that additional layers, if
inserted, would offer. We consider fully connected feedforward networks with
selected activation functions as well as residual neural networks. In numerical
experiments, the proposed sensitivity-based layer insertion technique exhibits
improved training decay, compared to not inserting the layer. Furthermore, the
computational effort is reduced in comparison to inserting the layer from the
beginning. The code is available at
\url{https://github.com/LeonieKreis/layer_insertion_sensitivity_based}.
