  Training an artificial neural network involves an optimization process over
the landscape defined by the cost (loss) as a function of the network
parameters. We explore these landscapes using optimisation tools developed for
potential energy landscapes in molecular science. The number of local minima
and transition states (saddle points of index one), as well as the ratio of
transition states to minima, grow rapidly with the number of nodes in the
network. There is also a strong dependence on the regularisation parameter,
with the landscape becoming more convex (fewer minima) as the regularisation
term increases. We demonstrate that in our formulation, stationary points for
networks with $N_h$ hidden nodes, including the minimal network required to fit
the XOR data, are also stationary points for networks with $N_{h} +1$ hidden
nodes when all the weights involving the additional nodes are zero. Hence,
smaller networks optimized to train the XOR data are embedded in the landscapes
of larger networks. Our results clarify certain aspects of the classification
and sensitivity (to perturbations in the input data) of minima and saddle
points for this system, and may provide insight into dropout and network
compression.
