  We show that deep convolutional neural networks (CNN) can massively
outperform traditional densely-connected neural networks (both deep or shallow)
in predicting eigenvalue problems in mechanics. In this sense, we strike out in
a new direction in mechanics computations with strongly predictive NNs whose
success depends not only on architectures being deep, but also being
fundamentally different from the widely-used to date. We consider a model
problem: predicting the eigenvalues of 1-D and 2-D phononic crystals. For the
1-D case, the optimal CNN architecture reaches $98\%$ accuracy level on unseen
data when trained with just 20,000 samples, compared to $85\%$ accuracy even
with $100,000$ samples for the typical network of choice in mechanics research.
We show that, with relatively high data-efficiency, CNNs have the capability to
generalize well and automatically learn deep symmetry operations, easily
extending to higher dimensions and our 2D case. Most importantly, we show how
CNNs can naturally represent mechanical material tensors, with its convolution
kernels serving as local receptive fields, which is a natural representation of
mechanical response. Strategies proposed are applicable to other mechanics'
problems and may, in the future, be used to sidestep cumbersome algorithms with
purely data-driven approaches based upon modern deep architectures.
