Deep Convolutional Neural Networks for Eigenvalue Problems in Mechanics.

In this paper 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 novel direction in mechanics computations with strongly predictive neural networks whose success depends not only neural architectures being deep but also being fundamentally different from neural architectures which have been used in mechanics till now. To show this, we consider a model problem: predicting the eigenvalues of a 1-D phononic crystal, however, the general observations pertaining to the predictive superiority of CNNs over MLPs should extend to other problems in mechanics as well. In the present problem, the optimal CNN architecture reaches $98\%$ accuracy level on unseen data when trained with just 20,000 training samples. Fully-connected multi-layer perceptrons (MLP) - the network of choice in mechanics research - on the other hand, does not improve beyond $85\%$ accuracy even with $100,000$ training samples. We also show that even with a relatively small amount of training data, CNNs have the capability to generalize well for our problems and that they automatically learn deep symmetry operations such as translational invariance. Most importantly, however, we show how CNNs can naturally represent mechanical material tensors and that the convolution operation of CNNs has the ability to serve as local receptive fields which is a natural representation of mechanical response. Strategies proposed here may be used for other problems of mechanics and may, in the future, be used to completely sidestep certain cumbersome algorithms with a purely data driven approach based upon deep architectures of modern neural networks such as deep CNNs.