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Learning the mapping $\mathbf{x}\mapsto \sum_{i=1}^d x_i^2$: the cost of finding the needle in a haystack
When such structural information is not available, and we may only use a dense neural network, the optimization procedure to find the sparse network embedded in the dense network is similar to finding the needle in a haystack, using a given number of samples of the function.
Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networks
By directly targeting at the self-consistent electron density, we demonstrate that the adapted network architecture, called the Deep Density, can effectively represent the electron density as the linear combination of contributions from many local clusters.
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