  Kohn-Sham Density Functional Theory (KS-DFT) has been traditionally solved by
the Self-Consistent Field (SCF) method. Behind the SCF loop is the physics
intuition of solving a system of non-interactive single-electron wave functions
under an effective potential. In this work, we propose a deep learning approach
to KS-DFT. First, in contrast to the conventional SCF loop, we propose to
directly minimize the total energy by reparameterizing the orthogonal
constraint as a feed-forward computation. We prove that such an approach has
the same expressivity as the SCF method, yet reduces the computational
complexity from O(N^4) to O(N^3). Second, the numerical integration which
involves a summation over the quadrature grids can be amortized to the
optimization steps. At each step, stochastic gradient descent (SGD) is
performed with a sampled minibatch of the grids. Extensive experiments are
carried out to demonstrate the advantage of our approach in terms of efficiency
and stability. In addition, we show that our approach enables us to explore
more complex neural-based wave functions.
