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Including prior knowledge is important for effective machine learning models in physics, and is usually achieved by explicitly adding loss terms or constraints on model architectures. Prior knowledge embedded in the physics computation itself rarely draws attention. We show that solving the Kohn-Sham equations when training neural networks for the exchange-correlation functional provides an implicit regularization that greatly improves generalization. Two separations suffice for learning the entire one-dimensional H2 dissociation curve within chemical accuracy, including the strongly correlated region. Our models also generalize to unseen types of molecules and overcome self-interaction error. 

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.