  Many phenomena in physics, including light, water waves, and sound, are
described by wave equations. Given their coefficients, wave equations can be
solved to high accuracy, but the presence of the wavelength scale often leads
to large computer simulations for anything beyond the simplest geometries. The
inverse problem, determining the coefficients from a field on a boundary, is
even more demanding, since traditional optimization requires a large number of
forward problems be solved sequentially. Here we show that the free-form
inverse problem of wave equations can be solved with machine learning. First we
show that deep neural networks can be used to predict the optical properties of
nanostructured materials such as metasurfaces. Then we demonstrate the
free-form inverse design of such nanostructures and show that constraints
imposed by experimental feasibility can be taken into account. Our neural
networks promise automated design in several technologies based on the wave
equation.
