I am a physicist and I don't have much background on machine learning or deep learning except taking a couple of courses on statistics. In physics, we often simulate a model by means of two-way coupled systems where each system is described by a partial differential equation. The equations are generally unfolded in a numerical grid of interest and then solved iteratively until a self-consistent solution is obtained.
A well-known example is the Schrödinger-Poisson solver. Here, for a given nano/atomic structure, we assume an initial electron density. Then we solve the Poisson equation for that electron density. The Poisson equation tells us the electrostatics (electric potential) of the structure. Given this information, we solve the Schrödinger equation for the structure which tells us about the energy levels of the electrons and their wave functions in the structure. But one would then find that this energy levels and wavefunctions correspond to a different electron density (than the one we initially guessed). So we iterate the process with the new electron density and follow the above mentioned procedures in a loop until a self-consistent solution is obtained.
Often times the iteration processes are computationally expensive and extremely time-consuming.
My question is this: Would the use of deep learning algorithms offer any advantage in modeling such problems where iteration and self-consistency are involved? Are there any study/literature where researchers explored this avenue?