After a talk by Matias Zaldarriaga (IAS) about making simulations faster, I had the following possibly stupid idea: It is possible to speed up simulations of cosmological structure formation by simulating not the full growth of structure, but just the departures away from a linear or quadratic approximation to that growth. As structure grows, smooth initial conditions condense into very high-resolution and informative structure. First observation: That growth looks like some kind of deconvolution. Second: The better you can approximate it with fast tools, the faster you can simulate (in principle) the departures or errors in the approximation. So let's fire up some machine learning!
The idea is to take the initial conditions, the result of linear perturbation theory, the result of second-order perturbation theory, and a full-up simulation, and try to infer each thing from the other (with some flexible model, like a huge, sparse linear model, or some mixture of linear models or somesuch). Train up and see if we can beat other kinds of approximations in speed or accuracy. Then see if we can use it as a basis for speeding full-precision simulations. Warning: If you don't do this carefully, you might end up learning something about gravitational collapse in the Universe!. My advice, if you want to get started, is to ask Zaldarriaga for the inputs and outputs he used, because he is sitting on the ideal training sets for this, and may be willing to share.