machine learning and ODEs

Today Soledad Villar (JHU) and I discussed different ways to structure a machine-learning method for a cosmological problem: The idea is to use the machine-learning method to replace or emulate a cosmological simulation. This is just a toy problem; of course I'm interested in data analysis, not theory, in the long run. But we realized today that we have a huge number of choices about how to structure this. Since the underlying data come from an ordinary differential equation, we can structure our ML method like an ordinary differential equation, and see what it finds! Or we can give it less structure (and more freedom) and see if it does better or worse. That is, you can build a neural network that is, on the inside, a differential equation. That's crazy. Obvious in retrospect but I've never thought this way before.