Can machine learning methods be somehow helpful in solving differential equations? I noted that regression task in machine learning is somehow related to solving differential equations approximately - both are trying to approximate unknown function.
Then, my question is: Can ML be somehow helpful in solving differential equations?
 A: Absolutely!
Here is information on the "shooting method". (link)
For much harder problems than the example given, the "root finding" takes more work.  It is useful to stick some machine learning on top of the output in order to determine which initial conditions are appropriate for the solution of interest.
EDIT:
Neural Networks (NN) are used to (profoundly) improve computation time for combustion.  The networks are trained on the thermo-chemical model and approximate the chemical reactions so that instead of solving (insane) complexity coupled fluid-dynamic and chemistry differential equations, the numeric solver has a reduced set of solves, and the NN with its very short run time, fills in the gaps "well enough".  Here is a link.  Here is another.
EDIT 2:
See physics-informed neural networks.
https://www.youtube.com/watch?v=hKHl68Fdpq4
A: I believe so, Archembau, Cornford, Opper, Shawe-Taylor, Girolami, Lawrence and Rattray are all excellent researchers in machine learning, so these would probably be good places to start.
