It doesn’t really work to discretize this way in general, though it can sometimes in specific circumstances. The issue is that control problems have a very peculiar backwards/forwards structure, which leads to specialized discretization schemes where some parts of the equation are discreitized forwards and others backwards. Then things converge to the viscosity solution. Not to say that other approaches can’t work, but the convergence of these things is very complicated… You can’t just throw equations at an optimizer.
With spectral-style methods (or just using a neural network with a function) rather than finite difference methods you probably can get away just minimizing residuals with all of your equations with some neural network approximation, but I can say with experience that they are very finicky for HJBE problems with a control. Boundary conditions are tricky… If you want to do transition dynamics without a control choice, in which case it is kind of like a HJBE without much of an “H”, then you can use simpler discretizations.
For something along the lines of what you are describing I think it is a little premature in terms of the theory. For one thing, you are describing a finite time-horizon setup, which is of limited usefulness to most economists, but mostly it is that there is a vast literature on this due to the peculiarities of these equations. Also, I think it very unlikely that neural networks will beat the standard algorithms (e.g. codes - HACT - Benjamin Moll - University of Princeton ) for this dimensionality.