Do numbers of this form have a name?
$$48625 = 4^{5} + 8^{2} + 6^{6} + 2^{8} + 5^{4}$$
Notice that the digits of the number are in digit order, while the exponents are the digits of the number in reverse order.
Do numbers of this form have a name?
They are "tabulated" at OEIS. I use the quotes because there are only two of them, the other one being $397612$ (I don't count $1$). No name is given at that site, but there are some links there that you might follow up. If you find anything, be sure to let us know.
Inverted Munchausen numbers in base 10.