Thanks for your question, Jhtariq, but there is no such assumption. We are literally adding the base case into the loss. So, when t=0, g(t) =u0, respecting the initial condition.

When t!=0, the training will occur with a "shift" of u0. The NN(t) will adapt itself to this shift and, at the end, g(t) will approximate the real function (at least theoretically).

Please, let me know if you need more details.