We will get a slightly different trial solution, as the boundary conditions are different compared to the case for exponential decay.
A possible trial solution satisfying the condition \( g(0) = g_0 \) could be
$$ h_1(t) = g_0 + t \cdot N(t,P) $$
with \( N(t,P) \) being the output from the neural network with weights and biases for each layer collected in the set \( P \).
The analytical solution is
$$ g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)} $$