To begin with, a trial solution \( g_t(t) \) must be chosen. A general trial solution for ordinary differential equations could be
$$ g_t(x, P) = h_1(x) + h_2(x, N(x, P)) $$with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and \( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the neural network \( N(x,P) \) with \( P \) being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.