I'm attempting to numerically approximate a simple ODE, I'm using it to describe the motion of a gradient descent search, but it could easily have physical interpretation. In particular, $$ x'(t) = \nabla f(x)x(t) $$ with IV $x(t_0) = x_0$, and where $f : \mathbb{R}^d \to \mathbb{R}$ is my objective function which may be evaluated at any point (for a price), we also may evaluate the gradient, $\nabla f(x)$, at any point, and also, and the point fo this question, the Hessian, $\nabla^2f(x)$. There are a number of different methods for solving this ODE, but none of them actually use the fact that the Hessian is available. Is it any use to have this?
One could, for example, essentially "fit" a quadratic to the initial point, $x_0$, with the second-order Taylor, solve the ODE for this approximate function, then propose a step forward. Or maybe I should be trying to approximate this ODE istead? $$ x'(t) = \nabla f(x(t)) \\ x''(t) = \nabla^2 f(x(t)) $$
