Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a real analytic function with $\inf_{x\in\mathbb{R}^n} f(x) > -\infty$. If we know the solution $x:[0,\infty)\rightarrow\mathbb{R}^n$ to the vanilla gradient flow $$\dot{x}=-\nabla f(x), \quad x(0)=x_0$$ is bounded for every $x_0$. Can we say the solution $y:[0,\infty)\rightarrow\mathbb{R}^n$ to the preconditioned gradient flow $$\dot{y}=-A^{-1}\nabla f(y), \quad y(0)=y_0, $$ where $A\in\mathcal{S}_{++}^n$ is a real symmetric positive definite matrix, is also bounded? Is there a simple relation between the solution of these two systems?
My research on this problem:
I found in a paper (p.2, after equation (4)), the authors said that "which can also be thought of as the gradient flow dynamics on a reparametrization...". However, I don't know how the reparametrization leads a solution of the vanilla gradient flow to a preconditioned one.