# gradient descent in space of functions

Differential equations of the form $$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$ can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, then the solution $\vec{x}(t)$ evolve towards it. These special points are called fixed points or stagnation points. If we start from them then the vector would not change in time. Here I want to ask about the similar problem regarding the space of functions, namely consider the equation: $$\frac{d}{dt}\psi(\vec{x},t) = -\frac{\delta E[\psi, \psi^*]}{\delta \psi^*}$$ or $$\frac{d}{dt}(\psi_1, \psi_2, \psi_3) = -\left(\frac{\delta E}{\delta \psi_1^*}, \frac{\delta E}{\delta \psi_2^*}, \frac{\delta E}{\delta \psi_3^*}\right)$$ where $\psi^*$ is the complex conjugate of $\psi$. Can we state that the function $\psi$ evolves towards the minimum of the energy $E[\psi,\psi^*]$? (These kind of equations arise in the study of ground state of the dilute Bose gases and come with a name: imaginary-time evolution).

• There is a huge literature on this, with many flavors. A classical problem where this idea is applied is the question of existence of closed geodesics. A keyword is calculus of variations. – Thomas Rot Dec 7 '15 at 22:16