This sounds pretty close to what people doing reduced order modeling for nonlinear differential equations are working on. They have a reduced basis for expressing (and computing) an approximation $\tilde y$ of the solution $y$, but evaluation of the nonlinearities $f(\tilde y)$ in the equation still have to be performed in the full space. To circumvent this, they construct a reduced model $\widetilde{f(\tilde y)}$ from a basis $\{f(\tilde y_1),f(\tilde y_2),\dots\}$.

The keywords here are "nonlinear proper orthogonal decomposition (POD)" and "(discrete) empirical interpolation methods ((D)EIM)". The latter is the topic of Saifon Chaturantabut's thesis.

This sounds pretty close to what people doing reduced order modeling for optimization of nonlinear differential equations are working on. They have a reduced basis for expressing (and computing) an approximation $\tilde y$ of the solution $y$, but evaluation of the nonlinearities $f(\tilde y)$ in the equation still have to be performed in the full space. To circumvent this, they construct a reduced model $\widetilde{f(\tilde y)}$ from a basis $\{f(\tilde y_1),f(\tilde y_2),\dots\}$.

In this context, some possible keywords are "nonlinear proper orthogonal decomposition (POD)" and "(discrete) empirical interpolation methods ((D)EIM)" (the latter is the topic of Saifon Chaturantabut's thesis).