Generate examples with high targets

Suppose you have trained a Feedforward Neural Network with labeled examples ($x$,$y$) in a regression task to learn a function $\hat{f}$ to predict $y$ from $x$ with some error $\epsilon$ as $\hat{f}(x)=y+\epsilon$. Assuming the performance of the model is adequate, is it possible to generate synthetic $x$ examples that have high values for $\hat{f}(x)$? Has this application of neural networks been researched in the literature?

For instance, each example $x$ is vector of product features and the target $y$ is the popularity score of the product. After the model has been trained on these data, $\hat{f}(x)$ would be typically used to predict the popularity of existing products.

However, another idea is to look for synthetic $x$ that have high predicted popularity scores e.g. a global maximum $arg_{x}max(\hat{f}({x}))$ or other local maxima of $\hat{f}$, as these would correspond to the discovery of popular products (as long as we trust the model).