What is Pooled panel model, permutation-invariant network?

A pooled panel model fits one function $f_\theta(x_{it})$ to every asset $i$ and date $t$ of a panel, so that assets share parameters through their characteristics rather than each having its own. A permutation-invariant network maps a set of vectors to an output that does not depend on their order, typically $\rho(\sum_i\phi(x_i))$ for networks $\phi$ and $\rho$ (Zaheer et al., 2017).