Before revision:  \mu \in P \} associated with the parameter domain P \subset R^P and the parametric field \mu \mapsto u_{\mu} \in L^2(\Omega), our approach takes as input a set of snapshots \{ u^k \}_{k=1}^{n_{\rm train}} \subset M and returns a parameter-dependent bijective mapping \Phi : \Omega \times P \to R^2: the mapping is designed to make the mapped manifold \{ u_{\mu} \circ \Phi _{\mu}: \, \mu \in P \} more amenable for linear compression methods.
After revision:  \mu \in P \} associated with the parameter domain P \subset R^P and the parametric field \mu \mapsto u_{\mu} \in L^2(\Omega), our approach takes as input a set of snapshots \{ u^k \}_{k=1}^{n_{\rm train}} \subset M and returns a parameter-dependent bijective mapping \Phi : \Omega \times P \to R^2: the mapping is designed to make the mapped manifold \{ u_{\mu} \circ \Phi _{\mu}: \, \mu \in P \} more amenable for linear compression methods.