Learning Contact Maps: Universality and Inversion Through Folded Fronts

Symplectic map networks combine exact geometric structure with approximation guarantees. We develop the corresponding theory for contact maps, which also represent dissipative and action-dependent dynamics, and connect it to learned inversion. We propose two architectures: LegendreNet composes strict lifts of Henon-like symplectic maps, while ContactNet alternates these lifts with prolongations of invertible base maps. We prove that LegendreNet is dense in smooth strict contact maps on compact sets. For general contact Hamiltonian flow maps, we establish a modular approximation theorem: any strict family that is dense in $C^r$, combined with prolongations of any base-map family that is dense in $C^{r+1}$, gives $C^r$ approximation, with nonsingular factors on the relevant compact sets and convergence of local inverses. This turns contact splitting into a criterion for choosing architectural blocks. Smooth neural-flow bases satisfy the theorem's hypotheses, while triangular bases provide algebraic inverses for recovery. We benchmark the architectures against structure-preserving and generic baselines on damped Hamiltonian systems to quantify finite-capacity approximation. Finally, we use the contact structure of the generalized Hamilton-Jacobi equation to enable recovery of initial smooth profiles from terminal folded fronts.