Lax–Oleinik Aubry contact probe

Attach this module between neural PDE training iterations, after the network has produced a preliminary value function. Evaluate u_theta on a dense periodic candidate grid and approximate T_t^-u_theta at each endpoint. For a first MVP, discretize time into M steps of size Delta t and restrict velocities to a finite stencil V. Initialize A_0(y)=u_theta(y), then apply the dynamic-programming update A_{r+1}(x)=min_{v in V}[A_r(x-Delta t v)+Delta t L(v,x-Delta t v)], using periodic indexing. Set A_M(x) as the approximation to T_t^-u_theta(x), compute g_t(x)=u_theta(x)-A_M(x)+ct, and subtract its median to remove numerical additive drift. Define contact weights q_i=exp(-|g_t(x_i)|/tau). Use a 50/50 mixture of uniform collocation points and points sampled proportional to q_i; alternatively, add an auxiliary penalty mean_i min(|g_t(x_i)|,epsilon)^2 only after the base Hamiltonian loss is stable. The paper supplies the contact-set characterization, while the velocity stencil, finite horizon, grid interpolation, and tolerance are empirical approximations. Test first on a 1D or 2D periodic mechanical Hamiltonian with a reference Aubry set from a high-resolution solver. Compare uniform sampling with contact-guided sampling at equal H evaluations. Measure Aubry-set Hausdorff distance, dense-grid Hamiltonian violation, and value error. Success is faster error reduction near separatrices and improved contact-set localization at equal sample count.