Tail observability and fourth-order closure recovery in physics-informed neural networks for Bhatnagar-Gross-Krook normal shocks

Closure-level accuracy in neural kinetic shock solvers is not guaranteed by accurate density, velocity and temperature profiles, because the relevant observables are velocity-weighted projections of the nonequilibrium distribution. We study this observability problem for one-dimensional Bhatnagar--Gross--Krook (BGK) shock waves using a positive macro--micro physics-informed neural network (PINN) in which the distribution is represented as a local Maxwellian multiplied by a bounded exponential correction. Independent discrete-velocity method (DVM) references are used for validation. Shock-tube tests show that sparse joint anchoring of heat flux and normal stress stabilises the primary nonequilibrium layer, whereas residual-only, macro-only and single-moment variants fail in distinct ways. In a stationary Mach-2 normal shock, a flux-locked compact model recovers $\rho$, $u_x$, $T$, $q_x$, $\sigma_{xx}$ and $m_{xxx}^{cl}$, but leaves $R_{xx}^{cl}$ with order-unity error. DVM diagnostics show that $R_{xx}^{cl}$ is controlled by a sign-changing, tail-weighted cancellation weakly observed by lower moments. A shock-local closure correction aligned with this missing projection reduces the relative $R_{xx}^{cl}$ error to $1.12\times10^{-1}$ while preserving the lower moments. A common-initialisation ablation shows that optional distribution-function probe losses are diagnostic rather than constitutive. A supplementary DVM--PINN comparison for the scalar fourth-order excess $\Delta$ shows that the obstruction is anisotropic, sign-changing tail weighting rather than fourth-order polynomial degree alone.