Certified neural approximations for parabolic obstacle problems

Residual-trained neural surrogates carry no error guarantee, a limitation for free-boundary problems such as phase change and contact. We derive guaranteed error bounds for neural approximations of parabolic obstacle problems in first-order state–flux–multiplier form. The main result bounds the flux error of the network and the energy error of its feasible projection $\max \{v,g\}$ by the constitutive residual, the initial mismatch, the complementarity defect, and the PDE residual weighted by $c_F^2\mathrm{/}\alpha$, without the coercivity and norm-equivalence constants of the first-order formulation. It extends the Apushkinskaya–Repin majorant for the heat operator to general coefficients, free multipliers and a flux bound. A penetration-consistency result shows that neural triples with arbitrarily small bound exist. Outward-rounded cellwise Taylor-model enclosures turn the bound into a finite computation valid on any adaptive partition. The flux and gradient errors of the projected output are certified below $0.27$ and $0.33$ of the solution scale for a one-dimensional Stefan problem and, after adaptive refinement, below one half for all three trained networks on a two-dimensional expanding disk. On an expanding ellipse the gradient bound stays above one half; substitution diagnostics trace the floor of the trained networks to the projected state near the free boundary, and the remaining gap to enclosure overhead. Without a closed-form solution, a validated lower bound replaces the solution norm; there the enclosure is nearly exact, but the trained networks have errors above the solution scale and no useful accuracy is certified. The bounds control a state–flux norm, not the multiplier or the free-boundary location.