Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation

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 $c_F^2/α$, 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.

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do not provide enough information to derive tight deterministic and guaranteed bounds on function space norms. In this work, we move beyond a purely black-box setting and exploit the neural network structure directly. We present a framework for the certified and accurate computation of integral quantities of neural networks, including Lebesgue and Sobolev norms, by combining interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement and quadrature-based aggregation. On each box, we compute guaranteed lower and upper bounds for function values and derivatives, and propagate these local certificates to global lower and upper bounds for the target integrals. Our analysis provides a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, yielding certified computation of $L^p$, $W^{1,p}$, and $W^{2,p}$ norms. We further show how these ingredients lead to practical certified bounds for PINN interior residuals. Numerical experiments illustrate the accuracy and practical behavior of the proposed methods.