Certified Envelope Bounds and an Envelope-Informed Learning Surrogate for Road Network Resilience Under Stochastic Capacity Degradation

Road networks are routinely exposed to partial capacity losses caused by incidents, work zones, flooding and pavement failures, and planners need fast, trustworthy estimates of how such losses propagate through route choice. This paper develops a framework for user-equilibrium (UE) traffic networks with degradable link capacities that combines variational analysis with a learned surrogate. Writing the Beckmann potential in the coordinates ua=ca−p makes it affine in u, so standard results (the concavity of a pointwise minimum of affine functions and Danskin’s theorem) show that the optimal potential is concave, non-decreasing and differentiable, with a gradient available from one equilibrium solution. The resulting criticality index equals p/(p+1) times the congestion delay of each link; the theory adds to this classical indicator a derivative interpretation and certified bounds. We derive two-sided certificates on the potential change for any degradation, the identity TSTT=(p+1)V−pFFT, a solver-free certificate for the total system travel time (TSTT), and an additive certificate for restoring sets of links, whose per-link weight is the damaged-state delay of the link times a severity factor, and we use these quantities as features of an envelope-informed surrogate (EIGS), a gradient-boosted model on per-link envelope features diffused over the line graph. Experiments on the Sioux Falls and Anaheim benchmarks show where the results are useful and where they are not. The potential certificate is within a factor of 1.45 of the exact change at 25% capacity loss on Sioux Falls (1.03 on the 100 most critical Anaheim links) but loosens quickly with severity (factor 3.28 at 50% loss on Sioux Falls; 574 and 74 at 90% loss), and the solver-free TSTT bound exceeds the true change by a median factor of 3.8–12 in the multi-link Monte Carlo scenario sets. No single cheap indicator ranks link criticality best on both networks. On Sioux Falls, EIGS predicts the TSTT increase with R2=0.942 (0.913–0.960 over ten splits) against 0.919 for ridge regression and 0.924 for gradient boosting on the diffused severities, and with R2=0.909 against 0.862 on a spatially correlated test set generated after the model was fixed. Under shifts in demand (±15%) and nominal capacity, refreshing its envelope features from one equilibrium of the shifted network gives R2=0.86–0.91; using the same equilibrium to rescale the predictions of raw-input baselines gives them 0.84–0.89, so the gain from the per-link envelope features is modest. On Anaheim, whose losses are extremely heavy-tailed, EIGS ranks scenarios well (Spearman ρ=0.92) but does not predict their magnitude reliably, while ridge regression with a cross-validated penalty ranks them poorly (ρ=0.16). For restoring three of about fifteen damaged links, the certified envelope rule, which needs a single equilibrium, recovers on average 93.6% of the loss recovered by the exact optimum over all three-link sets.