GAIN-Cert: Gap-Aware Certification of Disturbance-to-State Gain Curves

Formal stability at one disturbance budget gives limited guidance for a learned system exposed to disturbances of varying magnitude. In inverter-rich power systems and other cyber-physical settings, the relevant question is which disturbance magnitudes keep the eventual state deviation within a prescribed operating limit. Repeating Lyapunov verification at selected budgets yields isolated certificates: interpolating them lacks a formal guarantee, whereas endpoint constructions can be needlessly conservative. We study certified disturbance-to-state gain-curve synthesis for nonlinear and learned ordinary differential equations. At each disturbance magnitude in a prescribed range, the curve bounds the ultimate state deviation for every admissible disturbance signal of up to that magnitude. For a fixed Lyapunov certificate and regional verifier, we define a disturbance-dependent threshold profile and separate conservatism caused by the certificate, state-norm comparison, and verifier from that introduced by representing and verifying a curve. Targeting the latter, we train a monotone integral network against a pointwise branch-and-bound reference and allocate knots adaptively where the bound changes rapidly. A piecewise envelope is checked jointly over state and disturbance regions, repaired where verification fails, and tightened after certification. We prove soundness of the resulting curve under explicit operating-region assumptions and measure the contributions of neural approximation, verification repair, and tightening to its gap. We evaluate on six classical ODEs and two learning-enabled systems against pointwise branch-and-bound and staircase, affine-level, and neural curve baselines. The comparisons show smaller normalized bound area at competitive end-to-end cost.