Surrogate-accelerated Landweber iteration for stiffness identification

Abstract

Identifying stiffness degradation from sparse sensor data is a central inverse problem in structural health monitoring, and neural surrogates are increasingly used to replace the finite element solves it requires. Such replacements are rarely accompanied by guarantees that the resulting identification procedure converges or that its error is controlled. We study a Landweber iteration in which both the parameter-to-observation map and its derivative are replaced by a neural surrogate trained on finite element data. For an elliptic structural model with a finite-dimensional damage parameterization, we establish explicit Lipschitz bounds on the forward map and its derivative in terms of material bounds, load and sensor functionals. We then show that a surrogate whose empirical value and derivative errors are small on a random design attains uniform accuracy on the whole admissible set, with an explicit dependence on the fill distance of the design and on the mesh size. Under a tangential cone condition, which we prove holds locally whenever the sensor layout renders the linearized problem identifiable, the surrogate-based iteration is monotone, terminates after finitely many steps under a discrepancy principle that accounts for the surrogate error, and returns an estimate whose error is bounded by a constant multiple of the measurement noise, the empirical training error, the fill distance and the squared mesh size, divided by the smallest singular value of the linearized sensor map. The analysis yields quantitative criteria for sensor placement and for derivative-informed training. A numerical study on a bridge-deck model problem confirms the predicted trends and quantifies the conservatism of the certificate.