Inverse problems stop learning the inverse map: a differentiable neural-operator surrogate drives Landweber iteration, yet three of four surrogates still semiconverge on exact data without preconditioning
Because an inverse map cannot be uniformly approximated by a continuous network near a discontinuity, the work learns a differentiable forward operator and its derivative and uses them as the surrogate inside Landweber iteration. On reaction-coefficient identification in an elliptic equation, four surrogates reconstruct both an in-distribution truth and one with sharp inclusions, but without preconditioning three of them semiconverge early and develop high-frequency spurious structure even on exact data; derivative supervision along fixed directions barely changes the best reconstruction.