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
Synopsis
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.
Interpretation
Instead of training a network to map observations directly to parameters, a differentiable neural-operator surrogate takes over the forward operator and its derivative that Landweber iteration evaluates at every step. Learning the inverse map directly fails near discontinuities: no continuous network approximates a discontinuous map uniformly across a jump, and a fixed-depth FNO must then have diverging weights. On reaction-coefficient identification in an elliptic equation, the four surrogates reach oracle coefficient errors of 5.825% to 7.651% on the in-distribution truth T1 and 8.574% to 9.703% on T2 with sharp inclusions, against 4.118% and 6.936% for the exact finite-element iteration.
The surrogate is a linear encoder, a finite-dimensional network and a linear decoder, and forward values and derivatives come from automatic differentiation of the same network rather than a separately trained derivative model. A generalised DeepONet uses PCA input coordinates and POD output modes, and a generalised Fourier operator uses truncated Fourier coefficients; both realise this encoder–network–decoder class exactly. Training uses a heterogeneous Matérn random-field measure with derivative supervision along empirical KL directions; on independent test fields all four surrogates have forward errors below 0.5% and derivative errors below 0.8%.
Iteration stability depends on the preconditioning geometry, not only on derivative accuracy along the training directions. Removing the Matérn preconditioner makes DeepONet, GenFNO and GenDeepONet semiconverge early and develop high-frequency spurious structure even with exact data, while the exact iteration and the FNO do not. On T2 the DeepONet oracle error of 9.703% occurs at step 10000 and the GenFNO minimum of 8.574% at step 8859; with preconditioning GenFNO and GenDeepONet stay within 0.5% of their respective minima over the budget, at the cost of slower recovery of sharp features.
Perspective
The demonstrated scope is nonlinear Landweber iteration in Hilbert spaces, with numerical validation limited to reaction-coefficient identification in an elliptic equation and transient Darcy flow; the convergence result requires a compact preconditioner and a regularity condition, which neither truth satisfies under the baseline Matérn geometry, so the authors repeat the noiseless experiments with geometries that do satisfy it.
The theoretical error level relies on uniform error bounds that cannot be computed, and thresholds built from held-out test errors failed to identify when to stop in the noiseless, surrogate-limited regime; the small-model and reduced-data study lowers forward and derivative accuracy together, so it cannot separate their contributions, and each model is trained only once.