2026/07/24 by David Hämmerling, Lukas Pieronek, Andreas Rieder
paper · doi:10.1088/1361-6420/ae9004
Abstract Nonlinear inverse problems are typically solved by minimizing a data‑misfit functional, which is often non-convex that leads minimization algorithms to stagnate at a local minimum. A typical example is the cycle‑skipping phenomenon in full waveform inversion (FWI) of seismic reflection or transmission data. To overcome this difficulty, distance functionals are constructed by training neural networks to emulate a convex distance measure. Two construction strategies are discussed: (i) a data-converter network that simplifies the forward map so that a standard quadratic loss becomes convex, and (ii) a scalar-valued distance‑network based on the data residual. Training samples can be generated from measured data exploiting an approximate invariance of the forward operator. 
Under a set of structural assumptions, it is proven that applying the Landweber iteration to the learned functional is a well-defined regularization method that guarantees monotone error reduction and convergence to the exact solution as the noise level vanishes. Given certain restrictions on the trained network and the nonlinear forward operator, it is validated that these structural assumptions are satisfied by the learned distance.
This methodology is numerically validated on the Camembert benchmark model for FWI in the acoustic regime. Replacing the conventional L2-‑misfit with the learned convexified functional considerably mitigates cycle‑skipping and enlarges the domain of convergence for gradient‑based optimization. Numerical experiments show that the Landweber scheme with the learned misfit reaches a lower reconstruction error than the standard least-squares approach. For comparison, in some experiments convergence has been accelerated by using a limited‑memory BFGS optimizer leading to slightly larger reconstruction errors compared to the Landweber scheme.
Overall, the work provides a systematic way to embed learned, convex distance measures into inverse problem solvers, supplies a solid theoretical foundation for their use, and demonstrates practical gains in a seismic imaging benchmark model.