This preprint studies whether a theorem-guided constructive forward operator — the finite-K Chernoff–Remizov trajectory — can serve as a differentiable engine inside a gradient-based inverse loop, without building a discretization matrix and without hand-deriving an adjoint equation. Using a 1D Darcy coefficient-recovery problem as a controlled validation setting, we treat the K-step constructive trajectory as an approximate differentiable forward model and optimize the coefficient field directly through torch. autograd. On top of this, we run joint forward/inverse K-scans on four representative coefficient families (fourierₛmooth, fourierₛharp, gaussianbump, tanhₛtep) and also study a learned residual corrector trained with supervised L2 regression. Main findings: • Joint forward/inverse K-scans reveal three distinct mechanistic regimes of the inverse pipeline: (I) forward-limited — inverse error tracks forward error at near O (1/K), (II) approximation-limited — same coupling but slower, extra constant driven by how well the constructive trajectory resolves the true coefficient, (III) parameterization-limited — inverse error saturates while forward error keeps decreasing, identifying the coefficient ansatz (not the solver) as the dominant bottleneck. • On smooth Fourier fields the Chernoff–Remizov inverse solver recovers the coefficient reliably and tracks the finite-difference adjoint baseline within a fraction of a percentage point. • A derivative-level boundary for learned correction: a residual neural corrector improves forward-state accuracy but *degrades* inverse recovery by disturbing the Jacobian ∂ucorr/∂a that the optimizer actually uses. Standard supervised L2 regression on state error is not a sufficient training target when the trained surrogate is plugged into a gradient-based inverse loop. • Robust to 1% observational noise; at 5% it degrades in a comparable way to the FD baseline. • In this 1D setting the method is not yet runtime-competitive with a classical FD solver — the contribution is mechanistic clarity (when it works, why it slows down, how its failures should be interpreted), not raw 1D benchmark leadership. Contents: • main. pdf — English preprint (26 pp. ) • mainᵣu. pdf — Russian translation (29 pp. )
Sergey Shpital (Sat,) studied this question.