Spectral Evaluation of 2D Chernoff–Remizov Trajectories: Resolving Interpolation Diffusion in 2D Darcy Inverse Problems

This is the 2D follow-up to a companion preprint on differentiable Chernoff–Remizov trajectories for inverse problems (10.5281/zenodo.19637120). The companion preprint established that the finite-K Chernoff–Remizov trajectory can serve as a differentiable forward model inside a gradient-based inverse loop on a 1D Darcy coefficient-recovery testbed, and exposed three mechanistic regimes (forward-limited, approximation-limited, parameterization-limited) that govern when such recovery succeeds, slows down, or saturates. The present work carries that program into 2D. Main findings. (1) A direct, grid-based 2D extension is forward-bias-limited on coarse grids. A Lie–Trotter composition of the 1D Chernoff–Remizov step is differentiable end-to-end but yields 10–26% inverse error across four coefficient families on 32x32 grids, with a non-monotonic K-dependence and collapse at high K. (2) The bottleneck is interpolation diffusion, not the Chernoff approximation. A per-mode sinc-attenuation analysis, supported by a resolution-disambiguation study, identifies accumulated bilinear resampling at coefficient-dependent shift locations as the dominant source of forward-model bias on coarse grids; the same analysis explains both the observed K-sweet-spot and the catastrophic high-K degradation. (3) A spectral variant of the Chernoff–Remizov step removes the bottleneck. Representing the state in a truncated DST-I sine basis and evaluating the variable shift analytically on that basis (instead of through grid interpolation) lowers inverse error to 1–13% across the four families (a 2.0x–9.2x improvement), restores strictly monotonic K-convergence on broadband features, and is 7–18% faster than the grid variant on all four families on our hardware. (4) The 1D decomposition framework of the companion preprint transports to 2D in a refined form. Two of its three regimes reappear cleanly on this testbed: a forward-limited regime visible as a sub-1% residual on smooth families (traced to a tau-dependent pseudo-time fixed-point bias via a Strang-vs-Lie–Trotter diagnostic), and a parameterization-limited regime refined into an identifiability limit on localized features (confirmed by an M-ablation and a closed-form softplus–L2 basis-capacity baseline: the retained Fourier basis can represent the true coefficient to within 0.12%, so the 10.98% inverse-error floor at K=1024 is a 90x gap from basis capacity). The third 1D regime (approximation-limited) does not separate into a distinct empirical signature on the four families studied here. What transports cleanly is the decomposition itself: failure modes separate into solver bias, basis capacity, and information-theoretic limits, and each component is directly measurable. Scope. 2D Darcy with smooth positive coefficients, a single forcing and a single noisy observation, on a 32x32 grid. The FD solver is used only to generate synthetic observations; we do not run a paired FD-adjoint inverse benchmark in this manuscript. The contribution is accuracy and diagnostics, not wall-clock leadership against tuned classical solvers. Companion preprint. Differentiable Chernoff–Remizov Trajectories for Inverse Problems, 10.5281/zenodo.19637120. Code. An accompanying code release is in preparation; the link will be added to this record once the repository is public.