Full waveform inversion benchmark

Given: A positive velocity map, the acoustic pressure-wave equation and source signal with zero initial pressure and velocity, the receiver sampling operator, a free-surface top and absorbing other boundaries of width $80$ grid points, $16$ sources at depth $1270\mathrm{m}$, $129$ receivers at depth $10\mathrm{m}$, the stated discrete solver and mesh, and the finite CurveFaultB split.

Benchmark risk, compute budgets, and uncertainty records: a benchmark instance includes its forward solver, admissible inputs, noise, compute protocol, and test population.

Scope of InverseBench findings: the reported FWI outcomes concern this selected solver, tuning procedure, and test set only.

For an explicit conditional finite-data benchmark, supply the actual nonempty training, validation, and test lists of the stated sizes, including realized observations, target images, operator metadata, and valid discrete forward values. Let $X=U=C$ be the finite set of labelled cases, with its full power-set sigma-algebra; labels distinguish repeated images. Let $Q={\mathbb{R}}^{{128}^2}$ with Euclidean distance and Borel sets, and let $Y$ be the finite-dimensional observation-and-operator-metadata space with Borel sets. For case $i$, set $q(i)=a_i$ (its target image), $F(i)=f_i$ (its discrete noiseless forward data), and $K(\cdot \mid i)={\delta }_{y_i}$ (its recorded data, with the same metadata). Thus $K$ describes replay conditional on these realized lists, not a population of newly sampled velocity fields. Require $y_i=f_i$ because this case is noise-free; the forward values are outputs of the specified discrete solver on admissible inputs, with its actual settings recorded. Set $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}=N^{-1}{\sum }_{i\in T}{\delta }_{(y_i,a_i)}$, where $T$ is the test index set and $N=10$; use the analogous uniform empirical validation law. Take $P_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}$ to be the point mass at the entire supplied fitting record, including training/tuning lists and seeds. This conditional specialization supplies laws, not just sample counts, and is not a reconstruction of the paper's full evaluation protocol.

Supply each candidate's complete fitting/tuning procedure and deployed measurable rule $g:Y\to Q$, with all seeds fixed. Fix the loss $\mathrm{\ell }(a,b)=\mathrm{\parallel }a-b{\mathrm{\parallel }}_2^2$ and a radius $r>0$ before testing. The uncertainty diagnostic is empirical coverage $N^{-1}{\sum }_{i\in T}\mathbf{1}\{\mathrm{\parallel }g(y_i)-a_i{\mathrm{\parallel }}_2\le r\}$, targeting coverage of the balls of radius $r$ centered at $g(y)$ under $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$. Evaluate every test case once and average its loss. Before fitting or evaluation, supply finite upper bounds for every applicable forward, adjoint, gradient, ODE-evaluation, iteration/sampling, and runtime count, marking inapplicable entries as not applicable, together with hardware, precision, software/model versions, seeds, tuning data, and stopping tolerances. A procedure must return within these bounds or record failure; its deployed rule returns the zero image on failure. Retain that case in both aggregates and report every status and realized count. These supplied procedures and budget records are parameters of this mathematical specialization; the coverage statistic is not a reported InverseBench calibration result.