From Fast Surrogates to Credible Solvers: Verification, Validation, and Reliability of Neural Operators in Computational Mechanics

Neural operators, including Fourier neural operators, deep operator networks, graph- and geometry-based operators, wavelet operators, and physics-informed variants, are increasingly promoted as fast alternatives or complements to classical solvers for the Partial Differential Equations (PDEs) of computational mechanics. The supporting evidence, however, is still dominated by low prediction error, inference-time speed-up, in-distribution interpolation, and visually convincing field reconstructions. These criteria are necessary but not sufficient to establish numerical credibility: a model used as an engineering solver must additionally be examined through consistency, stability, convergence, conservation, boundary condition fidelity, validation against reference solutions, calibrated uncertainty, robustness under physically meaningful distribution shift, honest accuracy-matched cost, and reproducibility. This review reorganizes the neural operator literature around credibility rather than architecture or application. We introduce NO-CRED, a structured evidence framework with seven credibility dimensions and a five-level evidence ladder that assigns each study a transparent credibility profile rather than a single conflated score. NO-CRED translates established verification-and-validation practice to operator learning. Its specific additions are an operator-specific evidence ladder, a distribution shift dimension in which the applicability domain is defined by a training distribution rather than a solver’s fixed regime of validity, profile-based reporting justified by the weak coupling we measure between dimensions, and an executed auditable coding of the literature. We test the rubric used by the latter for stability; an intra-coder test–retest on a seed-selected subset of 14 studies gives per-dimension quadratic-weighted κ from 0.63 to 1.00, median 0.91, with all disagreements at adjacent levels, though no external double coding has been performed yet. Application to a curated released sample of 32 primary studies shows that within these 32 studies, the evidence is concentrated on accuracy. Uncertainty is absent in most (25/32), while genuine extrapolation (2/32) and accuracy-matched cost (3/32) are rare. Code is widely released (30/32), but full reproducibility is not (1/32). Strength on one dimension barely predicts strength on another. The sample supports a hypothesis that credibility appears to depend more on evaluation practice than on architecture, but more is needed before this can be proven. We assemble a failure mode atlas and an engineering-relevant taxonomy of distribution shift. In addition, we analyze when reported speed increases survive a fair accuracy-matched comparison with optimized classical and reduced-order methods. Finally, the paper closes with a credibility reporting checklist and a benchmark protocol.