Generalization of learned reconstruction (deconvolution) operators across regimes and discretizations

Determine how neural network-based reconstruction (deconvolution) operators R_θ that map a reduced field q̄ to a reconstructed full field u generalize across different physical regimes, numerical schemes, and discretizations when used for closure modeling by applying the original high-fidelity model F to R_θ(q̄).

Background

The paper discusses a reconstruction-based approach to closure in which a learned operator R_θ reconstructs the full field u from the reduced field q̄, enabling application of the original governing model F to the reconstructed state. Such reconstruction (also termed deconvolution) is an ill-posed inverse problem, for which recent machine learning methods have shown promise over classical approximate deconvolution.

However, the authors explicitly note that a key unresolved issue is whether these data-driven reconstruction operators can robustly generalize when the physical regime, numerical scheme, or discretization changes. This limits the portability and reliability of learned reconstructions for practical multiscale simulations.

References

These methods can be considered an improvement over approximate deconvolution techniques because they do not need the assumption of an invertible filter, but open challenges include how such data-driven methods can generalize across physical regimes, numerical schemes, and discretizations.

— Scientific machine learning for closure models in multiscale problems: a review  (2403.02913 - Sanderse et al., 2024) in Section 3.3 (Reconstruction)

This is an architectural transfer property, not a claim of exact equality across resolutions or arbitrary array geometries. Approximation quality under a changed discretization remains an empirical question.

— GCNO: Gramian Chebyshev Neural Operator for Physics-Based Compression of Wireless Channels  (2608.18522 - Murshed et al., 19 Aug 2026) in Appendix B, Section "Neural-Operator Characterization"

Several directions remain open for future work. The reconstruction of the smallest scales, particularly in unseen geometries, may be improved by training on a broader range of geometries, mesh topologies, and Reynolds numbers. To improve the capabilities of the model on strongly non-orthogonal grids, one option is to incorporate non-orthogonal corrections into the finite-volume basis operators, following standard practice in finite-volume CFD methods. Alternatively, training the network on meshes with strong non-orthogonality may enable the model to learn the necessary corrections implicitly, without modifying the basis operators. Since the FVCU is formulated for arbitrary polyhedral cells, its assessment on more complex geometries and on meshes with mixed cell types, such as tetrahedral, prismatic, and polyhedral cells, is a natural next step. Moreover, because the FVCU is based on standard finite-volume discretizations that are already implemented in established CFD codes, the integration of the method into existing solvers is facilitated. More broadly, although the present work focused on super-resolution reconstruction, the FVCU provides a general framework for applying deep learning on finite-volume meshes. Its application to other tasks, such as turbulence modelling, flow prediction, or data assimilation on unstructured grids, is a promising direction for future research.

— Extending convolutional neural networks to finite-volume meshes for fine-scale reconstruction of wall-bounded turbulence  (2610.12141 - Tofighian et al., 8 Oct 2026) in Section Conclusion, final paragraph