Establish stability guarantees for learned global MsHDG systems

Establish operator-norm or solution-relevant trace-direction conditions on learned local Dirichlet-to-Neumann operators that guarantee stability of the assembled global NN-MsHDG skeleton system and control the resulting global solution error in the high-contrast regime.

Background

The conclusion states that a rigorous treatment of the high-contrast regime is still required. In particular, the authors seek a relationship between local DtN approximation errors, conditioning of the assembled skeleton operator, and global solution error.

The proposed problem is more specific than merely improving numerical accuracy: it asks for mathematically verifiable operator norms or trace directions whose control would ensure stability of the learned global system. Such conditions would provide guarantees for replacing exact local HDG solves with neural-network surrogates.

References

A rigorous study of the high-contrast regime remains necessary. In particular, future analysis should clarify the relationship between local DtN approximation errors, conditioning of the assembled skeleton operator, and the resulting global solution error. It would also be useful to identify operator norms and solution-relevant trace directions whose control guarantees stability of the learned global system.

— A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media  (2608.25850 - Haines et al., 26 Aug 2026) in Section 7, “Concluding Remarks”