Determine whether the lowest-order MDPG scalar error is caused by locking

Determine whether the anomalously large scalar error observed for the lowest-order monotone discontinuous Petrov–Galerkin finite element approximation is caused by a locking phenomenon.

Background

The numerical experiments compare MDPG and monotone first-order system least-squares (MFOSLS) loss functions for neural approximations of nonlinear diffusion solutions. At polynomial degree zero, the MDPG approximation exhibits a substantially larger scalar error than the MFOSLS approximation, and the same discrepancy is already present in the corresponding finite element solution.

Because the scalar error persists in the finite element benchmark, the authors do not attribute it solely to neural-network training. They identify the possibility of locking but leave unresolved whether locking is the underlying mechanism.

References

These observations motivate using degree $k\ge 1$ for the present example and suggest that further studies are needed to examine whether the lowest-order case suffers from locking phenomena.

— Global error estimators for parametric monotone nonlinearities and neural approximations  (2610.05767 - Castillo et al., 5 Oct 2026) in Section 6.2, Section 7 (Conclusions)