Reliable adaptive constitutive-degree selection

Determine a reliable observation-based procedure for adaptively selecting the constitutive degree and associated weak-test supports in Allen–Cahn potential identification without allowing noise-sensitive coefficients to degrade recovery accuracy.

Background

The method treats the Bernstein constitutive degree as a prescribed modeling choice because the tested validation-based selection rules do not consistently identify the most accurate representation. A richer family can reduce approximation error for some constitutive laws, but it can also amplify sensitivity to poorly determined coefficients and observation noise.

The paper specifically reports that validation-based selection improves aggregate error under one noise level but worsens it under another, and can select an excessively low degree for a law that benefits from additional coefficients. Thus, a dependable adaptive complexity-selection criterion remains unresolved.

References

The present implementation therefore prescribes the degree. This is a practical conclusion about the tested criteria and observation regime, not a proof that constitutive complexity cannot be inferred from data. Numerical rank, support sensitivity, and phase coverage remain useful diagnostics, but reliable adaptive degree selection requires further study.

— Constrained weak identification of Allen--Cahn free energies with surface-tension calibration  (2609.18403 - Jung et al., 16 Sep 2026) in Section 4.4, Section “Limits of observation-based degree selection”