Determine non-inferiority of value-only metric recovery relative to endpoint-eikonal supervision

Determine whether scalar-value-only training is non-inferior to endpoint-eikonal supervision for recovering P-wave and S-wave velocity structure from learned travel-time fields across the tested velocity realizations and optimization seeds.

Background

The paper compares travel-time fields trained solely on scalar arrival values with fields additionally trained using endpoint eikonal residuals and known endpoint velocities. In the prospectively locked second realization, values alone pass the predefined metric-emergence gates, while the differences between value-only and eikonal-supervised velocity errors have bootstrap intervals that include zero.

The unresolved issue is whether the apparent value-only performance can formally meet the prespecified non-inferiority margins. The reported experiment does not establish that conclusion because the confidence intervals are compatible with both non-inferiority and a practically meaningful disadvantage for value-only training.

References

Non-inferiority remains unresolved across three seeds, while scalar secants provide resolved refinements (Supplementary Information).

— Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations  (2609.31238 - Yu, 25 Sep 2026) in Results, subsection “Direction and metric emerge without equation supervision”; Supplementary Information, Section “Independent Same-Data Confirmation of Metric Recovery”

Its observational benefit remains unresolved, and restricting epicentral distance to 10--50~km does not itself improve recovery.

— Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations  (2609.31238 - Yu, 25 Sep 2026) in Discussion; Supplementary Information, subsection “Fixed-receiver averaging and training-time norm regularization”