Bilevel optimization for learning the activation schedule

Develop a bilevel objective that evaluates the entire optimization trajectory, rather than only the current data residual, to learn the coarse-to-fine hash-level activation schedule for C2F-IFWI without prematurely opening fine-resolution levels.

Background

C2F-IFWI uses a fixed activation schedule to progressively introduce multi-resolution hash levels from coarse to fine. An ablation experiment shows that directly learning the level-opening times from the instantaneous data misfit causes the schedule to open faster and produces a higher final model error than the fixed schedule.

The authors explain that the benefit of delaying fine levels is a property of the full training dynamics—specifically, avoiding poorly conditioned early updates—and is not captured by the instantaneous residual. Consequently, learning the schedule requires an objective that assesses the complete optimization trajectory rather than the current misfit alone; this remains identified for future work.

References

It is not reflected in the instantaneous data misfit, so a gradient on that misfit always favors opening the levels sooner. Learning the schedule would therefore require a bilevel objective that scores the whole optimization trajectory rather than the current residual, which we leave to future work.

— Coarse-to-fine multi-resolution hash encoding for implicit full waveform inversion  (2610.01081 - Wang et al., 1 Oct 2026) in Section 5.4, “Fixed versus learned schedule” (Discussion)