Validity of the physics mapping beyond quadratic order

Establish whether the mapping between neural learning dynamics and the associated physical or ecological models survives beyond the quadratic neural quadratic form approximation.

Background

The paper interprets neural quadratic forms through analogies with Landau theory, interacting populations, hydrodynamic instabilities, and polymer flow. These analogies are developed for the leading quadratic approximation around small initialization.

Higher-order terms are acknowledged as potentially responsible for plateaus not captured by the neural quadratic form. The authors therefore leave unresolved whether the correspondence with the proposed physical descriptions persists when cubic, quartic, or higher-order terms are included.

References

Whether the mapping survives beyond quadratic order, and whether the ecological picture of competition between features predicts anything about generalization rather than optimization, are the questions we regard as most worth pursuing.

— Neural Quadratic Forms: A Unified Minimal Model for Sudden Learning and Scaling Laws  (2608.13335 - Ziyin et al., 13 Aug 2026) in Section 6, Discussion, paragraph “Outlook”