Extending the analysis to Gaussian training distributions

Extend the holomorphic function and operator-learning analysis from bounded uniform-coordinate training distributions to Gaussian measures, including Gaussian random fields with unbounded support.

Background

The analysis assumes independent uniform coordinates on [−1,1][-1,1], which permits the use of Legendre polynomial approximation and bounded-support arguments. Gaussian measures instead have unbounded support and produce standard-normal coordinates.

Gaussian and lognormal random fields are common in operator learning and are used in the numerical experiments, but the theoretical results do not cover them. Extending the analysis to this setting is explicitly left unresolved.

References

As we discuss below, extending our theory to deal with the latter -- and consequently Gaussian random fields -- is an open problem.

— Into the danger zone: stable extrapolation in high-dimensional function and operator learning  (2609.36709 - Adcock et al., 29 Sep 2026) in Remark following Assumption 2.1; Section 7, subsection “Gaussian distributions”