Statistical gains under misspecified physical constraints

Determine whether the statistical gains established for Physics Informed Kernel MethodS persist when the imposed physical constraints are misspecified, meaning that the differential observations or constraints do not exactly correspond to the differential structure of the target function.

Background

The paper assumes consistency of the differential observations: their conditional expectation equals the differential operator applied to the target. Under this well-specified setting, differential information can accelerate prediction and eventually attain the physical-oracle rate. The authors leave open whether such gains survive when the physical constraint is inaccurate or otherwise misspecified.