Statistical gains for nonlinear differential operators

Establish whether learning with nonlinear differential operators can provide statistical gains analogous to those proved for linear differential operators in Physics Informed Kernel MethodS, including improved prediction rates and a saturation regime relative to value-only kernel regression.

Background

The analysis in the paper assumes that the physical information is generated by a linear differential operator and formalizes its effect through covariance operators associated with value and differential observations. The authors explicitly leave unresolved whether comparable improvements in statistical learning rates can be obtained when the differential operator is nonlinear.