Minimax optimality of PIKS learning rates

Determine whether the finite-sample learning rates established for Physics Informed Kernel MethodS (PIKS), combining noisy function-value observations with noisy differential observations under the value-derivative capacity decomposition, are minimax optimal.

Background

The paper derives two-regime high-probability prediction rates for PIKS. When the number of differential observations is below a critical threshold, the rate depends jointly on the numbers of value and differential samples; beyond that threshold, the rate saturates at the physical-oracle rate associated with exact differential information. The authors do not establish whether these rates are optimal over corresponding statistical function classes, leaving their minimax status unresolved.