  This study investigates the impact of Sobolev Training on operator learning
frameworks for improving model performance. Our research reveals that
integrating derivative information into the loss function enhances the training
process, and we propose a novel framework to approximate derivatives on
irregular meshes in operator learning. Our findings are supported by both
experimental evidence and theoretical analysis. This demonstrates the
effectiveness of Sobolev Training in approximating the solution operators
between infinite-dimensional spaces.
