  Although neural operator networks theoretically approximate any operator
mapping, the limited generalization capability prevents them from learning
correct physical dynamics when potential data biases exist, particularly in the
practical PDE solving scenario where the available data amount is restricted or
the resolution is extremely low. To address this issue, we propose and
formulate the Physical Trajectory Residual Learning (DeltaPhi), which learns to
predict the physical residuals between the pending solved trajectory and a
known similar auxiliary trajectory. First, we transform the direct operator
mapping between input-output function fields in original training data to
residual operator mapping between input function pairs and output function
residuals. Next, we learn the surrogate model for the residual operator mapping
based on existing neural operator networks. Additionally, we design helpful
customized auxiliary inputs for efficient optimization. Through extensive
experiments, we conclude that, compared to direct learning, physical residual
learning is preferred for PDE solving.
