  Neural operators have been explored as surrogate models for simulating
physical systems to overcome the limitations of traditional partial
differential equation (PDE) solvers. However, most existing operator learning
methods assume that the data originate from a single physical mechanism,
limiting their applicability and performance in more realistic scenarios. To
this end, we propose Physical Invariant Attention Neural Operator (PIANO) to
decipher and integrate the physical invariants (PI) for operator learning from
the PDE series with various physical mechanisms. PIANO employs self-supervised
learning to extract physical knowledge and attention mechanisms to integrate
them into dynamic convolutional layers. Compared to existing techniques, PIANO
can reduce the relative error by 13.6\%-82.2\% on PDE forecasting tasks across
varying coefficients, forces, or boundary conditions. Additionally, varied
downstream tasks reveal that the PI embeddings deciphered by PIANO align well
with the underlying invariants in the PDE systems, verifying the physical
significance of PIANO. The source code will be publicly available at:
https://github.com/optray/PIANO.
