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 the physical invariant attention neural operator (PIANO) to decipher and integrate the physical invariants 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. 

Publication: 

National Science Review, Volume 11, Issue 4, April 2024, nwad336 

https://doi.org/10.1093/nsr/nwad336

Author: 

Rui Zhang 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing 100190, China 

Qi Meng 

Microsoft Research, Beijing 100080, China 

现为中国科学院数学与系统科学研究院副研究员 

Email: meq@amss.ac.cn 

Zhi-Ming Ma 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing 100190, China 

Email: mazm@amt.ac.cn