In a recent publication in the National Science Review, a team of researchers has introduced a novel operator learning framework named PIANO. This new approach utilizes self-supervised learning to extract physical invariants from partial differential equation (PDE) systems with varying physical mechanisms, thus enhancing the generalizability of neural operators to different physics scenarios. The study, led by
Neural operators are a method of training PDE solvers by learning a mapping from one function to another, resulting in a faster simulation process. Compared to traditional techniques, neural operators overcome the limitations of space-time discretization, significantly accelerate the inference speed, and show great potential in fields such as inverse design and physical simulation. However, most existing methods only consider data from systems driven by a single equation, which restricts the application of neural operators in multi-physics scenarios.
To address this issue, the researchers have proposed a new operator learning framework called
PIANO incorporates two key designs: the use of self-supervised learning methods to acquire representations containing physical invariants, and the integration of these representations into neural operators through dynamic convolution layers. Additionally, the researchers have introduced three types of physics-aware cropping techniques based on prior knowledge to align with the attributes of different PDE systems.
The effectiveness and physical significance of PIANO have been demonstrated through several benchmark problems, including Burgers' equation, convection-diffusion equation, and Navier-Stokes equation. The results of these experiments have shown that PIANO outperforms existing methods in terms of accuracy and generalizability when learning neural operators from PDE datasets with varying physical mechanisms.
The results of six experiments have revealed that PIANO can reduce the relative error rate by 15.1% – 82.2% by effectively deciphering and integrating the physical information of the PDE system. Furthermore, a series of downstream tasks have confirmed the physical significance of the extracted PI representation by PIANO.