Generalized Lie Symmetries in Physics-Informed Neural Operators
This work addresses a specific bottleneck in PINOs for solving PDEs, offering an incremental improvement in training efficiency for researchers in scientific machine learning.
The paper tackled the problem of limited training signal from point symmetries in physics-informed neural operators (PINOs) by proposing a novel loss augmentation strategy using evolutionary representatives of generalized symmetries, resulting in improved data efficiency and accuracy during training.
Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.