Coordinate Transform Fourier Neural Operators for Symmetries in Physical Modelings

Abstract

Symmetries often arise in many natural sciences; rather than relying on data augmentation or regularization for learning these symmetries, incorporating these inherent symmetries directly into the neural network architecture simplifies the learning process and enhances model performance. The laws of physics, including partial differential equations (PDEs), remain unchanged regardless of the coordinate system employed to depict them, and symmetries sometimes can be natural to illuminate in other coordinate systems. Moreover, symmetries often are associated with the underlying domain shapes. In this work, we consider physical modelings with neural operators (NOs), and we propose an approach based on coordinate transforms (CT) to work on different domain shapes and symmetries. Canonical coordinate transforms are applied to convert both the domain shape and symmetries. For example, a sphere can be naturally converted to a square with periodicities across its edges. The resulting CT-FNO scheme barely increases computational complexity and can be applied to different domain shapes while respecting the symmetries.