Frame invariance and scalability of vector cloud neural network for partial differential equations*

Solving partial differential equations (PDEs) often requires prohibitively high computational costs, especially when multiple evaluations are to be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, a recently proposed vector cloud neural network (VCNN) has been assessed to emulate the invariance properties and non-local dependencies of transport PDEs. First, the invariance properties and computational complexity of VCNN have been examined for transport PDE of a scalar quantity. For comparison purposes, an alternate neural operator based on graph kernel network (GKN) is considered, for which a modified formulation of GKN has been presented to ensure frame invariance. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN. Furthermore, VCNN is presented as a robust tool to emulate transport equations for tensorial quantities. We demonstrate its performance on Reynolds stress transport equations, showing that the VCNN can effectively emulate the Reynolds stress transport model for Reynolds-averaged Navier–Stokes (RANS) equations. The VCNN respects all the invariance properties desired by constitutive model and faithfully reflects the region of influence in physics.