Equivariant geometric convolutions for emulation of dynamical systems

TL;DR

This paper introduces equivariant geometric convolutions in neural network architectures to create coordinate-free surrogate models for dynamical systems, improving accuracy and stability in fluid dynamics simulations.

Contribution

It presents a novel approach to enforce coordinate invariance in CNN-based surrogate models using geometric convolutions across multiple architectures.

Findings

Enhanced accuracy in emulating 2D Navier-Stokes equations

Improved stability over baseline models

Easy integration of coordinate freedom in existing architectures

Abstract

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied to an appropriate class of problems