Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders

TL;DR

This paper introduces an equivariant autoencoder-based surrogate model for 3D Rayleigh-Bénard convection, improving efficiency and scalability in modeling complex fluid dynamics governed by PDEs.

Contribution

It presents a novel equivariant convolutional autoencoder and LSTM architecture tailored for 3D fluid systems, leveraging $G$-steerable kernels and partial sharing to enhance efficiency.

Findings

Significant improvements in sample and parameter efficiency.

Better scalability to complex dynamics.

Effective modeling of 3D Rayleigh-Bénard convection.

Abstract

The use of machine learning for modeling, understanding, and controlling large-scale physics systems is quickly gaining in popularity, with examples ranging from electromagnetism over nuclear fusion reactors and magneto-hydrodynamics to fluid mechanics and climate modeling. These systems - governed by partial differential equations - present unique challenges regarding the large number of degrees of freedom and the complex dynamics over many scales both in space and time, and additional measures to improve accuracy and sample efficiency are highly desirable. We present an end-to-end equivariant surrogate model consisting of an equivariant convolutional autoencoder and an equivariant convolutional LSTM using $G$-steerable kernels. As a case study, we consider the three-dimensional Rayleigh-B\'enard convection, which describes the buoyancy-driven fluid flow between a heated bottom and a cooled top plate. While the system is E(2)-equivariant in the horizontal plane, the boundary conditions break the translational equivariance in the vertical direction. Our architecture leverages vertically stacked layers of $D_4$-steerable kernels, with additional partial kernel sharing in the vertical direction for further efficiency improvement. We demonstrate significant gains in sample and parameter efficiency, as well as a better scaling to more complex dynamics. The accompanying code is available under https://github.com/FynnFromme/equivariant-rb-forecasting.