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

2025-05-19

AI Summary

This work addresses three-dimensional Rayleigh–Bénard convection—a canonical PDE-driven turbulent system exhibiting horizontal E(2) symmetry but broken vertical translational symmetry due to boundary conditions. We propose the first end-to-end equivariant surrogate model for this setting. Methodologically, we introduce a novel architecture integrating D₄-steerable equivariant convolutions with partial kernel sharing along the vertical direction, explicitly encoding geometric symmetries and boundary inhomogeneity. Temporal dynamics are captured via an equivariant convolutional autoencoder coupled with an equivariant ConvLSTM. Compared to standard CNNs, our model achieves significantly higher parameter and sample efficiency, maintains high predictive accuracy even under strong turbulence at high Rayleigh numbers, and demonstrates superior generalization and computational scalability over baseline models.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: 3D Computer Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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. Our results demonstrate significant gains both in sample and parameter efficiency, as well as a better scaling to more complex dynamics, that is, larger Rayleigh numbers. The accompanying code is available under https://github.com/FynnFromme/equivariant-rb-forecasting.

Problem

• Modeling 3D Rayleigh-Benard convection with equivariant autoencoders
• Improving accuracy and efficiency in large-scale physics systems
• Addressing E(2)-equivariance challenges in fluid dynamics modeling

Innovation

• Equivariant convolutional autoencoder for surrogate modeling
• G-steerable kernels enhance symmetry handling
• Partial kernel sharing improves vertical efficiency