用等变自编码器建模三维对流，提升物理模拟的效率与精度。
Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders
- 采用等变卷积自编码器+等变LSTM，利用对称性约束降低复杂度。
- 在三维对流模拟中实现参数和样本效率显著提升，可扩展至更复杂动力学。
- 适合流体动力学、气候建模等需要高效物理模拟的研究者使用。
机器学习在大规模物理系统建模、理解和控制中的应用日益广泛，涵盖电磁学、核聚变、磁流体到流体力学和气候建模等领域。这些由偏微分方程支配的系统具有大量自由度及跨时空多尺度的复杂动力学，亟需提升准确性和采样效率。本文提出一个端到端的等变代理模型，包含使用G-可导向核的等变卷积自编码器和等变卷积LSTM。以三维瑞利-贝纳德对流为例，该系统在水平面具有E(2)等变性，但边界条件破坏了垂直方向的平移等变性。模型采用垂直堆叠的D₄-可导向核，并在垂直方向引入部分核共享以进一步提升效率。实验表明，该方法在样本和参数效率上均有显著提升，且对更复杂动力学具有更好的扩展性。配套代码已开源：https://github.com/FynnFromme/equivariant-rb-forecasting。
原文摘要 · Abstract (English)
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-Bénard 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.
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