用任意维机器学习热启动偏微分方程求解器
Warm-starting PDE solvers with any-dimensional machine learning
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中文总结 AI 辅助
本文提出利用对称性条件，使低维训练的PDE求解器可零样本或热启动求解高维PDE，并在多个方程上验证了性能提升与数据效率优势。
中文摘要 AI 辅助
任意维机器学习模型，如图神经网络（GNNs），可以自然地在不同大小和维度的输入上进行训练和评估。受GNN可迁移性文献的启发，我们展示了在何种数学条件下，一个基于学习的偏微分方程（PDE）求解器可以在小维度上训练，并直接以零样本方式应用于求解更高维度的PDE。这些条件基于偏微分方程和初始数据中的对称性。当方程满足对称性但数据不满足时（许多来自物理的PDE即如此），我们表明我们的理论为低维PDE求解器热启动高维PDE提供了一种有原则的方法。我们将此方法应用于热方程、Burgers方程和可压缩Navier-Stokes方程，在高维数据上的零样本和典型训练场景中均提升了性能。例如，我们在2D Navier-Stokes数据上训练一个代理模型，在3D测试数据上取得的结果优于在3D数据上训练的基线代理模型，而仅使用12%的计算量和20%的总数据量。
英文摘要
Any-dimensional machine learning models, such as graph neural networks (GNNs), can be naturally trained and evaluated on inputs of different sizes and dimensions. Inspired by the GNN transferability literature, we show mathematical conditions under which a partial differential equation (PDE) learning-based solver can be trained in small dimensions and directly applied to solve a higher dimensional PDE in a zero-shot fashion. These conditions are based on symmetries in both the partial differential equation and the initial data. When the equations satisfy the symmetries but the data does not, which is the case for many PDEs arising from physics, we show that our theory gives a principled way of warm-starting low-dimensional PDE solvers for higher dimensional PDEs. We apply this method on the heat equation, Burgers' equation, and the compressible Navier--Stokes equations, improving the performance in both zero-shot and typical training regimes on high dimensional data. For example, we train a surrogate model on 2D Navier--Stokes data and achieve better results on 3D test data than a baseline surrogate model trained on 3D data, while only using 12$\%$ of the flops and 20$\%$ of the total data size.
发表机构
- Columbia University（哥伦比亚大学）
- Brown University（布朗大学）
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
