用物理残差指导采样，让神经算子少用数据也能学好微分方程。
Data-Efficient Neural Operator Training via Physics-Based Active Learning
- 基于方程残差选择最需训练的数据点，实现高效迭代采样。
- 在1D布格尔斯和2D纳维-斯托克斯方程上，数据效率达当前最优。
- 自动聚焦模型薄弱环节，兼具物理先验与低样本需求优势。
用神经算子求解偏微分方程可显著降低计算成本，但受限于高训练数据需求。主动学习通过迭代选取最具信息量的样本，自然缓解此问题。本文提出一种基于物理的采样策略——利用偏微分方程残差指导数据选择。我们在1D布格尔斯方程和2D可压缩纳维-斯托克斯方程上验证该方法。实验表明，该方法在各项指标上均优于随机采样，并达到当前最优的数据效率。同时，它能将物理归纳偏置注入训练过程，确保模拟成本集中在模型物理理解最弱的区域。
原文摘要 · Abstract (English)
Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.
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