PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning, explaining why these methods can be data-efficient. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of 3D elliptic PDEs from input-output data and achieves an exponential convergence rate with respect to the size of the training dataset with an exceptionally high probability of success.

翻译：偏微分方程（PDE）学习是一个结合物理学与机器学习的新兴领域，旨在从实验数据中恢复未知物理系统。尽管传统深度学习模型需要大量训练数据，但近期PDE学习技术能在有限数据条件下取得显著成果。然而，这些成果仍停留于经验层面。本研究为PDE学习所需的输入-输出训练对数量提供了理论保证，解释了此类方法为何能实现数据高效性。具体而言，我们利用随机数值线性代数与PDE理论，推导出一种可证明数据高效的算法，该算法能从输入-输出数据中恢复三维椭圆型PDE的解算子，并以极高的成功概率实现相对于训练数据集规模的指数级收敛速率。