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.

本文介绍了一种结合了物理与机器学习的新兴领域：PDE 学习。我们提出了一种理论上保证数据效率的算法，可以从有限的输入输出数据中恢复 3D 椭圆型偏微分方程的解算子，并以极高的成功概率呈指数收敛率。

椭圆型偏微分方程学习可证明具有数据效率

Elliptic PDE learning is provably data-efficient

Given input-output pairs of an elliptic partial differential equation (PDE)
in three dimensions, we derive the first theoretically-rigorous scheme for
learning the associated Green's function $G$. By exploiting the hierarchical
low-rank structure of $G$, we show that one can construct an approximant to $G$
that converges almost surely and achieves a relative error of
$\mathcal{O}(\Gamma_\epsilon^{-1/2}\log^3(1/\epsilon)\epsilon)$ using at most
$\mathcal{O}(\epsilon^{-6}\log^4(1/\epsilon))$ input-output training pairs with
high probability, for any $0<\epsilon<1$. The quantity $0<\Gamma_\epsilon\leq
1$ characterizes the quality of the training dataset. Along the way, we extend
the randomized singular value decomposition algorithm for learning matrices to
Hilbert--Schmidt operators and characterize the quality of covariance kernels
for PDE learning.

通过利用低秩结构，我们构建了一种近似于关联绿函数的构建方案，其中相对误差为 $\mathcal {O}(\Gamma_\epsilon^{-1/2}\log^3 (1/\epsilon)\epsilon)$，使用高概率的 $\mathcal {O}(\epsilon^{-6}\log^4 (1/\epsilon))$ 个输入 - 输出训练对在任何 $0<\epsilon<1$ 的情况下可以保证收敛，从而获得了三维椭圆型偏微分方程相关的绿函数的学习的第一个具有理论依据的方案，我们还扩展了用于学习矩阵的随机奇异值分解算法到 Hilbert-Schmidt 算子，并刻画了该算法中偏微分方程协方差核的质量。