  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.
Specifically, we exploit randomized numerical linear algebra and PDE theory to
derive a provably data-efficient algorithm that recovers solution operators of
3D uniformly elliptic PDEs from input-output data and achieves an exponential
convergence rate of the error with respect to the size of the training dataset
with an exceptionally high probability of success.
