# Elliptic PDE learning is provably data-efficient

## TL;DR

This paper provides the first theoretical guarantees for data efficiency in elliptic PDE learning, demonstrating an exponentially fast convergence rate with high probability.

## Contribution

It introduces a provably data-efficient algorithm for learning solution operators of 3D elliptic PDEs using randomized linear algebra and PDE theory.

## Key findings

- Achieves exponential convergence rate of error with respect to training data size
- Provides high-probability guarantees on data efficiency
- Demonstrates theoretical bounds for PDE learning performance

## Abstract

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.