Elliptic PDE learning is provably data-efficient

• URL: http://arxiv.org/abs/2302.12888v2
• Date: Tue, 19 Sep 2023 09:35:41 GMT
• Title: Elliptic PDE learning is provably data-efficient
• Authors: Nicolas Boull\'e, Diana Halikias, Alex Townsend
• Abstract summary: PDE learning combines physics and machine learning to recover unknown physical systems from experimental data. 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.
• Score: 7.097838977449412
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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