Elliptic PDE learning is provably data-efficient

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.

25 pages, 2 figures

References in corpus (6)

• Fourier Neural Operator for Parametric Partial Differential Equations
• Data-driven discovery of Green's functions with human-understandable deep learning
• Learning elliptic partial differential equations with randomized linear algebra
• Elliptic PDE learning is provably data-efficient
• Learning Green's functions associated with time-dependent partial differential equations
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Cited by in corpus (4)

• A Mathematical Guide to Operator Learning
• Operator Learning Using Random Features: A Tool for Scientific Computing
• Elliptic PDE learning is provably data-efficient
• An operator learning perspective on parameter-to-observable maps