What is it about?

Partial differential equations (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 amount of training required in PDE learning and shows that machine learning models can understand complex equations in real-world situations while using far less data than is normally expected.

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Why is it important?

Many scientific breakthroughs have come from deriving new partial differential equations (PDEs) from first principles to model real-world phenomena and simulating them on a computer to make predictions. However, many crucial problems currently lack an adequate mathematical formulation. It is not clear how to derive PDEs to describe how turbulence sheds off the wing of a hypersonic aircraft, how Escherichia coli bacteria swim in unison to form an active fluid, or how atomic particles behave with long-range interactions. Rather than working from first principles, scientists are now looking to derive PDEs from real-world data using deep learning techniques. However, there is a lack of understanding of the efficiency of PDE learning methods with limited training data. This work provides theoretical insights by constructing a provably data-efficient algorithm,

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This page is a summary of: Elliptic PDE learning is provably data-efficient, Proceedings of the National Academy of Sciences, September 2023, Proceedings of the National Academy of Sciences,
DOI: 10.1073/pnas.2303904120.
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