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Promising directions of machine learning for partial differential equations

Nature Computational Science volume 4, pages 483–494 (2024)Cite this article

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Abstract

Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multiscale physics in a compact and symbolic representation. Here, we examine several promising avenues of PDE research that are being advanced by machine learning, including (1) discovering new governing PDEs and coarse-grained approximations for complex natural and engineered systems, (2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and (3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.

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Fig. 1: Overview of three areas where machine learning is advancing PDE research.
Fig. 2: Sparse regression procedure to discover PDEs from data, demonstrated on the incompressible Navier–Stokes equations.
Fig. 3: Illustration of sparse Bayesian PDE discovery applied to LES closure modeling in large-scale geophysical fluid simulations.
Fig. 4: Schematic of a neural network architecture used to discover a Koopman linearizing coordinate transformation.
Fig. 5: Machine-learned interpolation from coarse-grained to high-resolution flow fields.

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Acknowledgements

We acknowledge support from the National Science Foundation AI Institute in Dynamic Systems (grant no. 2112085). S.L.B. acknowledges support from the Army Research Office (ARO W911NF-19-1-0045).

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  1. Department of Mechanical Engineering, University of Washington, Seattle, WA, USA

    Steven L. Brunton

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, USA

    J. Nathan Kutz

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S.L.B. and J.N.K. contributed equally to this Review.

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Brunton, S.L., Kutz, J.N. Promising directions of machine learning for partial differential equations. Nat Comput Sci 4, 483–494 (2024). https://doi.org/10.1038/s43588-024-00643-2

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