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The Weak Form Is Stronger Than You Think

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arxiv 2409.06751 v1 pith:KNMUV6WS submitted 2024-09-10 cs.LG cs.CEcs.NAmath.NAstat.ML

Daniel A. Messenger , April Tran , Vanja Dukic , David M. Bortz This is my paper · ORCID
classification cs.LGcs.CEcs.NAmath.NAstat.ML
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The weak form is a ubiquitous, well-studied, and widely-utilized mathematical tool in modern computational and applied mathematics. In this work we provide a survey of both the history and recent developments for several fields in which the weak form can play a critical role. In particular, we highlight several recent advances in weak form versions of equation learning, parameter estimation, and coarse graining, which offer surprising noise robustness, accuracy, and computational efficiency. We note that this manuscript is a companion piece to our October 2024 SIAM News article of the same name. Here we provide more detailed explanations of mathematical developments as well as a more complete list of references. Lastly, we note that the software with which to reproduce the results in this manuscript is also available on our group's GitHub website https://github.com/MathBioCU .

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Cited by 3 Pith papers

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    cs.RO 2025-09 conditional novelty 5.0 of 10

    A simulation study showing that online weak-form learning of drag and thrust coefficients combined with a boundary-value optimal controller substantially reduces MicroASV tracking error.

  3. Learning Physically Interpretable Atmospheric Models from Data with WSINDy

    physics.geo-ph 2025-01 conditional novelty 5.0 of 10

    WSINDy discovers interpretable PDE models from simulated and ERA5 atmospheric data, with strong results on simulations and partial results on assimilated data.

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