Data-driven, ML-assisted approaches to problem well-posedness [* Cross-Reference *]

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Tom Bertalan; George A. Kevrekidis; Eleni D. Koronaki; Siddhartha Mishra; Elizaveta Rebrova; Ioannis G Kevrekidis, 2026, "Data-driven, ML-assisted approaches to problem well-posedness [* Cross-Reference *]", https://lore.list.lu/citation?persistentId=perma:LIST.R3HWDE, LIST Open Repository, V1
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Description
Classically, to solve differential equation problems, it is necessary to specify sufficient initial condition and/or boundary condition (BC) so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves studying the existence and uniqueness of solutions, and their dependence on such prespecified conditions. However, in part due to mathematical necessity, these conditions are usually specified “to arbitrary precision” only on (appropriate portions of) the boundary of the space-time domain. This does not mirror how data acquisition is performed in realistic situations, where one may observe entire “patches” of solution data at arbitrary space-time locations; alternatively one might have access to more than one solution stemming from the same differential operator. In our short work, we demonstrate how standard tools from machine and manifold learning can be used to infer, in a data driven manner, certain well-posedness features of differential equation problems, for initial condition/BC combinations under which rigorous existence/uniqueness theorems are not known. Our study naturally combines a data assimilation perspective with an operator-learning one. (2026-05-30)
***This entry has been automatically imported via OpenAlex by LIST harvest scripts. Please refer to https://doi.org/10.1093/pnasnexus/pgag175 for the original and latest version of the publication*** (2026-07-01)
Subject Astronomy and Astrophysics; Engineering; Mathematical Sciences; Physics
Keyword Boundary value problem, Differential equation, Uniqueness, Universal differential equation, Boundary (topology), Exact differential equation, Differential (mechanical device)
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