How Much Can One Learn a Partial Differential Equation from Its Solution?

In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive appr...

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Bibliographic Details
Published in:Foundations of computational mathematics 2024-10, Vol.24 (5), p.1595-1641
Main Authors: He, Yuchen, Zhao, Hongkai, Zhong, Yimin
Format: Article
Language:English
Subjects:
Adaptive algorithms
Applications of Mathematics
Computer Science
Economics
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Partial differential equations
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Summary:In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-023-09620-z