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  3. Data-driven, Ml-assisted Approaches To Problem Well-posedness.
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  3. Data-driven, Ml-assisted Approaches To Problem Well-posedness.

Related Experiment Videos

Data-driven, ML-assisted approaches to problem well-posedness.

Tom Bertalan1, George A Kevrekidis2,3, Eleni D Koronaki4

  • 1Transformative Digital Capabilities, Amgen, Cambridge, MA 02142, USA.

PNAS Nexus
|June 19, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

This study uses machine and manifold learning to infer differential equation well-posedness from data patches, even without traditional boundary conditions. This data-driven approach aids in understanding complex problems where standard theorems are unknown.

Keywords:
differential equationsphysics-informed neural networksrandomized linear algebrasolution manifoldswell-posedness

Related Experiment Videos

Area of Science:

  • Computational Mathematics
  • Machine Learning
  • Differential Equations

Background:

  • Traditional differential equation solving requires precise initial and boundary conditions (BCs) for unique solutions.
  • Real-world data acquisition often involves solution 'patches' at arbitrary locations, not just boundaries.
  • Rigorous existence and uniqueness theorems are not always available for all condition combinations.

Purpose of the Study:

  • To demonstrate data-driven inference of differential equation well-posedness features.
  • To explore methods for cases lacking traditional existence/uniqueness theorems.
  • To bridge the gap between mathematical theory and practical data acquisition.

Main Methods:

  • Application of standard machine learning tools.
  • Utilization of manifold learning techniques.
  • Integration of data assimilation and operator learning perspectives.

    • Main Results:

    • Successful inference of well-posedness features from observed data patches.
    • Demonstration of a data-driven approach for differential equations.
    • Application to problems where standard theorems are not established.

    Conclusions:

    • Machine and manifold learning offer powerful tools for analyzing differential equations with non-traditional data.
    • This data-driven methodology can provide insights into well-posedness beyond established theoretical frameworks.
    • The study highlights the potential of combining data assimilation with operator learning.