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

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.