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Coarse-scale PDEs from fine-scale observations via machine learning.

Seungjoon Lee1, Mahdi Kooshkbaghi2, Konstantinos Spiliotis3

  • 1Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

This study introduces a data-driven framework using machine learning to discover macroscopic partial differential equations (PDEs) from microscopic data, solving the closure problem for complex systems.

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Area of Science:

  • Computational Physics
  • Data Science
  • Chemical Engineering

Background:

  • Physicochemical processes are modeled at micro and macro levels.
  • Deriving macroscopic models (closure problem) is challenging.
  • Data science offers new approaches to learn macroscopic models.

Purpose of the Study:

  • Develop a data-driven framework to identify coarse-scale PDEs from microscopic observations.
  • Utilize machine learning algorithms for uncovering macroscopic evolution laws.
  • Address the challenge of deriving accurate macroscopic descriptions.

Main Methods:

  • Employed machine learning algorithms: Gaussian processes, artificial neural networks, and diffusion maps.
  • Framework learns the relationship between macroscopic fields and their temporal evolution.
  • Applied to a reaction/transport process using a lattice Boltzmann model.

Main Results:

  • Successfully identified unavailable macroscopic, concentration-level PDEs.
  • Demonstrated the discovery of multiple, equally representative macroscopic laws.
  • Showcased the framework's ability to generate long-term macroscopic predictions.

Conclusions:

  • The data-driven framework effectively identifies coarse-scale PDEs from microscopic data.
  • Machine learning provides a powerful alternative to traditional methods for model discovery.
  • The approach facilitates accurate long-term macroscopic predictions.