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  3. Particles To Partial Differential Equations Parsimoniously.
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  3. Particles To Partial Differential Equations Parsimoniously.

Related Experiment Video

Setting Limits on Supersymmetry Using Simplified Models
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Particles to partial differential equations parsimoniously.

Hassan Arbabi1, Ioannis G Kevrekidis2

  • 1Department of Mechanical Engineering, MIT, Cambridge, Massachusetts 02139, USA.

Chaos (Woodbury, N.Y.)
|April 3, 2021

View abstract on PubMed

Summary
This summary is machine-generated.

This study introduces a novel framework for discovering coarse-grained partial differential equations (PDEs) from microscopic simulations. It combines neural networks with equation-free numerics and manifold learning to efficiently extract macro-scale dynamics, reducing computational costs.

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Area of Science:

  • Computational Physics
  • Data Science
  • Applied Mathematics

Background:

  • Physico-chemical processes are typically described by microscopic equations.
  • Effective equations at macroscopic scales (meso- or macroscopic) can simplify complex systems.
  • Discovering these coarse-grained equations aids in computational prediction and control.

Purpose of the Study:

  • To develop an efficient framework for discovering macro-scale partial differential equations (PDEs) from microscopic simulations.
  • To reduce the computational cost associated with data collection for training machine learning models.
  • To identify suitable macro-scale variables for formulating coarse-grained effective PDEs.

Main Methods:

  • Combining artificial neural networks with equation-free numerics for multiscale computation.
  • Employing manifold learning techniques, including unnormalized optimal transport and moment-based distribution descriptions.
  • Utilizing sparse data collection in the space-time domain.

    • Main Results:

    • Efficient discovery of coarse-grained PDEs directly from microscopic simulations.
    • Significant reduction in the computational effort required for data collection.
    • Successful identification of macro-scale variables, corroborating or introducing new candidates.

    Conclusions:

    • The proposed framework effectively extracts coarse-grained evolution equations from particle-based simulations.
    • The integration of equation-free numerics and data-driven manifold learning offers a computationally parsimonious approach.
    • This method facilitates the discovery of effective macro-scale PDEs for systems with unknown macroscopic variables.