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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Machine-learning-based data-driven discovery of nonlinear phase-field dynamics.

Elham Kiyani1,2, Steven Silber2,3, Mahdi Kooshkbaghi4

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This study introduces machine learning models to discover complex partial differential equations (PDEs) from data, even those with high-order derivatives. These data-driven PDEs accurately describe and predict the behavior of phase-field models.

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Area of Science:

  • Complex Systems
  • Computational Physics
  • Machine Learning

Background:

  • Complex systems require coarse-grained models to simplify microscopic details.
  • Discovering governing partial differential equations (PDEs) from data is challenging, especially for those with high-order derivatives.
  • Machine learning (ML) offers powerful tools for data-driven scientific discovery.

Purpose of the Study:

  • To develop and evaluate data-driven architectures for discovering nonlinear equations of motion in phase-field models.
  • To investigate ML approaches for identifying PDEs with high-order derivatives.
  • To assess the predictive capability of discovered PDEs for system evolution.

Main Methods:

  • Utilized multilayer perceptron (MLP), convolutional neural networks (CNNs), and CNN-LSTM architectures.
  • Applied data-driven methods to phase-field models, including Allen-Cahn, Cahn-Hilliard, and phase-field crystal models.
  • Implemented two approaches: one guided by physical intuition and a black-box approach without assumptions.

Main Results:

  • Successfully learned time derivatives for phase-field models using data-driven architectures.
  • Demonstrated the ability to discover nonlinear PDEs, including those with high-order derivatives.
  • Validated the discovered PDEs by accurately propagating fields in time, showing good agreement with original models.

Conclusions:

  • Data-driven ML architectures, including CNNs and LSTMs, are effective for discovering complex PDEs from observational data.
  • The developed methods can uncover governing equations for phase-field models under different levels of physical guidance.
  • The discovered PDEs can reliably predict system dynamics, advancing the field of data-driven scientific modeling.