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Area of Science:

  • Mathematical Physics
  • Computational Science
  • Machine Learning

Background:

  • Eigenmode localization is crucial for understanding quantum systems with random potentials.
  • Schrödinger Hamiltonians with random potentials present complex challenges in spectral analysis.
  • Effective confining potentials are key to describing localized states.

Purpose of the Study:

  • To develop accurate deep learning models for predicting eigenmode localization.
  • To investigate the use of physics-informed deep dense networks for higher-dimensional problems.
  • To ensure the interpretability of the deep learning models as reduced-order models.

Main Methods:

  • Utilized deep network architectures to predict localized bounded states from potential samples.
  • Employed physics-informed deep dense networks for high-dimensional reaction-diffusion operators.
  • Implemented deep networks as reduced-order models linking potentials to ground states.
  • Incorporated an error estimator for performance control and model updating.

Main Results:

  • Demonstrated the ability of deep networks to predict eigenmode localization accurately.
  • Showcased the effectiveness of physics-informed networks for complex, high-dimensional problems.
  • Validated the interpretability of deep networks as reduced-order models.
  • Presented experimental results confirming the algorithm's accuracy and performance.

Conclusions:

  • Deep learning offers a powerful tool for analyzing eigenmode localization in complex systems.
  • Physics-informed neural networks provide an interpretable and efficient approach.
  • The developed methods accurately predict localized states and serve as effective reduced-order models.