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Improved Physics-Informed Neural Networks Combined with Small Sample Learning to Solve Two-Dimensional Stefan

Jiawei Li1, Wei Wu1, Xinlong Feng1

  • 1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China.

Entropy (Basel, Switzerland)
|May 16, 2023
PubMed
Summary

Deep neural networks offer a novel approach to solving the Stefan problem, even with limited data. This enhanced deep learning method accurately predicts moving boundaries and dynamic interfaces in scientific modeling.

Keywords:
Stefan problemdeep neural networksefficient calculation methodsmall sample learning

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Area of Science:

  • Computational Science
  • Applied Mathematics
  • Numerical Analysis

Background:

  • The Stefan problem, a class of free boundary problems, is crucial in various scientific fields like phase transitions.
  • Traditional numerical methods face challenges with moving boundaries and require significant computational resources.
  • Deep learning presents a promising alternative for tackling complex partial differential equations.

Purpose of the Study:

  • To propose a novel deep neural network framework for solving the two-dimensional Stefan problem.
  • To enhance the model's performance and prediction accuracy using small sample learning.
  • To validate the method's efficacy by solving both forward and inverse problems.

Main Methods:

  • A general deep learning framework is adapted for the Stefan problem.
  • Small sample learning techniques are integrated to improve model performance with limited data.
  • The modified deep neural network is applied to solve the two-dimensional single-phase Stefan problem.

Main Results:

  • The proposed deep learning approach accurately solves the two-dimensional Stefan problem.
  • Improved prediction accuracy is achieved even with a reduced amount of sample data.
  • The method successfully predicts solutions for partial differential equations governing moving boundaries and dynamic interfaces.

Conclusions:

  • Deep neural networks, augmented with small sample learning, provide an effective solution for the Stefan problem.
  • This approach offers a robust and accurate method for modeling dynamic interfaces in scientific applications.
  • The validated framework demonstrates the potential of deep learning in addressing complex free boundary problems.