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Machine-learning-based spectral methods for partial differential equations.

Brek Meuris1, Saad Qadeer2, Panos Stinis3,4

  • 1Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA.

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|January 31, 2023
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Summary

This study introduces a novel method combining deep neural networks with spectral methods for solving partial differential equations (PDEs). The approach uses Deep Operator Networks (DeepONet) to create custom, hierarchical basis functions, enhancing PDE solution accuracy.

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Area of Science:

  • Scientific Computing
  • Machine Learning
  • Numerical Analysis

Background:

  • Spectral methods are crucial for solving partial differential equations (PDEs).
  • The effectiveness of spectral methods relies heavily on the selection of appropriate basis functions.
  • Deep learning has emerged as a powerful tool for function representation.

Purpose of the Study:

  • To integrate deep neural networks with spectral methods for solving PDEs.
  • To develop a novel approach for generating custom basis functions using deep learning.
  • To enhance the capabilities and applicability of spectral methods in scientific computing.

Main Methods:

  • Utilized Deep Operator Networks (DeepONet), a deep learning technique, to identify candidate basis functions.
  • Constructed orthonormal and hierarchical basis functions using DeepONet-identified candidates.
  • Applied these custom basis functions to approximate solutions of linear and nonlinear time-dependent PDEs.

Main Results:

  • Demonstrated the approximation capabilities of the newly developed basis functions.
  • Successfully applied the basis functions to solve various PDEs.
  • Showcased the effectiveness of combining deep learning with spectral methods.

Conclusions:

  • The proposed approach enhances the state-of-the-art in spectral methods for solving PDEs.
  • The synergy between traditional scientific computing and machine learning is promoted.
  • The custom-built, hierarchical basis functions offer improved versatility and accuracy.