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The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

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Physics-Informed Neural Networks (PINN) show promise for solving linear systems from PDEs like the Poisson equation. Hybrid approaches integrating PINNs with traditional solvers offer competitive accuracy and performance.

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Area of Science:

  • Computational Science and Engineering
  • Artificial Intelligence in Scientific Computing
  • Numerical Analysis

Background:

  • Physics-Informed Neural Networks (PINNs) integrate governing equations into neural network architectures.
  • PINNs offer a novel approach for solving complex problems, including linear systems from Partial Differential Equations (PDEs).
  • Traditional methods for linear systems are well-established, but challenges remain for large-scale scientific computing.

Purpose of the Study:

  • To evaluate the efficacy of PINNs as standalone linear solvers for the Poisson equation.
  • To analyze the impact of network configurations and transfer learning on PINN solver performance.
  • To explore hybrid strategies combining PINNs with traditional solvers for improved efficiency and accuracy.

Main Methods:

  • Characterization of PINN linear solvers across various network depths, activation functions, and data distributions.
  • Investigation of transfer learning's role in PINN-based linear system solutions.
  • Development and evaluation of hybrid solvers integrating PINNs with established methods like PETSc conjugate gradient.

Main Results:

  • PINNs demonstrate rapid convergence for low-frequency components but struggle with high-frequency accuracy.
  • Transfer learning significantly influences PINN solver performance.
  • Hybrid solvers achieve performance and accuracy comparable to high-performance traditional solvers.

Conclusions:

  • Direct application of PINNs as linear solvers is limited by accuracy and computational performance.
  • Hybrid strategies combining PINNs with traditional linear solvers represent a promising direction for next-generation computational tools.
  • Integrating deep learning with established numerical methods enhances problem-solving capabilities in scientific computing.