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Finite Element Modelling of a Cellular Electric Microenvironment
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Can physics-informed neural networks beat the finite element method?

Tamara G Grossmann1, Urszula Julia Komorowska2, Jonas Latz3

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.

IMA Journal of Applied Mathematics
|June 27, 2024
PubMed
Summary

This study compares numerical methods for solving partial differential equations (PDEs). Physics-informed neural networks (PINNs) were not found to outperform the established finite element method (FEM) in terms of accuracy and speed.

Keywords:
deep learningfinite element methodpartial differential equationsphysics-informed neural networks

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Partial differential equations (PDEs) are essential for modeling scientific phenomena.
  • Numerical methods approximate PDE solutions for simulations.
  • Physics-informed neural networks (PINNs) are a recent deep learning approach for PDEs.

Purpose of the Study:

  • To systematically compare the performance of PINNs and the finite element method (FEM).
  • To evaluate computational costs and approximation accuracies of both methods on various PDEs.

Main Methods:

  • Solving 1D, 2D, and 3D Poisson equations.
  • Solving 1D Allen-Cahn and 1D/2D semilinear Schrödinger equations.
  • Utilizing both PINNs and the FEM for numerical approximation.

Main Results:

  • The FEM generally achieved superior accuracy and solution times compared to PINNs.
  • PINNs demonstrated faster evaluation of the solved PDE in specific experimental setups.
  • No significant advantage of PINNs over FEM was observed in this comparative study.

Conclusions:

  • The finite element method remains a robust and efficient choice for solving a wide range of PDEs.
  • While promising, physics-informed neural networks require further development to consistently match or exceed FEM performance.
  • This study highlights the need for continued research in both numerical methods and deep learning for scientific computing.