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Physics-Informed Neural Networks for Solving Forward and Inverse PDEs with Limited and Noisy Data: Application to Solar Corona Modeling

Hubert Baty

TL;DR

This work tackles forward and inverse PDE problems with scarce and noisy data in the solar corona context by applying Physics-Informed Neural Networks (PINNs). It highlights how PINNs can fuse data with PDE residuals to reconstruct velocity and magnetic fields in 2D incompressible MHD, and demonstrates robustness to boundary-data limitations as well as data located inside the domain. The study also shows inverse learning of unknown dissipation coefficients, achieving accurate recovery of $\\nu$ and $\\eta$ while maintaining solution fidelity. Overall, the results indicate that PINNs offer a data-efficient, flexible approach for coronal MHD modeling and parameter inference, with potential practical impact for solar physics and space weather applications.

Abstract

I will demonstrate the effectiveness of Physics-Informed Neural Networks (PINNs) in solving partial differential equations (PDEs) when training data are scarce or noisy. The training data can be located either at the boundaries or within the domain. Additionally, PINNs can be used as an inverse method to determine unknown coefficients in the equations. This study will highlight the application of PINNs in modeling magnetohydrodynamic processes relevant to strongly magnetized plasmas, such as those found in the solar corona.

Physics-Informed Neural Networks for Solving Forward and Inverse PDEs with Limited and Noisy Data: Application to Solar Corona Modeling

TL;DR

This work tackles forward and inverse PDE problems with scarce and noisy data in the solar corona context by applying Physics-Informed Neural Networks (PINNs). It highlights how PINNs can fuse data with PDE residuals to reconstruct velocity and magnetic fields in 2D incompressible MHD, and demonstrates robustness to boundary-data limitations as well as data located inside the domain. The study also shows inverse learning of unknown dissipation coefficients, achieving accurate recovery of and while maintaining solution fidelity. Overall, the results indicate that PINNs offer a data-efficient, flexible approach for coronal MHD modeling and parameter inference, with potential practical impact for solar physics and space weather applications.

Abstract

I will demonstrate the effectiveness of Physics-Informed Neural Networks (PINNs) in solving partial differential equations (PDEs) when training data are scarce or noisy. The training data can be located either at the boundaries or within the domain. Additionally, PINNs can be used as an inverse method to determine unknown coefficients in the equations. This study will highlight the application of PINNs in modeling magnetohydrodynamic processes relevant to strongly magnetized plasmas, such as those found in the solar corona.

Paper Structure

This paper contains 14 sections, 11 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Vanilla PINNs Method
  3. Gradient Descent Method for Training PINNs
  4. Training data points and collocation points
  5. The neural network architecture
  6. The algorithm and implementation
  7. Direct PINNs method for MHD reconnection
  8. The MHD equations, spatial domain, and exact solutions
  9. PINNs results for a direct solver with standard BCs
  10. PINNs results for a forward solver with limited BCs
  11. PINNs results for a forward solver with noisy BCs
  12. PINNs results for a forward solver with internal conditions
  13. PINNs results for an inverse solver with unknown resistivity and viscosity coefficients
  14. Conclusion

Figures (11)

  • Figure 1: Schematic representation of an arbitrary 2D integration domain, and distribution of the training data points (red dots) situated on the boundary (left panel) and within the domain (right panel).
  • Figure 2: Schematic representation of an arbitrary 2D integration domain, and distribution of the collocation points (blue dots) within the integration domain.
  • Figure 3: Example of a neural network architecture used in the PINNs method, featuring three hidden layers with five neurons per layer
  • Figure 4: The solution predicted using the PINNs solver is shown, with magnetic field lines plotted as iso-contours and flow velocity represented by black arrows. The locations of the training and collocation data points are marked with red and blue dots, respectively.
  • Figure 5: Colored iso-contours of the $B_y$ and $B_x$ magnetic field components predicted by PINNs solver, and associated absolute error distributions.
  • ...and 6 more figures