VAE-like MHD equilibrium solver

• Magneto-Hydro-Dynamics (MHD) is the physics of electromagnetically interacting fluids.
• Solving the MHD force balance equation is the first step for analyzing fusion plasmas in a tokamak.
• This repository describes a variational auto-encoder (VAE)-like neural network to solve ideal MHD equilibrium.

Installation

• You can install by

$ git clone https://github.com/jaem-seo/vae_mhd_solver.git
$ cd vae_mhd_solver

Try it out

$ python predict.py

• This solves 0D-2D quantities of the MHD equilibrium, from 1D input profiles (pressure and current density) and the boundary coordinates.
• The below are sample predictions for 2D magnetic flux structure, ψ and φ.

Description

• The input profiles (pressure, current density, and plasma boundary) are

• Then, the 2D outputs (ground truth and prediction) are

• Other 0D and 1D physical quantities are also calculated.

Note

• The current version has a limitation in that the reconstructed outputs are slightly jagged.
• The model is similar to VAE, but the input and output are only physically consistent with each other, not the same structure.
• A simple physical constraint has been added to the loss calculation.
• The physical quantities for input/outputs are normalized according to CHEASE convention.

References

• Kingma, Diederik P., and Max Welling. "Auto-encoding variational bayes." arXiv preprint arXiv:1312.6114 (2013).
• Miller, R. L., et al. "Noncircular, finite aspect ratio, local equilibrium model." Physics of Plasmas 5.4 (1998): 973-978.
• Lütjens, Hinrich, Anders Bondeson, and Olivier Sauter. "The CHEASE code for toroidal MHD equilibria." Computer physics communications 97.3 (1996): 219-260.

TODO

• Physics (Grad-Shafranov equation)-informed loss applied
• Additional encoder for general boundary coordinates