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Reconstructing Relativistic Magnetohydrodynamics with Physics-Informed Neural Networks

Corwin Cheung, Marcos Johnson-Noya, Michael Xiang, Dominic Chang, Alfredo Guevara

TL;DR

This work tackles the computational challenge of RMHD by introducing a physics-informed neural-network surrogate that operates on primitive variables via a Jacobian/characteristic formulation and enforces the divergence-free constraint. By bypassing conservative-variable inversions and employing the MUON optimizer, the approach achieves rapid training and extrapolates RMHD dynamics from limited early-time data in both 1D and 2D settings. A residual-guided correction stage further reduces PDE violations, enhancing accuracy. The framework sets the stage for extensions to GRMHD in curved spacetime and for use as priors in Bayesian inference, offering a computationally efficient RMHD surrogate with strong physical grounding.

Abstract

We construct the first physics-informed neural-network (PINN) surrogates for relativistic magnetohydrodynamics (RMHD) using a hybrid PDE and data-driven workflow. Instead of training for the conservative form of the equations, we work with Jacobians or PDE characteristics directly in terms of primitive variables. We further add to the trainable system the divergence-free condition, without the need of cleaning modes. Using a novel MUON optimizer implementation, we show that a baseline PINN trained on early-time snapshots can extrapolate RMHD dynamics in one and two spatial dimensions, and that posterior residual-guided networks can systematically reduce PDE violations.

Reconstructing Relativistic Magnetohydrodynamics with Physics-Informed Neural Networks

TL;DR

This work tackles the computational challenge of RMHD by introducing a physics-informed neural-network surrogate that operates on primitive variables via a Jacobian/characteristic formulation and enforces the divergence-free constraint. By bypassing conservative-variable inversions and employing the MUON optimizer, the approach achieves rapid training and extrapolates RMHD dynamics from limited early-time data in both 1D and 2D settings. A residual-guided correction stage further reduces PDE violations, enhancing accuracy. The framework sets the stage for extensions to GRMHD in curved spacetime and for use as priors in Bayesian inference, offering a computationally efficient RMHD surrogate with strong physical grounding.

Abstract

We construct the first physics-informed neural-network (PINN) surrogates for relativistic magnetohydrodynamics (RMHD) using a hybrid PDE and data-driven workflow. Instead of training for the conservative form of the equations, we work with Jacobians or PDE characteristics directly in terms of primitive variables. We further add to the trainable system the divergence-free condition, without the need of cleaning modes. Using a novel MUON optimizer implementation, we show that a baseline PINN trained on early-time snapshots can extrapolate RMHD dynamics in one and two spatial dimensions, and that posterior residual-guided networks can systematically reduce PDE violations.
Paper Structure (12 sections, 26 equations, 9 figures, 3 tables)

This paper contains 12 sections, 26 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. From Conservative Scheme to Characteristics
  3. Physics-Informed Neural Networks for RMHD
  4. Architecture
  5. Training
  6. Numerical Experiments
  7. 1D shocktube
  8. 2D cylindrical explosion
  9. 2D shock tube
  10. Residual-guided Correction Networks
  11. Discussion and Outlook
  12. RMHD Equations and Jacobians

Figures (9)

  • Figure 1: Representative 1D training process for 1D RMHD shocktube. The left panel shows the training convergence of the total loss function (green) and its weighted constituents --- contributions from the boundary condition (orange) and data loss at times $t{=}0.0$ (red), $t{=}0.036$ (purple) and $t{=}0.10$ (brown). The unweighted domain loss (blue) is also shown. The right panel shows the mass density $\rho$ over the domain as predicted by the model.
  • Figure 2: A tradeoff between fitting and extrapolating. The two upper panels shows the network prediction early into training (${\sim} 1000$ epochs). We see good fitting of data conditions at $t{=}0.036$ (upper-left) and poor extrapolation to $t{=}0.22$ (upper-right). The bottom two panels shows how the networks behaviour changes as the training progresses (${\sim} 7000$ epochs). Now, the quality of the network's fit to the data conditions slightly loosen, but the networks extrapolation quality increases.
  • Figure 3: 2D cylindrical test. The left panel shows the evolution of the loss functions during training. Shown are the total loss function (green), the weighted boundary loss (orange), the weighted data losses at times $t{=}0.0$ (red), $t{=}0.080$ (purple) and $t{=}0.20$ (brown), and the unweighted PDE loss (blue). The right panel is a heatmap of our PINN's prediction of the density at $t=0.40$. We refer to the RMHD-NN repository for more details.
  • Figure 4: Density levels for 2D cylindrical explosion test. Model snapshots at $\textbf{t{=}0.0}$ (left), $\textbf{t{=}0.2}$ (middle) and $\textbf{t{=}0.4}$ (right).
  • Figure 5: Comparison before (left) and after (right) imposing $\partial_i B^i =0$ constraint. We find that the constraint not only increases convergence but helps resolve the internal shock.
  • ...and 4 more figures