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# Using Artificial Intelligence to speed up radiative hydrodynamics simulations

Gonzague Radureau, Claire Michaut

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## To cite this version:

Gonzague Radureau, Claire Michaut. Using Artificial Intelligence to speed up radiative hydrodynamics simulations. SF2A-2025, Jul 2025, Toulouse, France. hal-05409347

HAL Id: hal-05409347

https://hal.science/hal-05409347v1

Submitted on 10 Dec 2025

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## Context

Radiative hydrodynamics describes the interaction between hypersonic hot plasmas and radiation, whether self-emitted or external. The HADES code was specifically developed to model photon transport accurately in such contexts (Michaut et al. 2017; Nguyen 2011), particularly in optically intermediate regimes often encountered in astrophysical phenomena, yet still poorly understood.

HADES couples hydrodynamics with the M1-multigroup radiation model (Dubroca & Feugeas 1999), allowing spectral resolution through frequency group partitioning (Turpault 2002). However, such simulations are computationally expensive, limiting the scope of systematic studies in this field.

## Research objectives

Two main factors drive the high computational cost of these simulations:

1. the M1-multigroup closure relation requires expensive numerical searches,

2. the CFL condition forces very small time steps in explicit schemes.

Objectives:

* develop a fast neural network-based method to compute the Eddington factor,

* explore Physics-Informed Neural Networks' use for radiation hydrodynamics simulations.

## The M1 multi-group model

The M1-multigroup model discretizes the electromagnetic spectrum into *G* frequency groups. Each group *g* is associated with a radiative energy $E_g$, flux $\vec{F}_g$, and pressure tensor $\mathbb{P}_g$, governed by the radiation moment equations:

$$ \forall g \in [0, G], \partial_t E_g + \vec{\nabla} \cdot \vec{F}_g = -c S_g^0 \quad (\text{Energy}) $$

$$ \forall g \in [0, G], \partial_t(c^{-2}\vec{F}_g) + \vec{\nabla} \cdot \mathbb{P}_g = -\vec{S}_g \quad (\text{Momentum}) $$

$$ \forall g \in [0, G], \mathbb{P}_g = \mathbb{D}_g E_g \quad (\text{Closure}) $$

The closure tensor $\mathbb{D}_g$ depends namely on the Eddington factor $\chi_g$, a scalar function. In the M1-multigroup model, $\chi_g$ has no analytical expression and must be estimated numerically.

## Dependencies of the Eddington factor

Let $\nu_1$ and $\nu_2$ denote the frequency bounds of a given group in the M1-multigroup model. In the M1-multigroup formulation, the Eddington factor depends on three key quantities:

1. **reduced flux:** $f_g = \|\vec{F}_g\|/(cE_g) \in [0, 1]$, quantifying the anisotropy of the radiation field,

2. **adimensionalized radiative temperature:** $T_g = k_B E_g^{1/4}/(h a_R^{1/4} \sqrt{\nu_1 \nu_2}) \in \mathbb{R}_+$, comparing the characteristic radiative temperature to the photon energies in the group,

3. **group narrowness:** $\delta_g = \nu_1/\nu_2 \in [0, 1]$, indicating the width of the frequency interval.

<figure><img src="image_3.png" /><figcaption>Figure 1. (Upper right) Variation of the Eddington factor with respect to $f_g$ (other figures) Variation of the Eddington factor according to $T_g$ for different values of $\delta_g$ at $f_g = 0.65$.</figcaption></figure>

## Al-method outlines

This method uses different methods according to the values of $\delta_g$ and $T_g$ (Radureau et al. 2025):

* polynomial approximations (domains L and H),

* M1-grey model analytical expression (domain $I_b$,

* neural networks (domains $I_a$, $I_c$, $I_d$, $I_e$, $I_f$.

This method enables to reduce the simulation time up to a factor 3 000, while maintaining a high precision!

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## Physics-Informed Neural Networks (PINNs)

PINNs assess model performance using three components (Raissi et al. 2019):

1. $\mathcal{L}_{IC}$: measures how well initial conditions are satisfied,

2. $\mathcal{L}_{BC}$: measures compliance with boundary conditions,

3. $\mathcal{L}_{eq}$: measures how well physics equations are respected,

The equation loss has the form:

$$ \mathcal{L}_{eq} = \frac{1}{Neq} \sum_{i=1}^{Neq} \lambda \left\{ \partial_t \mathbf{U}_i + \vec{\nabla} \cdot \mathbf{F}[\mathbf{U}_i] - \mathbf{S}[\mathbf{U}_i] \right\}^2 $$

The factor $\lambda$ helps capture discontinuities (Liu et al. 2023):

$$ \lambda = \frac{1}{1 + (|\vec{\nabla} \cdot \vec{v}| - |\vec{\nabla} \cdot \vec{v}_i|)} $$

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## Data repartition

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## Extrapolation results

Results:

* the PINN strategy extrapolates very well hydrodynamic shock simulations,

* the PINN strategy for the moment can reproduce the simulation data, but does not extrapolate radiative shock simulations.

<figure><img src="image_7.png" /><figcaption>Figure 2. Extrapolation of a hydrodynamic shock simulation.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 3. Extrapolation of a radiative shock simulation.</figcaption></figure>

## Conclusions

We developed a neural network achieving an average Eddington factor error of $10^{-3}\% - 10^{-2}\%$, running **3 000 times faster** than classical search algorithms in HADES.

To further cut simulation costs, we tested PINNs to extrapolate shocks. So far, this worked well in a purely hydrodynamic simulation, but still fails in a radiative hydrodynamics simulation.

## Acknowledgements

This work was supported by a French government grant managed by the Agence Nationale de la Recherche under the Investissement d'avenir program, reference ANR-19-P3IA-0002.

## References

Dubroca, B. & Feugeas, J.-L. 1999, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 329, 915

Liu, L., Liu, S., Xie, H., et al. 2023, Journal of Scientific Computing, 98, 22

Michaut, C., Di Menza, L., Nguyen, H., Bouquet, S., & Mancini, M. 2017, High Energy Density Physics, 22, 77

Nguyen, H. C. 2011, PhD thesis, Paris 11

Radureau, G., Michaut, C., & Comport, A. I. 2025, Phys. Rev. E, 111, 035301

Raissi, M., Perdikaris, P., & Karniadakis, G. 2019, Journal of Computational Physics, 378, 686

Turpault, R. 2002, Comptes Rendus. Mathématique, 334, 331