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# New computational method for multigroup radiative hydrodynamics using Artificial Intelligence: Optimisation of the Eddington factor calculation

Gonzague Radureau, Claire Michaut

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Gonzague Radureau, Claire Michaut. New computational method for multigroup radiative hydrodynamics using Artificial Intelligence: Optimisation of the Eddington factor calculation. HEDLA-2024, May 2024, Talahassee, United States. hal-05409318

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https://hal.science/hal-05409318v1

Submitted on 10 Dec 2025

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# New computational method for multigroup radiative hydrodynamics using Artificial Intelligence: Optimisation of the Eddington factor calculation

G. Radureau¹, C. Michaut¹

¹Université Côte d'Azur, CNRS, OCA, Lagrange, France

## Context

Radiative hydrodynamics models the coupling between the dynamics of a hypersonic hot plasma and the radiation it produces or external radiation. Almost every numerical codes use simplified models, that are in most cases either limited or wrong. To accurately model the photon transport the HADES 2D code was specifically developed Michaut et al. (2011, 2017); Nguyen (2011). Such a code is indispensable for studying astrophysical objects, in which optically intermediate regions are still poorly modeled, yet commonly encountered within such phenomena.

This code couples the hydrodynamics with the M1-multigroup model for radiation transfer Dubroca & Feugeas (1999), to accurately represent the spectral behavior of light, involving the partitioning of the electromagnetic spectrum into groups Turpault (2002). Nevertheless, simulating radiative hydrodynamics flows remains highly time-consuming, constraining our capacity to conduct comprehensive numerical studies within this field.

### The M1 multi-group model

The M1-multigroup model consists in discretizing the electromagnetic spectrum into groups of frequencies. To each group is assigned a radiative energy $E_g$, flux $\vec{F}_g$ and pressure $P_g$, whose evolution are described by the moment's equations for radiation:

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

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

$$ \forall g \in [0, G], \quad P_g = D_g E_g \quad \text{(Closure relation)} $$

$D_g$ : Eddington factor depending on the radiative flux and the Eddington factor $\chi_g$. In the M1-multigroup model, $\chi_g$ doesn't have an analytical expression.

## Research objectives

In the literature, there exists three methods to compute the Eddington factor in the M1 multi-group model:

* A precise, but time-consuming method using search algorithms (Nguyen (2011))

* A very fast, but inaccurate method consisting in using the analytical expression of the Eddington factor, valid only in the M1 grey model (Vaytet et al. (2011)),

* An approach based on the interpolation of pre-calculated Eddington factors (Turpault (2003); Nguyen (2011)), requiring prior physical knowledge on the radiative quantities.

Our goal is to develop a fast and accurate method to compute the Eddington factor using neural networks.

### Dependencies of the Eddington factor

Lets say $\nu_1$ and $\nu_2$ are the bounds of a considered group in the multi-group model. The radiative energy, the radiative flux and the radiative pressure of the group are defined by the expressions :

$$ \begin{aligned} E_g &= \frac{4\pi h \nu_m^4}{c^3} \int_{\sqrt{\nu_1/\nu_2}}^{\sqrt{\nu_2/\nu_1}} \int_{-1}^{1} \tilde{\nu}^3 [\exp(\tilde{\nu}(\alpha_{0,g} + \alpha_{1,g}\mu)) - 1]^{-1} d\tilde{\nu} \, d\mu \\ \|\vec{F}_g\| &= \frac{4\pi h \nu_m^4}{c^2} \int_{\sqrt{\nu_1/\nu_2}}^{\sqrt{\nu_2/\nu_1}} \int_{-1}^{1} \mu \tilde{\nu}^3 [\exp(\tilde{\nu}(\alpha_{0,g} + \alpha_{1,g}\mu)) - 1]^{-1} d\tilde{\nu} \, d\mu \\ P_g &= \frac{4\pi h \nu_m^4}{c^3} \int_{\sqrt{\nu_1/\nu_2}}^{\sqrt{\nu_2/\nu_1}} \int_{-1}^{1} \mu^2 \tilde{\nu}^3 [\exp(\tilde{\nu}(\alpha_{0,g} + \alpha_{1,g}\mu)) - 1]^{-1} d\tilde{\nu} \, d\mu \end{aligned} $$

Where $\nu_m = \sqrt{\nu_1\nu_2}$. We can compute the radiative temperature, the reduced flux and the Eddington factor using the expressions :

$$ T_g = \left( \frac{E_g}{a_R} \right)^{1/4} \qquad f_g = \frac{\|\vec{F}_g\|}{cE_g} \qquad \chi_g = \frac{P_g}{E_g} $$

According to Turpault (2002), there exists a unique couple $(\alpha_{0,g}, \alpha_{1,g})$. Thus the Eddington factor $\chi_g$ depends on the reduced flux $f_g$, the radiative temperature $\tilde{T}_g = k_B T_g / (h \nu_m)$ and the frequency ratio $\delta_\nu = \nu_1/\nu_2$.

### Multi-Layer Perceptrons (MLP)

Multi-Layer Perceptrons (MLP) are composed of elementary blocks, which are the *perceptrons*. These last are mathematical functions, containing parameters that need to be adjusted to solve a given problem. These elementary structures can be assembled into layers to reproduce more complex functions.

<figure><img src="image_2.png" /><figcaption>Multi-layer Perceptron (MLP).</figcaption></figure>

The prediction error of the MLP is evaluated via the *Loss function*, which is minimized during the training.

We have used the following loss functions :

$$ \mathcal{L}_1(f_g, \tilde{T}_g, \chi_{g,\text{data}}, \chi_{g,\text{pred}}) = \left( \frac{\chi_{g,\text{pred}} - \chi_{g,\text{data}}}{[\min(\chi_{g,\text{data}} - 1/3, 1 - f_g) + \epsilon_f][\min(1 - \log_{10}(\tilde{T}_g), \log_{10}(\tilde{T}_g)) + \epsilon_T]} \right)^2 $$

$$ \mathcal{L}_2(f_g, \tilde{T}_g, \delta_\nu, \chi_{g,\text{data}}, \chi_{g,\text{pred}}) = \frac{\mathcal{L}_1(f_g, \tilde{T}_g, \chi_{g,\text{data}}, \chi_{g,\text{pred}})}{[\min(1 - \delta_\nu, \delta_\nu) + \epsilon_\nu]^2} $$

Where $\tilde{T}_g$ and $\delta_\nu$ are the normalized radiative temperature and frequency ratio and $\epsilon_f$, $\epsilon_T$ and $\epsilon_\nu$ are constants that we successively set to $10^{-3}$, 0.1 and 0.1.

## Strategy to accurately compute the Eddington factor

We consider 4 cases according to the frequency ratio :

1. $0 \le \delta_\nu < 3.10^{-2}$ : We have cut the energy domain into 5 domains (Fig. 1a).

2. $3.10^{-2} \le \delta_\nu < 0.1$ : We have cut the energy domain into 4 domains (Fig. 1b).

3. $0.1 \le \delta_\nu < 0.9$ : We have cut the energy domain into 4 domains (Fig. 1c).

4. $0.9 \le \delta_\nu \le 1$ : We have cut the energy domain into 3 domains (Fig. 1d).

<figure><img src="image_3.png" /><figcaption>(a) Eddington factor at $f_g = 0.65$ and $\delta_\nu = 10^{-4}$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Eddington factor at $f_g = 0.65$ and $\delta_\nu = 7.10^{-2}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Eddington factor at $f_g = 0.65$ and $\delta_\nu = 0.3$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Eddington factor at $f_g = 0.65$ and $\delta_\nu = 0.99$.</figcaption></figure>

<caption>Figure 1. Radiative temperature domains : $P_1, P_2$ - Polynomial expressions depending on $f_g$; $\chi_{gr}$ - Grey case Eddington factor expression; $MLP_1, MLP_2$ - Neural networks depending on $f_g$ and $\tilde{T}_g$, trained with the loss function $\mathcal{L}_1$; $MLP_3, MLP_4$ - Neural networks depending on $f_g, \tilde{T}_g$ and $\delta_\nu$, trained with the loss function $\mathcal{L}_2$; $MLP_5$ - Neural networks depending on $f_g, \tilde{T}_g^*$ where $\tilde{T}_g^* = \tilde{T}_g / (\sqrt{10}(1 - \delta_\nu)^{1/4})$, trained with the loss function $\mathcal{L}_1$.</caption>

## Validation simulation : Marshak wave

Parameters of the Marshak shock : $T_0 = 300$ K, $\rho_0 c_v = 10^{-4}$ J.m$^{-3}$.K$^{-1}$. Imposed temperature on the left boundary : $T_l = 5780$ K. We have used five groups of frequencies.

<figure><img src="image_7.png" /><figcaption>(a) Comparison of the radiative pressure at time $t = 133$ ps, using different methods to calculate the Eddington factor in each groups.</figcaption></figure>

<table><caption>Table 1. Comparison of the simulation time using different methods for the Eddington factor.</caption><thead><tr><th>Methods</th><th>Line search</th><th>AI method</th><th>Interpolation</th><th>grey case expression</th></tr></thead><tbody><tr><td>Simulation time</td><td>16.6 days CPU</td><td>7.72 min CPU</td><td>6.88 min CPU</td><td>5.53 min CPU</td></tr></tbody></table>

## Conclusions

We have obtained a neural network approach that provides an average error on the prediction of the Eddington factor in the M1-multigroup case of $10^{-3}$ % to $10^{-2}$ % and is faster by a factor 3,000 compared to a line search algorithm and only slower by a factor 1 to 3 compared the using the grey case Eddington factor. Moreover, this approach does not need prior knowledge of the radiative energy and reduced flux, which is an advantage compared to the method using interpolations of pre-calculated Eddington factors.

Yet, the simulations we have done so far did not reveal significant differences in the hydrodynamics quantities between the different methods, even using a large number of groups or in 2D simulations. The impact of the lack of precision of the grey case Eddington factor could yet prove decisive in cases where the radiative pressure plays an important role and the reduced flux is close to 1.

However, we propose this method to serve as a reference to future work using the M1-multigroup, as it renders precise estimation of the Eddington factor computationally accessible.

## 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

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

Michaut, C., Nguyen, H., & Di Menza, L. 2011, Astrophysics and Space Science, 336, 175

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

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

Turpault, R. 2003, Theses, Université Sciences et Technologies - Bordeaux I

Vaytet, N., Audit, E., Dubroca, B., & Delahaye, F. 2011, Journal of Quantitative Spectroscopy and Radiative Transfer, 112, 1323