High Energy Density Radiative Transfer in the Diffusion Regime with Fourier Neural Operators

TL;DR

This work tackles the challenge of predicting Marshak-wave propagation in high-energy-density physics across diverse materials and drive conditions. It adopts Fourier Neural Operators (FNOs) to learn a family of solutions from input drives and material properties, introducing a base FNO aligned with the Hammer–Rosen analytic model and a Hammer–Rosen Correction FNO that maps the HR solution to a diffusion-solver solution. The results demonstrate strong generalization across unseen materials and drive functions, with the HR-Correction model achieving substantial accuracy gains over the HR baseline while remaining computationally efficient for real-time predictions. The approach provides fast, accurate surrogates for radiative transfer problems relevant to inertial confinement fusion and high-energy-density physics, enabling rapid design and analysis across wide parameter regimes.

Abstract

Radiative heat transfer is a fundamental process in high energy density physics and inertial fusion. Accurately predicting the behavior of Marshak waves across a wide range of material properties and drive conditions is crucial for design and analysis of these systems. Conventional numerical solvers and analytical approximations often face challenges in terms of accuracy and computational efficiency. In this work, we propose a novel approach to model Marshak waves using Fourier Neural Operators (FNO). We develop two FNO-based models: (1) a base model that learns the mapping between the drive condition and material properties to a solution approximation based on the widely used analytic model by Hammer & Rosen (2003), and (2) a model that corrects the inaccuracies of the analytic approximation by learning the mapping to a more accurate numerical solution. Our results demonstrate the strong generalization capabilities of the FNOs and show significant improvements in prediction accuracy compared to the base analytic model.

Figures (6)

• Figure 1: Fourier neural operator architecture for solving the Marshak wave problem. The input function $a(x)$ is projected to a higher representation $v_0(x)$ by the projection layer $P$. This is then processed through $l$ iterations of Fourier layers. Each Fourier layer consists of a Fourier transform $\mathcal{F}$ that maps $v_i(x)$ to the Fourier domain, multiplication with the weight tensor $R$ and filtering of higher Fourier modes, and an inverse Fourier transform $\mathcal{F}^{-1}$ to return to the spatial domain. The output is linearly transformed by $W$ and passed through a nonlinear activation function $\sigma$. This is added to the previous Fourier layer's output to produce the updated representation $v_{i+1}(x)$. After $l$ layers, the final representation $v_l(x)$ is mapped to the output solution $u(x)$. The boundary temperature drive (top left) and parameters (bottom left) represent the input functions and the front position (top right) and temperature distribution (bottom right) represent the output functions for the Marshak wave problem
• Figure 2: Comparison of the Hammer and Rosen approximation and the FNO model for a representative material under different boundary temperature drives (a) are characterized by a constant temperature followed by a linear ramp at different times and rates. The corresponding temperature solutions (b) obtained from the Hammer and Rosen approximation (solid lines) and the FNO model (dashed lines) show close agreement.
• Figure 3: Parameter values from the test set for four different cases to evaluate the performance of the Hammer and Rosen Correction model
• Figure 4: Comparison of the front position solutions over time for the Hammer and Rosen approximation, the Hammer and Rosen Correction model, and the diffusion solution for different sets of input parameters. The Hammer and Rosen approximation (orange lines), deviates from the diffusion solution (blue lines) over time, while the Hammer and Rosen Correction (dashed green lines) accurately predicts the diffusion solution.
• Figure 5: Comparison of the temperature profiles for the Hammer and Rosen approximation, the Hammer and Rosen Correction model, and the diffusion solution at the same time instance for different sets of input parameters. The Hammer and Rosen approximation (orange line) exhibits discrepancies compared to the diffusion solution (blue line), while the Hammer and Rosen Correction (dashed green lines) closely match the diffusion solution.