Learning interpretable closures for thermal radiation transport in optically-thin media using WSINDy

TL;DR

This work introduces a WSINDy-based framework to learn interpretable moment closures for 1D optically thin thermal radiation transport, ensuring hyperbolicity, rotational symmetry, black-body equilibria, and stability via convex constraints. By augmenting the state with an auxiliary opacity moment $oldsymbol{ ext{σ}}_E E$ and modeling evolution equations for $F$ and $oldsymbol{ ext{σ}}_E E$, the authors obtain a closed hyperbolic system that preserves key physics and mitigates ray effects through weak-form learning. The closures are learned from high-fidelity data and parametrized to extrapolate across drive temperature $T_{in}$ and opacity $oldsymbol{ ext{γ}}$, with a Knudsen-like number $oldsymbol{ ext{kappa}}_L$ guiding regime validity. Results show robust extrapolation within optically thin regimes, controlled by $oldsymbol{ ext{kappa}}_L$, and demonstrate reduced sensitivity to numerical artifacts, highlighting the interpretability and potential for integration with HOLO-based TRT solvers. Overall, the paper demonstrates that physics-informed, weak-form, data-driven closures can yield accurate, generalizable moment models for challenging TRT regimes while offering clear pathways for improvement and extension.

Abstract

We introduce an equation learning framework to identify a closed set of equations for moment quantities in 1D thermal radiation transport (TRT) in optically thin media. While optically thick media admits a well-known diffusive closure, the utility of moment closures in providing accurate low-dimensional surrogates for TRT in optically thin media is unclear, as the mean-free path of photons is large and the radiation flux is far from its Fickean limit. Here, we demonstrate the viability of using weak-form equation learning to close the system of equations for the energy density, radiation flux, and temperature in optically thin TRT. We show that the WSINDy algorithm (Weak-form Sparse Identification of Nonlinear Dynamics), together with an advantageous change of variables and an auxiliary equation for the radiation-energy-weighted opacity, enables robust and efficient identification of closures that preserve many desired physical properties from the high fidelity system, including hyperbolicity, rotational symmetry, black-body equilibria, and linear stability of black-body equilibria, all of which manifest as library constraints or convex constraints on the closure coefficients. Crucially, the weak form enables closures to be learned from simulation data with ray effects and particle noise, which then do not appear in simulations of the resulting closed moment system. Finally, we demonstrate that our closure models can be extrapolated in the key system parameters of drive temperature $T_{in}$ and scalar opacity $γ$, and this extrapolation is to an extent quantifiable by a Knudsen-like dimensionless parameter.

Figures (10)

• Figure 1: Ray effects in MuDDPaRT simulations. Left and right plots show spatial profiles of the total energy $e$ and radiation flux $F$ at times $t\in \{1$e-$11s, 2$e-$11s, 3$e-$11s, 4$e-$11s\}$ for problem parameters $(\gamma,T_{in}^3)=(10^9,10^9)$, see \ref{['eq:paramgrid']}. Black and magenta curves show simulations with $M_\Omega=8$ and $M_\Omega=48$ discrete polar angles, respectively. Increasing $M_\Omega$ serves to decrease jump sizes while increasing the total number of jumps.
• Figure 2: Coefficient values for closure model \ref{['eq:extrap_closures']} plotted in log space. Left and right show $\partial_t F$ and $\partial_t (\sigma_E E)$ coefficients. Cyan dots indicate coefficient values found for training simulations. The rest of the values are extrapolated from the training values in a log-linear fashion.
• Figure 3: Left: relative $L_1$ errors ($\text{err}_{L_1}$, see eq. \ref{['eq:metrics']}) between observed $(e,F,T,\sigma_E E)$ and learned $(\widehat{e},\widehat{F},\widehat{T},\widehat{\sigma_E E})$ solution fields over the spacetime domain $(x,t)\in [0,4\text{cm}]\times [0,2$e-$10\text{s}]$ for parameters $(\gamma, T_{in}^3) \in \{10^8,10^{8.5},10^9,10^{9.5},10^{10}\}^2$. Simulations used for training are indicated with red Xs. Right: $\kappa_L$ over the parameter range.
• Figure 4: Comparison between low-resolution high-fidelity data ($M_\Omega=8$, top row), with region in spacetime used for training boxed, and resulting closure solution (middle row) at parameters $(\gamma,T_{in}^3) = (10^9,10^9)$. The bottom row shows the relative spatial error as a function of time using the $\text{err}_{L_1,j}$ and $\text{err}_{Int,j}$ metrics (see eq. \ref{['eq:metrics']}).
• Figure 5: Comparison between high-resolution high-fidelity data ($M_\Omega=48$) and resulting closure solution at $(\gamma,T_{in}^3) = (10^9,10^9)$ (compare to Figure \ref{['fig:extrap_33']}).