Probabilistic Flux Limiters

TL;DR

This work tackles Gibbs-induced oscillations in shock capturing on coarse grids by introducing a probabilistic flux limiter mechanism that replaces a single limiter $\phi(r)$ with a mixture of limiters $\{(\phi_m, p_m)\}$ learned from high-resolution data for Burgers' equation. The method encodes epistemic and aleatoric uncertainty by sampling limiters during flux computations, with a training objective that minimizes the discrepancy between low-/high-resolution predictions over a parameter space defined by coarse-graining $CG$, stencil size $K$, viscosity $\mu$, and the Dirac-delta weights. Across Burgers' equation tests, probabilistic limiters with up to $N_D=3$ Dirac components outperform deterministic limiters like van Leer and van Albada 2, especially at low viscosity, and show robustness to variations in $CG$, $K$, and $\mu$, with diminishing returns as $N_D$ increases beyond 3. The approach suggests a practical route to improved shock capturing in more complex flows by leveraging modest mixtures to account for subgrid variability and uncertainty in model parameters.

Abstract

The stable numerical integration of shocks in compressible flow simulations relies on the reduction or elimination of Gibbs phenomena (unstable, spurious oscillations). A popular method to virtually eliminate Gibbs oscillations caused by numerical discretization in under-resolved simulations is to use a flux limiter. A wide range of flux limiters has been studied in the literature, with recent interest in their optimization via machine learning methods trained on high-resolution datasets. The common use of flux limiters in numerical codes as plug-and-play blackbox components makes them key targets for design improvement. Moreover, while aleatoric (inherent randomness) and epistemic (lack of knowledge) uncertainty is commonplace in fluid dynamical systems, these effects are generally ignored in the design of flux limiters. Even for deterministic dynamical models, numerical uncertainty is introduced via coarse-graining required by insufficient computational power to solve all scales of motion. Here, we introduce a conceptually distinct type of flux limiter that is designed to handle the effects of randomness in the model and uncertainty in model parameters. This new, {\it probabilistic flux limiter}, learned with high-resolution data, consists of a set of flux limiting functions with associated probabilities, which define the frequencies of selection for their use. Using the example of Burgers' equation, we show that a machine learned, probabilistic flux limiter may be used in a shock capturing code to more accurately capture shock profiles. In particular, we show that our probabilistic flux limiter outperforms standard limiters, and can be successively improved upon (up to a point) by expanding the set of probabilistically chosen flux limiting functions.

Figures (5)

• Figure 1: A probabilistic flux limiter expands the flux limiter concept from an individual, deterministic interpolating function (top panel) to a set of interpolating functions applied probabilistically with probabilities drawn with replacement from a distribution learned from high-resolution data (bottom panel). In the example shown here, the probabilistic limiter consists of two flux limiters randomly selected with probability $p$ and $1-p$ for each flux computation.
• Figure 2: Cartoon of the hypercube of inputs for the optimization of a probabilistic flux limiter. A mean constraint on viscosity ($\mu$) is visualized as a fulcrum, and a variance constraint on $\mu$ is seen as a spring. Inputs to a flux limiter are specified by coarse graining ($CG$), total degrees of freedom of the limiter ($K$), and probability-weighted viscosity $(\mu, p)$. In this figure, the probability distribution of possible flux limiters is modeled by a discrete distribution of $\mu$ composed of $N_D = 2$ Dirac delta functions optimized over the parameter space $(CG, K, \mu, p)$ with $\sum_{m}^{N_D} p_m = 1$. In the text, we explore distributions of $\mu$ with up to $N_D=3$.
• Figure 3: Optimization of a probabilistic flux limiter with $N_D = 3$. (a) The optimized forms of the $3$ flux limiting functions $\phi_1$, $\phi_2$, and $\phi_3$ after $60$ iterations, the thicker the line the larger the probability. The probability distribution of the $3$ corresponding viscosity values, $\mu_1, \mu_2$, and $\mu_3$ (also shown in (b)) as a function of training iteration are depicted in the inset. (b) A plot of the $3$ viscosity values $\mu_1, \mu_2$, and $\mu_3$, bin number $K$, and coarse graining $CG$ versus iteration step. (c) The value of the cost function versus iteration step. Optimization was performed on the entire parameter space $CG=[1,2,\dots,10], K=[2,3,\dots,38], \mu=[0.002, 0.03$] with $p=p_1 + p_2 + p_3 = 1$.
• Figure 4: Shock reconstruction in the case of Burgers' equation learned with $N_D \in [1, 2, 3]$ Dirac delta functions as compared to van Leer and van Albada 2 cases. Ground truth is plotted as blue connected circles (HD). The inset is a magnification of the solution at the upper left of the shock. Note that the probabilistic flux limiters in this figure are plotted as gray ($N_D=2$) and black ($N_D=3$) bands indicating mean plus and minus standard deviation. The probabilistic flux limiter with $N_D = 3$ used in this plot is given in Tables \ref{['tab:threeDiracs_r']} and \ref{['tab:threeDiracs_b']}.
• Figure A1: Probabilistic flux limiter with coarse graining constraint. A probabilistic flux limiter obtained with $N_D = 3$ and constraint $CG=8$. Top inset is parameter convergence with respect to optimization iteration and bottom inset is probability $p_1, p_2, p_3$ corresponding to the weights of the $3$ Dirac delta functions as a function of optimization iteration.