A TVD neural network closure and application to turbulent combustion

TL;DR

This work introduces a TVD-inspired neural-network closure that is strictly constrained to prevent spurious oscillations and unphysical boundedness violations when embedded in PDEs. By formulating the NN closure as $h(q; \vec{\theta})$ with a flux representation and enforcing a CFL-bound-based feasible set via projection (and iterative rescaling for a posteriori training), the approach yields oscillation-free solutions across scalar advection, Burgers, and Euler equations, as well as in anti-diffusion scenarios. The framework is further demonstrated in large-eddy simulations of turbulent premixed flames, where the TVD NN SGS closures suppress bound violations (e.g., $Y\in[0,1]$) and outperform unconstrained or penalized variants, albeit with a finite computational overhead. The results indicate that embedding strict physical-property constraints in ML closures can enhance robustness and physical fidelity in complex reacting-flow simulations, with broad applicability to hyperbolic and parabolic PDE closures.

Abstract

Trained neural networks (NN) have attractive features for closing governing equations. There are many methods that are showing promise, but all can fail in cases when small errors consequentially violate physical reality, such as a solution boundedness condition. A NN formulation is introduced to preclude spurious oscillations that violate solution boundedness or positivity. It is embedded in the discretized equations as a machine learning closure and strictly constrained, inspired by total variation diminishing (TVD) methods for hyperbolic conservation laws. The constraint is exactly enforced during gradient-descent training by rescaling the NN parameters, which maps them onto an explicit feasible set. Demonstrations show that the constrained NN closure model usefully recovers linear and nonlinear hyperbolic phenomena and anti-diffusion while enforcing the non-oscillatory property. Finally, the model is applied to subgrid-scale (SGS) modeling of a turbulent reacting flow, for which it suppresses spurious oscillations in scalar fields that otherwise violate the solution boundedness. It outperforms a simple penalization of oscillations in the loss function.

Figures (10)

• Figure 1: Simulations of three-dimensional turbulent premixed flames: (a, c) vorticity $\bm \omega$ magnitude for (a) DNS, and (c) LES with trained NN closure; and (b, d) mass fraction $Y$ of a reactant species for (b) DNS, and (d) LES with trained NN closure. Dashed contours in (d) indicate $Y=0$ and $Y=1$, and the constraint $Y \in [0,1]$ is violated within the enclosed regions. Contours are plotted on a two-dimensional slice of the three-dimensional domain. The plot (d) is identical to \ref{['fig:flame']}(b).
• Figure 2: Schematic illustrating the difference between the unconstrained and the constrained NN closure.
• Figure 3: Demonstration for the one-dimensional advection $\partial_t q + \partial_x q = 0$. (a) The unconstrained and the TVD NN predictions at $t=t_f$; (b) learning curves of both models; and (c) TV \ref{['eqn:tv-scalar']} deviations from initial values.
• Figure 4: Demonstration for the one-dimensional Burgers equation $\partial_t q + \partial_x (q^2/2) = 0$. (a) Model predictions at $t=t_f$; (b) learning curves of both models; and (c) TV \ref{['eqn:tv-scalar']} deviations from initial values.
• Figure 5: Demonstration for the one-dimensional Euler equations. (a) Numerical solutions of $\vec{q}(x,t_f)$ predicted by both models, initial conditions, and the analytical solutions; (b) learning curves of both models.