# A TVD neural network closure and application to turbulent combustion

arXiv: https://arxiv.org/abs/2408.03413 · HTML: https://mlsanity.com/paper/2408.03413/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 6 Aug 2024.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures](https://mlsanity.com/paper/2105.14410/) (2021)
- [Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds](https://mlsanity.com/paper/2109.00700/) (2021)

## Built on by (most cited first, 2 of 2)

- [Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws](https://mlsanity.com/paper/2507.01795/) (2025)
- [Improving Reduced-Order Rotating Detonation Engine Models with Data Assimilation and Machine Learning](https://mlsanity.com/paper/2609.16237/) (2026)

## Abstract (arXiv)

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.