Trained neural networks (NN) have attractive features for closing governing equations, but in the absence of additional constraints, they can stray from physical reality. 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.

本研究解决了训练神经网络在闭合控制方程时可能偏离物理现实的问题，通过引入总变差减小（TVD）方法的灵感，提出了一种约束的神经网络模型。研究表明，该模型在湍流反应流的子网格尺度建模中有效抑制了虚假振荡，显著提升了模型的物理可靠性。

一种TVD神经网络闭合及其在湍流燃烧中的应用