We introduce a data-driven learning framework that assimilates two powerful ideas: ideal large eddy simulation (LES) from turbulence closure modeling and neural stochastic differential equations (SDE) for stochastic modeling. The ideal LES models the LES flow by treating each full-order trajectory as a random realization of the underlying dynamics, as such, the effect of small-scales is marginalized to obtain the deterministic evolution of the LES state. However, ideal LES is analytically intractable. In our work, we use a latent neural SDE to model the evolution of the stochastic process and an encoder-decoder pair for transforming between the latent space and the desired ideal flow field. This stands in sharp contrast to other types of neural parameterization of closure models where each trajectory is treated as a deterministic realization of the dynamics. We show the effectiveness of our approach (niLES - neural ideal LES) on a challenging chaotic dynamical system: Kolmogorov flow at a Reynolds number of 20,000. Compared to competing methods, our method can handle non-uniform geometries using unstructured meshes seamlessly. In particular, niLES leads to trajectories with more accurate statistics and enhances stability, particularly for long-horizon rollouts.

翻译：我们提出了一种数据驱动的学习框架，融合了两大核心思想：来自湍流闭合建模的理想大涡模拟（LES）与用于随机建模的神经随机微分方程（SDE）。理想大涡模拟将每一全阶轨迹视为底层动力学的随机实现，从而对LES流进行建模，通过对小尺度效应进行边缘化处理，获得LES状态的确定性演化。然而，理想大涡模拟在解析上难以处理。在本工作中，我们采用潜变量神经SDE来建模随机过程的演化，并借助编码器-解码器对实现潜空间与目标理想流场之间的变换。这与将每条轨迹视为动力学确定性实现的其他类型闭合模型神经参数化方法形成鲜明对比。我们通过具有挑战性的混沌动力系统——雷诺数为20,000的科尔莫戈罗夫流——验证了所提方法（niLES——神经理想大涡模拟）的有效性。与竞争方法相比，本方法可利用非结构化网格无缝处理非均匀几何构型。特别地，niLES生成的轨迹具有更精确的统计特性，并增强了稳定性，尤其在长时域推演中表现突出。