  Simulating spatiotemporal turbulence with high fidelity remains a cornerstone
challenge in computational fluid dynamics (CFD) due to its intricate multiscale
nature and prohibitive computational demands. Traditional approaches typically
employ closure models, which attempt to represent small-scale features in an
unresolved manner. However, these methods often sacrifice accuracy and lose
high-frequency/wavenumber information, especially in scenarios involving
complex flow physics. In this paper, we introduce an innovative neural
differentiable modeling framework designed to enhance the predictability and
efficiency of spatiotemporal turbulence simulations. Our approach features
differentiable hybrid modeling techniques that seamlessly integrate deep neural
networks with numerical PDE solvers within a differentiable programming
framework, synergizing deep learning with physics-based CFD modeling.
Specifically, a hybrid differentiable neural solver is constructed on a coarser
grid to capture large-scale turbulent phenomena, followed by the application of
a Bayesian conditional diffusion model that generates small-scale turbulence
conditioned on large-scale flow predictions. Two innovative hybrid architecture
designs are studied, and their performance is evaluated through comparative
analysis against conventional large eddy simulation techniques with
physics-based subgrid-scale closures and purely data-driven neural solvers. The
findings underscore the potential of the neural differentiable modeling
framework to significantly enhance the accuracy and computational efficiency of
turbulence simulations. This study not only demonstrates the efficacy of
merging deep learning with physics-based numerical solvers but also sets a new
precedent for advanced CFD modeling techniques, highlighting the transformative
impact of differentiable programming in scientific computing.
