Learning Stochastic Closures via Conditional Diffusion Model and Neural Operator*

Closure models are widely used in simulating complex multiscale dynamical systems such as turbulence and Earth's climate, for which direct numerical simulation that resolves all scales is often too expensive. For those systems without a clear scale separation, deterministic and local closure models often lack enough generalization capability, which limits their performance in many real-world applications. In this talk, we present a data-driven modeling framework for constructing stochastic and nonlocal closure models via conditional diffusion model and neural operator. Specifically, the Fourier neural operator is incorporated into a score-based diffusion model, which serves as a data-driven stochastic closure model for complex dynamical systems governed by partial differential equations (PDEs). We also demonstrate how accelerated sampling methods can improve the efficiency of the data-driven stochastic closure model. The results show that the proposed methodology provides a systematic approach via generative AI to constructing generalizable data-driven stochastic closure models for multiscale dynamical systems with continuous spatiotemporal fields.