  Closure models are widely used in simulating complex multiscale dynamical
systems such as turbulence and the earth system, 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 work, we propose a data-driven modeling
framework for constructing stochastic and non-local 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 machine learning techniques to
construct data-driven stochastic closure models for multiscale dynamical
systems with continuous spatiotemporal fields.
