Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Generative Models

We propose a latent score-based generative AI framework for learning stochastic, non-local closure models and constitutive laws in nonlinear dynamical systems of computational mechanics. This work addresses a key challenge of modeling complex multiscale dynamical systems without a clear scale separation, for which numerically resolving all scales is prohibitively expensive, e.g., for engineering turbulent flows. While classical closure modeling methods leverage domain knowledge to approximate subgrid-scale phenomena, their deterministic and local assumptions can be too restrictive in regimes lacking a clear scale separation. Recent developments of diffusion-based stochastic models have shown promise in the context of closure modeling, but their prohibitive computational inference cost limits practical applications for many real-world applications. This work addresses this limitation by exploring various regularization techniques for the latent spaces, thereby discovering a latent space for successfully deploying the conditional diffusion models and significantly reducing the dimensionality of the sampling process while preserving essential physical characteristics. Numerical results demonstrate that the regularization techniques help discover a proper latent space that not only guarantees small reconstruction errors but also ensures good performance of the diffusion model in the latent space. Comparisons between score-based diffusion models and other popular generative AI approaches will also be presented. When integrated into numerical simulations, the proposed stochastic modeling framework via latent conditional diffusion models achieves significant computational acceleration while maintaining comparable predictive accuracy to standard diffusion models in physical spaces.