Learning Stochastic Multiscale Models

The physical sciences are replete with dynamical systems that require the resolution of a wide range of length and time scales. This presents significant computational challenges since direct numerical simulation requires discretization at the finest relevant scales, leading to a high-dimensional state space. In this work, we propose an approach to learn stochastic multiscale models in the form of stochastic differential equations directly from observational data. Drawing inspiration from physics-based multiscale modeling approaches, we resolve the macroscale state on a coarse mesh while introducing a microscale latent state to explicitly model unresolved dynamics. We learn the parameters of the multiscale model using a simulator-free amortized variational inference method with a Product of Experts likelihood that enforces scale separation. We present detailed numerical studies to demonstrate that our learned multiscale models achieve superior predictive accuracy compared to under-resolved direct numerical simulation and closure-type models at equivalent resolution, as well as reduced-order modeling approaches.

Recent work in weather forecasting has employed hierarchical graph neural networks operating on multiple mesh resolutions to learn auto-regressive models [33,34]. In contrast, our multiscale approach learns continuous-time coupled SDEs governing macroscale and microscale dynamics, performing test-time simulations only on a single coarse grid and using high-resolution grids solely as decoding targets.