Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.