  Predicting and understanding the chaotic dynamics in complex systems is
essential in various applications. However, conventional approaches, whether
full-scale simulations or small-scale omissions, fail to offer a comprehensive
solution. This instigates exploration into whether modeling or omitting
small-scale dynamics could benefit from the well-captured large-scale dynamics.
In this paper, we introduce a novel methodology called Neural Downscaling (ND),
which integrates neural operator techniques with the principles of inertial
manifold and nonlinear Galerkin theory. ND effectively infers small-scale
dynamics within a complementary subspace from corresponding large-scale
dynamics well-represented in a low-dimensional space. The effectiveness and
generalization of the method are demonstrated on the complex systems governed
by the Kuramoto-Sivashinsky and Navier-Stokes equations. As the first
comprehensive deterministic model targeting small-scale dynamics, ND sheds
light on the intricate spatiotemporal nonlinear dynamics of complex systems,
revealing how small-scale dynamics are intricately linked with and influenced
by large-scale dynamics.
