We propose a new delay neural operator applicable to both large-scale advective flow field prediction and corresponding subgrid-scale closure. Our operator optimizes computation by focusing solely on input fields and correcting for any unseen field influences due to model truncation, coarsening, or aggregation of the full-order model. Compressing input fields to a latent space efficiently enables arbitrary output resolution without storing a complete, discretized system state in memory. Additionally, unseen fields are never computed, unlike classic numerical and many deep-learning Markov process models.  

We construct the delay neural operator by extending neural delay differential equations to 2D and higher dimensions. Inspired by the Mori-Zwanzig formulation, neural delay differential equations and neural closure models perform temporal convolution or kernel integration to accumulate hidden processes (a distributed delay), approximating unseen field effects without storing additional variables. Linear, discretized versions of this distributed delay (discrete delays) have been used to develop effective reduced-order models. We extend these distributed and discrete delays to neural operators. In particular, we present discrete delayed RNNs as a superset of Picard iteration-performing neural operators. We explore multiscale and scale-invariant architectures, enabling arbitrary input and output resolution flow fields. We also investigate the origin and extension of physical representations—concepts of waves, eddies, and vortices—through network layers. Towards an efficient backpropagation with constant memory (which is independent of the number of layers), we simplify adjoint computation and explore integral-free alternatives, including Suzuki-Trotter operator-splitting and simple discretization. Tests are performed against simulated 2D viscous Burger’s equation with Smagorinsky closure, 2D homogeneous isotropic, quasi-geostrophic beta-plane turbulence (2D-HIT QG), and data-assimilated ocean surface velocity simulations.