Operator learning is an emerging field at the intersection of machine learning and scientific computing.
By learning mappings between function spaces, neural operators provide data-driven surrogate models for families of partial differential equations (PDEs).
Once trained, these models can evaluate solution operators efficiently, making them suitable for many-query applications such as real-time prediction and parameter sweeps.
However, maintaining high approximation accuracy and stable long-term predictions remains challenging for complex forward and inverse problems.
To address these challenges, we propose the Starter-Iterator Neural Operator (SINO), which incorporates the initialization and residual-correction structures of classical iterative solvers into neural operator learning.
The frequency-domain Starter captures dominant global spectral features and provides an informed initial approximation, while the latent-space Iterator applies successive residual-based corrections to refine local and multiscale solution structures.
Experiments on representative time-dependent PDEs, including the Navier-Stokes and acoustic wave equations, together with applications to image super-resolution and weather forecasting, show that SINO achieves competitive accuracy and stable performance across the benchmarks considered in this work.