Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems
Abstract
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.
Related
- In-Context Learning for Multi-Fidelity Multiscale Model Closure and Cross-Operator Out-Of-Distribution Generalization
- A new operator splitting method for coupled PDE problems, with an application to poromechanics
- Infinitely many solutions to critical problems involving the p-Grushin operator
- Positive solutions to semipositone problems with mean curvature operator in Minkowski space
- Fixed-Point Formulation and Parallel Averaged Operator Algorithms for Generalized Variational Inclusion Problems
BibTeX
@misc{qin2026starteriterator,
title = {{Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems}},
author = {Kuilin Qin and Lianfang Wang and Xu Sun and Jiwei Jia and Yu Wang and Yong Wang and Yuping Duan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.18305},
eprint = {2606.18305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems
AU - Kuilin Qin
AU - Lianfang Wang
AU - Xu Sun
AU - Jiwei Jia
AU - Yu Wang
AU - Yong Wang
AU - Yuping Duan
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2606.18305
UR - https://arxiv.org/abs/2606.18305
AB - 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.
ER -