Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems

Operator learning is an emerging interdisciplinary field that integrates machine learning with scientific computing. By mapping infinite-dimensional function spaces, this approach provides an efficient surrogate modeling framework for high-dimensional partial differential equations (PDEs). Compared to traditional numerical solvers, it achieves a superior trade-off between computational complexity and approximation accuracy, demonstrating significant advantages in many-query tasks such as real-time prediction and parameter sweeps. Given the stringent accuracy requirements of both forward simulation and inverse inference, as well as the precision bottlenecks of existing operator learning methods in handling complex boundaries or long-term evolution, we propose the Starter-Iterator Neural Operator (SINO). Our framework reinterprets the initialization strategies and iterative formats of traditional iterative methods through neural networks, establishing an efficient approach for spectral-spatiotemporal collaborative modeling. Specifically, the frequency-domain initialization module captures globally stable low-frequency features, while the time-domain learning module focuses on optimizing local solution residuals, thereby effectively overcoming the inherent limitations of conventional single-domain modeling approaches. Extensive experiments on typical dynamical systems such as the Navier-Stokes equations and acoustic wave equations, as well as practical applications including super-resolution imaging and weather forecasting, demonstrate that SINO achieves outstanding performance in numerical accuracy, generalization capability, and robustness.

Orthogonal greedy algorithm for linear operator learning with shallow neural network

Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial differential equations (PDEs). In this paper, we extend the application of OGA to the tasks of linear operator learning, which is equivalent to learning the kernel function through integral transforms. Firstly, a novel greedy algorithm is developed for kernel estimation rate in a new semi-inner product, which can be utilized to approximate the Green's function of linear PDEs from data. Secondly, we introduce the OGA for point-wise kernel estimation to further improve the approximation rate, achieving orders of accuracy improvement across various tasks and baseline models. In addition, we provide a theoretical analysis on the kernel estimation problem and the optimal approximation rates for both algorithms, establishing their efficacy and potential for future applications in PDEs and operator learning tasks.

Green Multigrid Network

GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves $3.8\%$ to $39.15\%$ accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about $10\%$ of the full grid data, resulting in a $55.9\%$ and $92.5\%$ reduction in training time and GPU memory cost for one-dimensional test problems, and a $37.7\%$ and $62.5\%$ reduction for two-dimensional test problems.