arXiv:2606. 18305v1 Announce Type: cross Abstract: Operator learning is an emerging interdisciplinary field that integrates machine learning with scientific computing.
By Kuilin Qin, Lianfang Wang, Xu Sun, Jiwei Jia, Yu Wang, Yong Wang, Yuping Duan
arXiv:2503. 05598v2 Announce Type: replace-cross Abstract: This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation.
By Prashant K. Jha
arXiv:2608. 05839v1 Announce Type: cross Abstract: Deep neural networks have shown great empirical success in the solution of a wide variety of ill-posed inverse problems in imaging.
By Alexander Auras, Martin Burger, Samira Kabri, Michael Moeller, Michael Schopf-Kuester
arXiv:2606. 30495v1 Announce Type: cross Abstract: Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors.
By Jiwei Jia, Xinliang Liu, Juntao Wang, Jinchao Xu
arXiv:2602. 06842v2 Announce Type: replace-cross Abstract: Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence.
By Yuhan Wu, Jan Willem van Beek, Victorita Dolean, Alexander Heinlein
The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.
By Zhiping Mao, Zhenye Wen, Yong Zhang, Xiaofei Zhao