The paper introduces the Frame Kernel Method, a multiscale operator learning approach for surrogate modeling of multiscale partial differential equations. It uses a novel multiscale kernel frame function approximation to cast the learning problem as one of estimating frame coefficients, enabling automatic multiscale decomposition of outputs. The authors provide interpolation proofs, error estimates, and demonstrate that the method outperforms popular neural operators on challenging PDE problems while offering a posteriori multiscale analysis.
By Branden Frieden, Ryan Whitehead, M. Keith Ballard, Robert M. Kirby, Varun Shankar
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
arXiv:2507.07292v2 Announce Type: replace
Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
By Jacob Hauck, Yanzhi Zhang
arXiv:2607. 02715v1 Announce Type: new Abstract: Recently, neural operators have shown promising outcomes for learning solution operators of differential equations directly from data.
By Himanshu Pandey, Subham Patel, Ratikanta Behera
The paper introduces the Local Gradient Neural Operator (LGNO), a lightweight and interpretable neural operator designed for field temporal evolution prediction and source identification in mechanical problems. LGNO leverages nonlinear gradient discretization priors and multilayer perceptron convolutional layers to learn translation‑invariant local kernels resembling discrete stencils, with a zero‑consistent stencil factorization that separates coefficient learning from field reconstruction. Experiments on a range of PDE benchmarks—including linear, nonlinear, static, dynamic, low‑ and high‑dimensional cases—demonstrate that LGNO achieves comparable accuracy to global neural operators while using fewer parameters and maintaining rollout stability across diffusion, flow, and quantum phenomena.
By Baiming Zhang, Jinsong Tang, Ying Xu, Lihua Chen, Shiying Xiong
Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing pa...