Local gradient neural operator
Abstract
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 partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.
Related
- CLNO: Conservative Local Neural Operator for Compressible Flow Prediction
- LGNO: A Local-Global Neural Operator for Hyperbolic Conservation Laws
- Explicit sharp gradient stability of the Lp-Poincaré inequality and fundamental gaps for the Dirichlet p-Laplacian operator
- Lane-Emden equation for the superposition of a local and nonlocal operator with the "wrong sign"
- Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation
BibTeX
@misc{zhang2026local,
title = {{Local gradient neural operator}},
author = {Baiming Zhang and Jinsong Tang and Ying Xu and Lihua Chen and Shiying Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07752},
eprint = {2609.07752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07752},
}
RIS
TY - UNPB
TI - Local gradient neural operator
AU - Baiming Zhang
AU - Jinsong Tang
AU - Ying Xu
AU - Lihua Chen
AU - Shiying Xiong
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07752
UR - https://arxiv.org/abs/2609.07752
AB - 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 partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.
ER -