Local gradient neural operator
Researchers propose LGNO, a lightweight interpretable neural operator using learnable local stencils, matching global-operator accuracy on PDE benchmarks with fewer parameters.
LGNO builds on nonlinear gradient discretization priors and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels resembling discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, and network folding shares equivalent components to cut parameter counts for symmetric problems. Evaluations on linear and nonlinear, static and dynamic, and low- and high-dimensional PDE benchmarks show maintained accuracy, parameter efficiency, and rollout stability, with applicability to diffusion, flow, and quantum problems.
- Targets field temporal prediction and source identification without full knowledge of governing PDEs
- Learns translation-invariant local kernels resembling discrete stencils via MLP convolutional layers
- Network folding reduces parameters for problems with symmetries
- Evaluated across diffusion, flow, and quantum mechanical PDE benchmark tasks
Full article195 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.07752