Local gradient neural operator predicts physical systems with fewer data
Local gradient neural operator
Machine Learning
Summary
Predicting how things like heat or fluid flow change over time usually needs complex math and lots of data. The authors created a new method called the local gradient neural operator that simplifies this by focusing on small local patterns instead of the whole system at once. This approach uses fewer parameters, is easier to understand, and works well even with limited data. They tested it on different types of problems like diffusion and quantum mechanics, showing it can predict these systems accurately and efficiently.
partial differential equationsneural operatorsgradient discretizationmultilayer perceptronconvolutional layersstencil factorizationparameter efficiencyfield temporal predictionsource identificationdynamical systems
Authors
Baiming Zhang, Jinsong Tang, Ying Xu, Lihua Chen, Shiying Xiong
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.