发表机构
Zhejiang University; Nanjing University of Science and Technology; Hefei University of Technology（浙江大学; 南京理工大学; 合肥工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对动力系统场预测与源识别，提出轻量可解释的局部梯度神经算子（LGNO），利用平移不变局部核与零一致模板分解，在多种PDE基准上实现高精度、参数高效且稳定的预测。
AI 中文摘要
场的时间预测和源识别构成了动力系统中的典型问题。解决这些问题的传统方法依赖于对控制偏微分方程（PDEs）的深入理解。近年来，以神经算子为代表的深度学习为处理此类任务提供了一种数据驱动的范式。然而，大多数现有的用于PDEs的全局神经算子需要大规模的训练数据集和大量的可学习参数，且可解释性和泛化能力有限。我们提出了局部梯度神经算子（LGNO），作为典型力学问题中场的时间演化预测和源识别的一种轻量级且可解释的替代方案。该方法基于非线性梯度离散化的先验知识，利用多层感知机卷积层学习类似离散模板的平移不变局部核。零一致模板分解将系数学习与场重建分离，使学习到的算子更加透明。对于具有对称性的问题，网络折叠共享等效组件并减少参数数量。我们在涵盖线性和非线性、静态和动态、低维和高维情况的PDE基准上评估了该方法。结果表明，LGNO在这些任务中保持了准确性、参数效率和滚动稳定性，并进一步展现出对包括扩散、流动和量子现象在内的力学问题的广泛适用性。
英文摘要
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.
Comments29 pages, 11 figures. Code available at https://github.com/baiming-zhang/LGNO
