[论文] PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network ...

论文概要

研究领域: ML
作者: Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee
发布时间: 2026-07-24
arXiv: 2507.18398

中文摘要

偏微分方程（PDE）的物理信息学习一直由多层感知器（MLP）主导，其光谱偏差和密集参数化限制了准确性和可解释性。Kolmogorov Arnold网络（KAN）缓解了这些限制，因为其可学习的样条激活在结构上与经典离散化的分段多项式基础对齐。然而，PDE被转化为损失泛函的方式与近似器的选择一样重要：强形式残差最小化需要高阶导数和重加权损失，能量（Bubnov-Galerkin）形式限制于自伴算子，而且正如我们所示，对于参数识别问题崩溃为平凡解，边界积分形式需要已知的基本解。我们提出了PG-KINN，一种基于Petrov-Galerkin公式的物理信息KAN，其中试验空间是KAN，测试空间是用Gauss-Legendre求积评估的独立、紧支撑、分段多项式空间。分部积分降低了微分阶数，同时保持对一般非自伴、非线性和逆问题的适用性；局部化测试函数将全局残差转变为一组具有良好条件的元素级弱残差。在一套涵盖裂纹奇异性、应力集中、Neo-Hookean超弹性、异质介质中的逆参数识别和复杂几何的基准测试中，PG-KINN始终优于传统MLP基线和最先进的基于KAN的强/能量/逆公式（PIKAN）。这些结果将KAN试验空间和多项式测试空间的Petrov-Galerkin耦合定位为AI计算力学的鲁棒且准确的路径。

原文摘要

Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a k...

自动采集于 2026-07-24

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