发表机构
Shahid Beheshti University; Institute for Research in Fundamental Sciences (IPM); Cyberspace Research Institute, Shahid Beheshti University; University of Hertfordshire（沙希德·贝赫什提大学; 基础科学研究所（IPM）; 沙希德·贝赫什提大学网络空间研究所; 赫特福德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出PG-KINN，一种基于Petrov-Galerkin公式的物理信息KAN，用于求解正向和反向偏微分方程。其试验空间是KAN，测试空间是分段多项式空间，通过分部积分等降低微分阶数，在多基准测试中优于传统方法，为计算力学提供准确途径。
AI 中文摘要
偏微分方程（PDEs）的物理信息学习一直由多层感知器（MLPs）主导，其频谱偏差和密集参数化限制了准确性和可解释性。柯尔莫哥洛夫-阿诺德网络（KANs）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对齐。然而，将PDE转换为损失函数的方式与近似器的选择一样具有决定性：强形式残差最小化需要高阶导数和权重很大的损失，能量（Bubnov-Galerkin）形式仅限于自伴算子，并且对于参数识别问题会退化为平凡解，边界积分形式需要已知的基本解。我们提出了PG-KINN，这是一种基于Petrov-Galerkin公式构建的物理信息KAN，其中试验空间是一个KAN，测试空间是一个独立的、紧支的、用高斯-勒让德求积法评估的分段多项式空间。分部积分降低了微分阶数，同时保留了对一般非自伴、非线性和反问题的适用性；局部测试函数将全局残差转换为一组具有良好条件的元素级弱残差。在一系列涵盖裂纹奇异性、应力集中、新胡克超弹性、非均匀介质中的逆参数识别和复杂几何形状的基准测试中，PG-KINN始终优于传统的MLP基线和基于KAN的最新强/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耦合定位为基于人工智能的计算力学的一条强大而准确的途径。
英文摘要
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 known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.
