We propose a localized physics-informed kernel function neural network (LPIKFNN), which is an improved physics-informed neural network (PINN) based on physics-informed kernel function (PIKF). In the LPIKFNN framework, the localized collocation scheme discretizes the physical quantities within the local domain, where the physical field is represented as a linear combination of PIKFs. Based on this representation, the multilayer perceptron is trained to iteratively learn the physical quantities. To overcome the computational challenges of conventional PINN in higher-order derivative and high wavenumber problems, the LPIKFNN constructs the loss function using the PIKF and a localized collocation scheme rather than relying on automatic differentiation. As a result, the costly derivative evaluations required to enforce governing equations during iterative training are eliminated, leading to significantly improved computational efficiency and training performance. Moreover, incorporating PIKFs into the loss function enables the proposed LPIKFNN to significantly improve computational accuracy in high-wavenumber problems characterized by highly oscillatory physical fields. To overcome the computational bottleneck of the physics-informed kernel function neural network (PIKFNN) in heterogeneous problems, the LPIKFNN introduces a localized collocation scheme that removes reliance on global PIKFs, enabling accurate predictions where global PIKFs are unavailable. The feasibility and accuracy of the proposed LPIKFNN are demonstrated through a series of benchmark studies, including high wavenumber problems, higher-order derivative problems, nonlinear problems, heterogeneous problems, and potential-based inverse electromyography. The numerical predictions obtained by LPIKFNN show excellent agreement with available analytical solutions and experimental measurements.

翻译：本文提出一种局域物理信息核函数神经网络（LPIKFNN），该网络是基于物理信息核函数（PIKF）的改进型物理信息神经网络（PINN）。在LPIKFNN框架中，局域配点格式对局域域内的物理量进行离散化处理，其中物理场被表示为PIKF的线性组合。基于该表示形式，多层感知机被训练以迭代学习物理量。为克服传统PINN在高阶导数和高波数问题中的计算挑战，LPIKFNN利用PIKF和局域配点格式而非依赖自动微分来构造损失函数。因此，迭代训练过程中执行控制方程所需的高成本导数评估得以消除，从而显著提升计算效率与训练性能。此外，将PIKF融入损失函数使所提出的LPIKFNN能够在以高振荡物理场为特征的高波数问题中大幅提高计算精度。为克服物理信息核函数神经网络（PIKFNN）在异质问题中的计算瓶颈，LPIKFNN引入一种局域配点格式，摆脱了对全局PIKF的依赖，从而在全局PIKF不可用时仍能实现精确预测。通过一系列基准研究（包括高波数问题、高阶导数问题、非线性问题、异质问题以及基于势的反向肌电图），论证了所提LPIKFNN的可行性与准确性。LPIKFNN获得的数值预测与现有解析解及实验测量结果高度吻合。