Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China; College of Mechanics and Materials, Hohai University, Nanjing 211100, China.
College of Mechanics and Materials, Hohai University, Nanjing 211100, China.
Neural Netw. 2024 Apr;172:106098. doi: 10.1016/j.neunet.2024.106098. Epub 2024 Jan 2.
This paper proposes an improved version of physics-informed neural networks (PINNs), the physics-informed kernel function neural networks (PIKFNNs), to solve various linear and some specific nonlinear partial differential equations (PDEs). It can also be considered as a novel radial basis function neural network (RBFNN). In the proposed PIKFNNs, it employs one-hidden-layer shallow neural network with the physics-informed kernel functions (PIKFs) as the customized activation functions. The PIKFs fully or partially contain PDE information, which can be chosen as fundamental solutions, green's functions, T-complete functions, harmonic functions, radial Trefftz functions, probability density functions and even the solutions of some linear simplified PDEs and so on. The main difference between the PINNs and the proposed PIKFNNs is that the PINNs add PDE constraints to the loss function, and the proposed PIKFNNs embed PDE information into the activation functions of the neural network. The feasibility and accuracy of the proposed PIKFNNs are validated by some benchmark examples.
本文提出了一种改进的物理信息神经网络（PINNs），即物理信息核函数神经网络（PIKFNNs），用于求解各种线性和一些特定的非线性偏微分方程（PDEs）。它也可以被认为是一种新颖的径向基函数神经网络（RBFNN）。在提出的 PIKFNNs 中，它采用了具有物理信息核函数（PIKFs）的单隐藏层浅层神经网络作为定制激活函数。PIKFs 完全或部分包含 PDE 信息，可以选择作为基本解、格林函数、T-完全函数、调和函数、径向 Trefftz 函数、概率密度函数，甚至是某些线性简化 PDE 的解等。PINNs 和所提出的 PIKFNNs 的主要区别在于，PINNs 将 PDE 约束添加到损失函数中，而所提出的 PIKFNNs 将 PDE 信息嵌入到神经网络的激活函数中。通过一些基准示例验证了所提出的 PIKFNNs 的可行性和准确性。
