高阶ReLU-KANs（HRKANs）用于更准确、更鲁棒、更快速地求解物理信息神经网络（PINNs）
Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and faster
AI总结：
本文提出以高阶ReLU作为KANs激活函数基础的HRKANs，通过简化激活函数并保留光滑非零高阶导数，在Poisson方程和Burgers方程上显著提升PINNs的拟合精度、鲁棒性并降低训练时间。
摘要：
求解偏微分方程（PDEs）是许多科学和工程发现中重要且必不可少的组成部分。深度学习赋能的一种常见方法是物理信息神经网络（PINNs）。最近，一种新型的基础神经网络模型——Kolmogorov-Arnold网络（KANs）被提出作为多层感知器（MLPs）的替代方案，并具有可训练的激活函数。为了提高KANs的拟合精度，有人提出了一种KANs的改进版本，即所谓的ReLU-KANs，它使用“ReLU的平方”作为其激活函数的基础。在这项工作中，我们提出了另一种激活函数基础，即高阶ReLU（HR），它比KANs中使用的激活函数基础（即B样条）更简单；允许高效的KAN矩阵运算；并且具有光滑且非零的高阶导数，这对物理信息神经网络至关重要。我们将这种以高阶ReLU（HR）作为激活函数的KANs命名为HRKANs。我们对两个著名且具有代表性的PDEs，即线性Poisson方程和带粘性的非线性Burgers方程，进行了详细实验，结果表明我们提出的高阶ReLU-KANs（HRKANs）在KANs、ReLU-KANs和HRKANs中显著实现了最高的拟合精度和训练鲁棒性以及最低的训练时间。复现我们实验的代码可在https://github.com/kelvinhkcs/HRKAN获取。
英文摘要：
Finding solutions to partial differential equations (PDEs) is an important and essential component in many scientific and engineering discoveries. One of the common approaches empowered by deep learning is Physics-informed Neural Networks (PINNs). Recently, a new type of fundamental neural network model, Kolmogorov-Arnold Networks (KANs), has been proposed as a substitute of Multilayer Perceptions (MLPs), and possesses trainable activation functions. To enhance KANs in fitting accuracy, a modification of KANs, so called ReLU-KANs, using "square of ReLU" as the basis of its activation functions, has been suggested. In this work, we propose another basis of activation functions, namely, Higherorder-ReLU (HR), which is simpler than the basis of activation functions used in KANs, namely, Bsplines; allows efficient KAN matrix operations; and possesses smooth and non-zero higher-order derivatives, essential to physicsinformed neural networks. We name such KANs with Higher-order-ReLU (HR) as their activations, HRKANs. Our detailed experiments on two famous and representative PDEs, namely, the linear Poisson equation and nonlinear Burgers' equation with viscosity, reveal that our proposed Higher-order-ReLU-KANs (HRKANs) achieve the highest fitting accuracy and training robustness and lowest training time significantly among KANs, ReLU-KANs and HRKANs. The codes to replicate our experiments are available at https://github.com/kelvinhkcs/HRKAN.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
