发表机构
Texas A&M University-Corpus Christi（德克萨斯农工大学科珀斯克里斯蒂分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基准测试了MLP与B样条/切比雪夫KAN作为非线性常微分方程求解器，发现KAN在多数问题上精度更高，且B样条KAN在Falkner-Skan问题上精度提升显著。
AI 中文摘要
本文比较了三种神经网络作为非线性常微分方程求解器的性能：多层感知器（MLP）、边为B样条的Kolmogorov-Arnold网络（KAN）以及边为切比雪夫级数的KAN。每个网络仅基于方程的残差及其初始条件或边界条件进行训练。精确解或精确到$10^{-13}$的数值解仅用于事后测量误差。研究了五个微分阶数从一阶到四阶的测试问题：逻辑斯蒂方程、奇异Lane-Emden型初值问题、奇异边值问题、带吸气的Falkner-Skan方程以及一个四阶问题。所有网络分配相同的训练预算和相同的五个激活函数选择。在\nRuns{}次训练运行中未遇到任何数值失败。对于每个问题，最佳KAN的精度至少与最佳MLP相当，而使用的参数数量为\paramsKANmin{}到\paramsKANmax{}，相比之下MLP使用\paramsMLP{}个参数；差距从持平到三倍不等。在早期研究的Falkner-Skan设置中，B样条KAN在每个已发布的测试点上精度高出34到220倍。更小的MLP也被发现具有竞争力，而在三阶和四阶问题上，KAN更频繁地遇到失败，需要粗糙的基函数。方程中的导数被B样条KAN的常规网格扩展所扭曲；因此，用节点插入替代网格扩展，使得逻辑斯蒂误差降低了一百多倍，尽管在四阶问题上没有改善。
英文摘要
Three neural networks are compared as solvers of nonlinear ordinary differential equations: a multilayer perceptron (MLP), a Kolmogorov-Arnold network (KAN) whose edges are B-splines, and a KAN whose edges are Chebyshev series. Each network is trained only on the residual of the equation and on its initial or boundary conditions. The exact solution, or a numerical solution accurate to $10^{-13}$, is used only afterward to measure the error. Five test problems of differential orders one to four are investigated: the logistic equation, a singular Lane-Emden type initial value problem, a singular boundary value problem, the Falkner-Skan equation with suction, a fourth-order problem. The same training budget and the same choice of five activation functions are allocated to all networks. No numerical failures are encountered in any of the \nRuns{} training runs. For every problem, the best KAN is found to be at least as accurate as the best MLP while \paramsKANmin{} to \paramsKANmax{} parameters are used instead of \paramsMLP{}; the margin ranges from a tie to a factor of three. In the Falkner-Skan setting of an earlier study, the B-spline KAN is shown to be 34 to 220 times more accurate at every published test point. Much smaller MLPs are also found to be competitive, whereas at orders three and four, failures are experienced more frequently by KANs, and a coarse basis is required. The derivatives in the equation are distorted by the usual grid extension of a B-spline KAN; consequently, it is replaced with knot insertion, by which the logistic error is lowered more than a hundredfold, though no improvement is provided at order four.
