用于常微分方程和偏微分方程的物理信息神经网络和柯尔莫戈罗夫-阿诺德网络统一基准
A Unified Benchmark of Physics-Informed Neural Networks and Kolmogorov-Arnold Networks for Ordinary and Partial Differential Equations
AI总结：
本文比较了基于MLP的PINNs与基于KAN的PIKANs，发现后者在求解微分方程时更准确、收敛更快且梯度估计更优。
摘要：
物理信息神经网络（PINNs）已发展为一种强大的无网格框架，用于通过将支配的物理定律直接嵌入损失函数来求解常微分方程和偏微分方程。然而，其经典公式依赖于多层感知机（MLPs），其固定的激活函数和全局近似偏差限制了在具有振荡行为、多尺度动态或陡峭梯度的问题中的性能。与此同时，柯尔莫戈罗夫-阿诺德网络（KANs）已被引入为一种基于可学习的单变量变换的函数适应架构，沿每条边提供更丰富的局部近似和改进的表达性。本文系统地比较了标准MLP基PINNs与其KAN基对应物，即物理信息柯尔莫戈罗夫-阿诺德网络（PIKANs），使用相同的物理信息公式和匹配的参数预算来隔离架构效应。两种模型在具有已知解析解的代表性ODEs和PDEs集合中进行评估，允许直接评估梯度重建精度。结果表明，PIKANs始终能够获得更精确的解，收敛于更少的迭代次数，并产生更优的梯度估计，突显了其在物理信息学习中的优势。这些发现强调了KAN基架构作为下一代科学机器学习方法的潜力，并提供了严格的证据来指导微分方程求解中的模型选择。
英文摘要：
Physics-Informed Neural Networks (PINNs) have emerged as a powerful mesh-free framework for solving ordinary and partial differential equations by embedding the governing physical laws directly into the loss function. However, their classical formulation relies on multilayer perceptrons (MLPs), whose fixed activation functions and global approximation biases limit performance in problems with oscillatory behavior, multiscale dynamics, or sharp gradients. In parallel, Kolmogorov-Arnold Networks (KANs) have been introduced as a functionally adaptive architecture based on learnable univariate transformations along each edge, providing richer local approximations and improved expressivity. This work presents a systematic and controlled comparison between standard MLP-based PINNs and their KAN-based counterparts, Physics-Informed Kolmogorov-Arnold Networks (PIKANs), using identical physics-informed formulations and matched parameter budgets to isolate the architectural effect. Both models are evaluated across a representative collection of ODEs and PDEs, including cases with known analytical solutions that allow direct assessment of gradient reconstruction accuracy. The results show that PIKANs consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates, highlighting their advantage for physics-informed learning. These findings underline the potential of KAN-based architectures as a next-generation approach for scientific machine learning and provide rigorous evidence to guide model selection in differential equation solving.
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