对比MLP与KAN在物理问题中的表现，发现浅层KAN更优。
MLPs and KANs for data-driven learning in physical problems: A performance comparison
- 用浅层网络比较MLP与KAN在物理建模中的表现
- 浅层KAN精度显著高于MLP，深层则无明显优势
- 适合追求效率与精度平衡的物理系统建模任务
求解偏微分方程（PDEs）逐渐转向机器学习框架。近年来，科尔莫戈罗夫-阿诺尔德网络（KANs）作为多层感知机（MLPs）的替代方案受到关注。尽管前景可观，其在物理问题中的性能优势仍不明确。本文通过对比KANs与MLPs在深度算子网络（DeepONet）和图网络模拟器（GNS）中的表现，测试了它们在不同尺度与复杂度物理问题上的能力。基于科尔莫戈罗夫表示定理，研究了浅层与深层架构下的行为差异。结果表明：虽然深层网络中KANs未持续优于MLPs，但在浅层设置下，其表达能力更强，多个测试案例中精度显著超越MLPs。这表明KANs是物理系统建模中兼顾效率与准确性的有力选择。
原文摘要 · Abstract (English)
There is increasing interest in solving partial differential equations (PDEs) by casting them as machine learning problems. Recently, there has been a spike in exploring Kolmogorov-Arnold Networks (KANs) as an alternative to traditional neural networks represented by Multi-Layer Perceptrons (MLPs). While showing promise, their performance advantages in physics-based problems remain largely unexplored. Several critical questions persist: Can KANs capture complex physical dynamics and under what conditions might they outperform traditional architectures? In this work, we present a comparative study of KANs and MLPs for learning physical systems governed by PDEs. We assess their performance when applied in deep operator networks (DeepONet) and graph network-based simulators (GNS), and test them on physical problems that vary significantly in scale and complexity. Drawing inspiration from the Kolmogorov Representation Theorem, we examine the behavior of KANs and MLPs across shallow and deep network architectures. Our results reveal that although KANs do not consistently outperform MLPs when configured as deep neural networks, they demonstrate superior expressiveness in shallow network settings, significantly outpacing MLPs in accuracy over our test cases. This suggests that KANs are a promising choice, offering a balance of efficiency and accuracy in applications involving physical systems.
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