Kolmogorov-Arnold Networks (KANs) were recently introduced as an alternative
representation model to MLP. Herein, we employ KANs to construct
physics-informed machine learning models (PIKANs) and deep operator models
(DeepOKANs) for solving differential equations for forward and inverse
problems. In particular, we compare them with physics-informed neural networks
(PINNs) and deep operator networks (DeepONets), which are based on the standard
MLP representation. We find that although the original KANs based on the
B-splines parameterization lack accuracy and efficiency, modified versions
based on low-order orthogonal polynomials have comparable performance to PINNs
and DeepONet although they still lack robustness as they may diverge for
different random seeds or higher order orthogonal polynomials. We visualize
their corresponding loss landscapes and analyze their learning dynamics using
information bottleneck theory. Our study follows the FAIR principles so that
other researchers can use our benchmarks to further advance this emerging
topic.

KANs 被引入作为 MLP 的替代表征模型，本研究中，我们应用 KANs 构建了 PIKANs 和 DeepOKANs 用于解决正向和逆向微分方程问题，与基于标准 MLP 表征的 PINNs 和 DeepONets 进行对比，发现虽然基于 B-splines 参数化的原始 KANs 精度和效率较低，但基于低阶正交多项式的修改版本性能与 PINNs 和 DeepONet 相当，尽管这些版本仍然缺乏稳健性，因为它们可能对不同的随机种子或更高阶的正交多项式发散。我们通过信息瓶颈理论可视化了相应的损失态势并分析了他们的学习动态。本研究符合 FAIR 原则，其他研究者可以利用我们的基准来进一步推动这个新兴领域。