We introduce the first method of uncertainty quantification in the domain of Kolmogorov-Arnold Networks, specifically focusing on (Higher Order) ReLUKANs to enhance computational efficiency given the computational demands of Bayesian methods. The method we propose is general in nature, providing access to both epistemic and aleatoric uncertainties. It is also capable of generalization to other various basis functions. We validate our method through a series of closure tests, including simple one-dimensional functions and application to the domain of (Stochastic) Partial Differential Equations. Referring to the latter, we demonstrate the method's ability to correctly identify functional dependencies introduced through the inclusion of a stochastic term. The code supporting this work can be found at https://github.com/wmdataphys/Bayesian-HR-KAN

翻译：本文首次在Kolmogorov-Arnold Networks领域引入不确定性量化方法，特别聚焦于（高阶）ReLUKANs以提升计算效率，这主要针对贝叶斯方法的高计算需求。我们所提出的方法具有普适性，能够同时获取认知不确定性和随机不确定性。该方法还可推广至其他多种基函数。我们通过一系列闭合测试验证了该方法的有效性，包括简单的一维函数测试以及在（随机）偏微分方程领域的应用。针对后者，我们展示了该方法能够正确识别因引入随机项而产生的函数依赖关系。本工作的支持代码可在https://github.com/wmdataphys/Bayesian-HR-KAN 获取。