Uncertainty Quantification with Bayesian Higher Order ReLU KANs

TL;DR

This paper presents a novel Bayesian uncertainty quantification method for Kolmogorov-Arnold Networks, specifically Higher Order ReLU KANs, capable of estimating both epistemic and aleatoric uncertainties and applicable to stochastic PDEs.

Contribution

It introduces the first Bayesian uncertainty quantification approach for Higher Order ReLU KANs, improving computational efficiency and generalizability to other basis functions.

Findings

Successfully quantifies uncertainties in simple functions

Accurately identifies functional dependencies in stochastic PDEs

Demonstrates computational efficiency improvements

Abstract

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