Kolmogorov-Arnold Networks have emerged as interpretable alternatives to traditional multi-layer perceptrons. However, standard implementations lack principled uncertainty quantification capabilities essential for many scientific applications. We present a framework integrating sparse variational Gaussian process inference with the Kolmogorov-Arnold topology, enabling scalable Bayesian inference with computational complexity quasi-linear in sample size. Through analytic moment matching, we propagate uncertainty through deep additive structures while maintaining interpretability. We use three example studies to demonstrate the framework's ability to distinguish aleatoric from epistemic uncertainty: calibration of heteroscedastic measurement noise in fluid flow reconstruction, quantification of prediction confidence degradation in multi-step forecasting of advection-diffusion dynamics, and out-of-distribution detection in convolutional autoencoders. These results suggest Sparse Variational Gaussian Process Kolmogorov-Arnold Networks (SVGP KANs) is a promising architecture for uncertainty-aware learning in scientific machine learning.

Kolmogorov-Arnold Networks (KANs) offer a promising alternative to Multi-Layer Perceptron (MLP) by placing learnable univariate functions on network edges, enhancing interpretability. However, standard KANs lack probabilistic outputs, limiting their utility in applications requiring uncertainty quantification. While recent Gaussian Process (GP) extensions to KANs address this, they utilize exact inference methods that scale cubically with data size N, restricting their application to smaller datasets. We introduce the Sparse Variational GP-KAN (SVGP-KAN), an architecture that integrates sparse variational inference with the KAN topology. By employing $M$ inducing points and analytic moment matching, our method reduces computational complexity from $O(N^3)$ to $O(NM^2)$ or linear in sample size, enabling the application of probabilistic KANs to larger scientific datasets. Furthermore, we demonstrate that integrating a permutation-based importance analysis enables the network to function as a framework for structural identification, identifying relevant inputs and classifying functional relationships.

The integration of multi-omics data presents a major challenge in precision medicine, requiring advanced computational methods for accurate disease classification and biological interpretation. This study introduces the Multi-Omics Graph Kolmogorov-Arnold Network (MOGKAN), a deep learning model that integrates messenger RNA, micro RNA sequences, and DNA methylation data with Protein-Protein Interaction (PPI) networks for accurate and interpretable cancer classification across 31 cancer types. MOGKAN employs a hybrid approach combining differential expression with DESeq2, Linear Models for Microarray (LIMMA), and Least Absolute Shrinkage and Selection Operator (LASSO) regression to reduce multi-omics data dimensionality while preserving relevant biological features. The model architecture is based on the Kolmogorov-Arnold theorem principle, using trainable univariate functions to enhance interpretability and feature analysis. MOGKAN achieves classification accuracy of 96.28 percent and demonstrates low experimental variability with a standard deviation that is reduced by 1.58 to 7.30 percents compared to Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs). The biomarkers identified by MOGKAN have been validated as cancer-related markers through Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) enrichment analysis. The proposed model presents an ability to uncover molecular oncogenesis mechanisms by detecting phosphoinositide-binding substances and regulating sphingolipid cellular processes. By integrating multi-omics data with graph-based deep learning, our proposed approach demonstrates superior predictive performance and interpretability that has the potential to enhance the translation of complex multi-omics data into clinically actionable cancer diagnostics.

Kolmogorov--Arnold Networks (KANs), a recently proposed neural network architecture, have gained significant attention in the deep learning community, due to their potential as a viable alternative to multi-layer perceptrons (MLPs) and their broad applicability to various scientific tasks. Empirical investigations demonstrate that KANs optimized via stochastic gradient descent (SGD) are capable of achieving near-zero training loss in various machine learning (e.g., regression, classification, and time series forecasting, etc.) and scientific tasks (e.g., solving partial differential equations). In this paper, we provide a theoretical explanation for the empirical success by conducting a rigorous convergence analysis of gradient descent (GD) and SGD for two-layer KANs in solving both regression and physics-informed tasks. For regression problems, we establish using the neural tangent kernel perspective that GD achieves global linear convergence of the objective function when the hidden dimension of KANs is sufficiently large. We further extend these results to SGD, demonstrating a similar global convergence in expectation. Additionally, we analyze the global convergence of GD and SGD for physics-informed KANs, which unveils additional challenges due to the more complex loss structure. This is the first work establishing the global convergence guarantees for GD and SGD applied to optimize KANs and physics-informed KANs.