利用稀疏变分高斯过程柯尔莫戈洛夫-阿诺德网络进行科学机器学习的不确定性量化
Uncertainty Quantification for Scientific Machine Learning using Sparse Variational Gaussian Process Kolmogorov-Arnold Networks (SVGP KAN)
AI总结：
本文提出一种结合稀疏变分高斯过程与柯尔莫戈洛夫-阿诺德拓扑的网络架构，用于科学机器学习中的不确定性量化，通过解析矩匹配实现不确定性传播并保持可解释性。
摘要：
柯尔莫戈洛夫-阿诺德网络已成为传统多层感知机的可解释性替代方案。然而，标准实现缺乏为许多科学应用所必需的系统性不确定性量化能力。我们提出一个框架，将稀疏变分高斯过程推断与柯尔莫戈洛夫-阿诺德拓扑相结合，实现了计算复杂度与样本量几乎成线性的可扩展贝叶斯推断。通过解析矩匹配，我们在深度加法结构中传播不确定性，同时保持可解释性。我们使用三个示例研究来展示该框架区分随机不确定性和epistemic不确定性的能力：在流体流动重建中校准异方差测量噪声，量化多步预测中对流-扩散动态预测置信度退化，以及在卷积自编码器中的分布外检测。这些结果表明稀疏变分高斯过程柯尔莫戈洛夫-阿诺德网络（SVGP KANs）是科学机器学习中不确定性感知学习的一种有前景的架构。
英文摘要：
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.
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