可扩展的贝叶斯物理信息Kolmogorov-Arnold网络
Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks
AI总结：
针对KANs在贝叶斯推断中计算效率低的问题，提出结合DTEKI与Chebyshev KANs的无梯度方法，并利用活动子空间降维，实现高效、稳定且可扩展的不确定性量化。
摘要：
不确定性量化（UQ）在科学机器学习中扮演着关键角色，尤其是在使用代理模型逼近复杂系统时。尽管多层感知器（MLPs）通常被用作代理模型，但由于其参数数量庞大，它们常常遭受过拟合的困扰。Kolmogorov-Arnold网络（KANs）提供了一种参数更少的替代方案。然而，基于梯度的推断方法，如哈密顿蒙特卡洛（HMC），在应用于KANs时可能因反向传播的高成本而导致计算效率低下，尤其是在大规模数据集上。为应对这些挑战，我们提出了一种新方法，将dropout Tikhonov集成卡尔曼反演（DTEKI）与Chebyshev KANs相结合。这种无梯度方法有效缓解了过拟合并增强了数值稳定性。此外，我们引入活动子空间方法来降低参数空间的维度，从而能够提高预测的准确性并获得更可靠的不确定性估计。大量实验证明了我们方法在各种测试案例中的有效性，包括大数据集和高噪声水平的情景。我们的结果表明，与HMC相比，新方法在实现相当或更好准确性的同时，具有更高的效率以及稳定性，并且具备可扩展性。此外，通过利用低维参数子空间，我们的方法在保持预测准确性的同时，进一步大幅降低了计算成本。
英文摘要：
Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov-Arnold networks (KANs) offer an alternative solution with fewer parameters. However, gradient-based inference methods, such as Hamiltonian Monte Carlo (HMC), may result in computational inefficiency when applied to KANs, especially for large-scale datasets, due to the high cost of back-propagation. To address these challenges, we propose a novel approach, combining the dropout Tikhonov ensemble Kalman inversion (DTEKI) with Chebyshev KANs. This gradient-free method effectively mitigates overfitting and enhances numerical stability. Additionally, we incorporate the active subspace method to reduce the parameter-space dimensionality, allowing us to improve the accuracy of predictions and obtain more reliable uncertainty estimates. Extensive experiments demonstrate the efficacy of our approach in various test cases, including scenarios with large datasets and high noise levels. Our results show that the new method achieves comparable or better accuracy, much higher efficiency as well as stability compared to HMC, in addition to scalability. Moreover, by leveraging the low-dimensional parameter subspace, our method preserves prediction accuracy while substantially reducing further the computational cost.
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