投影柯尔莫哥洛夫-阿诺德神经网络（P-KANs）：熵驱动的函数空间发现用于可解释机器学习
Projective Kolmogorov Arnold Neural Networks (P-KANs): Entropy-Driven Functional Space Discovery for Interpretable Machine Learning
AI总结：
针对KANs高维样条参数冗余导致的过拟合问题，提出P-KANs框架，通过熵最小化引导边函数收敛至低参数投影空间，实现高达80%参数压缩并增强可解释性。
摘要：
柯尔莫哥洛夫-阿诺德网络（KANs）将可学习的非线性从节点转移到边，在科学机器学习和可解释建模中展现出显著能力。然而，当前的KAN实现因高维样条参数空间中的冗余而存在根本性的低效问题，其中大量不同的参数化产生功能等价的行为。这种冗余表现为模型雅可比矩阵中的“干扰空间”，导致对过拟合的敏感性和较差的泛化能力。我们引入了投影柯尔莫哥洛夫-阿诺德网络（P-KANs），一种新颖的训练框架，通过信号分析和稀疏字典学习中的熵最小化技术，引导边函数发现走向可解释的函数表示。我们的方法不将函数限制在预定空间中，而是保持样条空间的灵活性，同时引入“引力”项以鼓励向最优函数表示收敛。我们的关键见解认识到，最优表示可以通过投影系数的熵分析来识别，将边函数压缩到低参数投影空间（傅里叶、切比雪夫、贝塞尔）。P-KANs在多个领域展现出优越性能，实现了高达80%的参数减少同时保持表示能力，相比标准KANs显著提高了对噪声的鲁棒性，并成功应用于工业自动化纤维铺放预测。我们的方法能够自动发现混合函数表示，其中不同的边收敛到不同的最优空间，为科学机器学习应用提供压缩收益和增强的可解释性。
英文摘要：
Kolmogorov-Arnold Networks (KANs) relocate learnable nonlinearities from nodes to edges, demonstrating remarkable capabilities in scientific machine learning and interpretable modeling. However, current KAN implementations suffer from fundamental inefficiencies due to redundancy in high-dimensional spline parameter spaces, where numerous distinct parameterisations yield functionally equivalent behaviors. This redundancy manifests as a "nuisance space" in the model's Jacobian, leading to susceptibility to overfitting and poor generalization. We introduce Projective Kolmogorov-Arnold Networks (P-KANs), a novel training framework that guides edge function discovery towards interpretable functional representations through entropy-minimisation techniques from signal analysis and sparse dictionary learning. Rather than constraining functions to predetermined spaces, our approach maintains spline space flexibility while introducing "gravitational" terms that encourage convergence towards optimal functional representations. Our key insight recognizes that optimal representations can be identified through entropy analysis of projection coefficients, compressing edge functions to lower-parameter projective spaces (Fourier, Chebyshev, Bessel). P-KANs demonstrate superior performance across multiple domains, achieving up to 80% parameter reduction while maintaining representational capacity, significantly improved robustness to noise compared to standard KANs, and successful application to industrial automated fiber placement prediction. Our approach enables automatic discovery of mixed functional representations where different edges converge to different optimal spaces, providing both compression benefits and enhanced interpretability for scientific machine learning applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
