Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement (2505.18131v1)

Abstract: Multilayer perceptrons (MLPs) are a workhorse machine learning architecture, used in a variety of modern deep learning frameworks. However, recently Kolmogorov-Arnold Networks (KANs) have become increasingly popular due to their success on a range of problems, particularly for scientific machine learning tasks. In this paper, we exploit the relationship between KANs and multichannel MLPs to gain structural insight into how to train MLPs faster. We demonstrate the KAN basis (1) provides geometric localized support, and (2) acts as a preconditioned descent in the ReLU basis, overall resulting in expedited training and improved accuracy. Our results show the equivalence between free-knot spline KAN architectures, and a class of MLPs that are refined geometrically along the channel dimension of each weight tensor. We exploit this structural equivalence to define a hierarchical refinement scheme that dramatically accelerates training of the multi-channel MLP architecture. We show further accuracy improvements can be had by allowing the $1$D locations of the spline knots to be trained simultaneously with the weights. These advances are demonstrated on a range of benchmark examples for regression and scientific machine learning.

Overview of Leveraging Kolmogorov-Arnold Networks for Expedient Training of Multichannel MLPs

The paper "Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement" explores the intersection of Multilayer Perceptrons (MLPs) with Kolmogorov-Arnold Networks (KANs) to enhance the training performance of the former. The authors focus on the convergence acceleration and accuracy improvements when utilizing KANs, demonstrating a method that exploits equivalent representations to precondition MLPs and enhance their geometric refinement.

Key Insights and Contributions

KANs have gained attention for their efficacy in scientific machine learning, particularly due to the Kolmogorov Superposition Theorem, which underpins their architecture. The paper recognizes the structural equivalence between fre-knot spline KAN architectures and a special class of multichannel MLPs. The following contributions are noteworthy:

1. Reformulation Using B-Splines and ReLU Basis: The paper reformulates KANs in the language of multichannel MLPs by utilizing B-splines, where the weights are constrained to correspond with spline knots. The edge here lies in understanding that a KAN expressed with a B-spline basis can be preconditioned to offer a more robust optimization landscape for training MLPs, alleviating issues like spectral bias that frequently arise due to the global nature of ReLU functions.
2. Preconditioning: Through a change of basis, KAN expressed in a ReLU basis provides preconditioning, which notably improves the condition number of the Hessian in the training system. This explains why architectures with localized geometric features, like image processing networks, often train more effectively and suggests a method to leverage this insight for multichannel MLPs.
3. Hierarchical Refinement: A hierarchical refinement strategy is introduced to exploit this equivalence, significantly speeding up the training of multichannel MLPs. KANs can offer faster convergence rates and improved training accuracy, illustrated by a method that refines the B-spline grids as training progresses, allowing for adaptive grid improvement during learning.
4. Free-Knot Spline Parameterization: The authors also propose a parameterization method for free-knot splines, allowing the one-dimensional locations of the spline knots to evolve during training. This approach addresses unresolved issues about the dependence of KAN solutions on the spline grid, which has been a concern in the academic community.

Numerical Results and Implications

The paper documents strong numerical results across several tasks, including regression problems and physics-informed neural networks (PINNs). The results indicate marked improvements in training efficiency and performance metrics when using hierarchical refinement schemes and preconditioning through KAN structures.

• One critical empirical result is the reduced mean squared error (MSE) when using the spline basis compared to both standard MLPs and traditional KAN formulations. The MSE reduction is significant in regression tasks, demonstrating the potential of this approach to generalize across various functions.
• Additionally, the adaptation of spline knots further decreases error rates, particularly evident in cases with non-smooth functions, suggesting practical applications where dynamic feature localization is paramount.

Theoretical and Practical Implications

From a theoretical viewpoint, this research opens avenues to further explore the potential of alternative neural network constructions like KANs to provide structured, efficient pathways to complex problem-solving. It challenges the prevailing reliance on ReLU-MLPs by offering a design that can potentially reduce computational burden while enhancing robustness in optimization.

Practically, the implications extend to diverse applications in scientific computing and areas reliant on precise data-driven models, where adaptation and efficient learning of spatially localized features are advantageous. The strategies laid out could spur further exploratory work into more adaptive network configurations.

Speculation on Future Developments

Looking ahead, the interplay between network design and bases in influencing training dynamics invites deeper investigation into alternative architectural choices that exploit underlying mathematical structures. Advances in understanding such networks' refinement schemes could contribute to scalable AI models that are more data-efficient and compute-efficient, aligning with evolving demands for resource-aware AI systems.

In summary, this paper presents compelling evidence to consider KAN-inspired architectures and methods for enhancing the training of network topologies, facilitating faster, more reliable, and accurate learning processes within neural networks.