Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems

2024-06-28

arXiv.org

AI Summary

Kolmogorov–Arnold Networks (KANs) suffer from high training costs and limited efficacy in multiscale modeling. Method: We propose Finite-Basis KANs (FBKANs), the first KAN variant integrating domain decomposition and finite-basis representation: it partitions the global domain into subdomains, trains local KANs in parallel, and employs learnable basis functions to enable parameter sharing and embed physical constraints. FBKAN synergistically combines Kolmogorov–Arnold-theorem-inspired learnable activations, adaptive domain partitioning, finite-basis expansions, and physics-informed loss optimization. Contribution/Results: Experiments demonstrate that FBKAN significantly outperforms standard KANs and MLPs in noisy function approximation and partial differential equation solving—achieving higher accuracy, accelerating training by multiple-fold, and exhibiting superior generalization, robustness to noise, and scalability across problem sizes and domains.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

Problem

• Develop domain decomposition for Kolmogorov-Arnold networks
• Enable parallel training of small KANs for multiscale problems
• Provide accurate solutions with noisy data and physics-informed training

Innovation

• Domain decomposition method for Kolmogorov-Arnold networks
• Parallel training of multiple small KANs
• Accurate solutions for multiscale problems