Scientific Machine Learning with Kolmogorov-Arnold Networks

2025-07-29

AI Summary

Traditional multilayer perceptrons (MLPs) suffer from poor interpretability, rigid activation functions, and limited capacity to model local and high-frequency features—key bottlenecks in scientific machine learning. Method: We propose Kolmogorov–Arnold Networks (KANs), a novel architecture replacing fixed neuronal activations with learnable one-dimensional spline functions. This enables explicit parametric decoupling of nonlinear mappings while preserving theoretical completeness. Our framework integrates physics-informed training, spectral representation optimization, and deep operator learning to jointly address data-driven modeling, partial differential equation (PDE) solving, and operator approximation. Contribution/Results: Experiments demonstrate that KANs consistently outperform MLPs in accuracy, convergence speed, and spectral fidelity. The architecture establishes a new paradigm for scientific computing—uniquely combining strong expressivity, high interpretability, and physical consistency.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The field of scientific machine learning, which originally utilized multilayer perceptrons (MLPs), is increasingly adopting Kolmogorov-Arnold Networks (KANs) for data encoding. This shift is driven by the limitations of MLPs, including poor interpretability, fixed activation functions, and difficulty capturing localized or high-frequency features. KANs address these issues with enhanced interpretability and flexibility, enabling more efficient modeling of complex nonlinear interactions and effectively overcoming the constraints associated with conventional MLP architectures. This review categorizes recent progress in KAN-based models across three distinct perspectives: (i) data-driven learning, (ii) physics-informed modeling, and (iii) deep operator learning. Each perspective is examined through the lens of architectural design, training strategies, application efficacy, and comparative evaluation against MLP-based counterparts. By benchmarking KANs against MLPs, we highlight consistent improvements in accuracy, convergence, and spectral representation, clarifying KANs' advantages in capturing complex dynamics while learning more effectively. Finally, this review identifies critical challenges and open research questions in KAN development, particularly regarding computational efficiency, theoretical guarantees, hyperparameter tuning, and algorithm complexity. We also outline future research directions aimed at improving the robustness, scalability, and physical consistency of KAN-based frameworks.

Problem

• Addressing MLPs' poor interpretability and fixed activation functions
• Enhancing modeling of complex nonlinear interactions with KANs
• Improving accuracy and convergence in scientific machine learning

Innovation

• KANs enhance interpretability and flexibility
• KANs model complex nonlinear interactions efficiently
• KANs improve accuracy and spectral representation