# LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior

arXiv: https://arxiv.org/abs/2502.17844 · HTML: https://mlsanity.com/paper/2502.17844/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 25 references that are in the map.
- First posted on arXiv 25 Feb 2025.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [On the Kolmogorov neural networks](https://mlsanity.com/paper/2311.00049/) (2023)
- [Kolmogorov-Arnold PointNet: Deep learning for prediction of fluid fields on irregular geometries](https://mlsanity.com/paper/2408.02950/) (2024)
- [Probabilistic Kolmogorov-Arnold Network](https://mlsanity.com/paper/2104.01714/) (2021)

## Built on by (most cited first, 12 of 12)

- [ChemKANs for Combustion Chemistry Modeling and Acceleration](https://mlsanity.com/paper/2504.12580/) (2025)
- [Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries](https://mlsanity.com/paper/2504.06327/) (2025)
- [SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks](https://mlsanity.com/paper/2603.18548/) (2026)
- [Making Gaussian Kolmogorov-Arnold Networks Reliable and Accurate](https://mlsanity.com/paper/2604.21174/) (2026)
- [A Graph Meta-Network for Learning on Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2602.16316/) (2026)
- [SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functions](https://mlsanity.com/paper/2607.18290/) (2026)
- [Kolmogorov-Arnold Chemical Reaction Neural Networks for learning pressure-dependent kinetic rate laws](https://mlsanity.com/paper/2511.07686/) (2025)
- [TruKAN: Towards More Efficient Kolmogorov-Arnold Networks Using Truncated Power Functions](https://mlsanity.com/paper/2602.03879/) (2026)
- [Partition-of-Unity Gaussian Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2604.23599/) (2026)
- [RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis](https://mlsanity.com/paper/2609.01729/) (2026)
- [Learning continuous state of charge dependent thermal decomposition kinetics for Li-ion cathodes using Kolmogorov-Arnold Chemical Reaction Neural Networks (KA-CRNNs)](https://mlsanity.com/paper/2512.15628/) (2025)
- [Evaluating the Effectiveness of SechKAN on 1D Data](https://mlsanity.com/paper/2609.25876/) (2026)

## Abstract (arXiv)

The recently proposed Kolmogorov-Arnold network (KAN) is a promising alternative to multi-layer perceptrons (MLPs) for data-driven modeling. While original KAN layers were only capable of representing the addition operator, the recently-proposed MultKAN layer combines addition and multiplication subnodes in an effort to improve representation performance. Here, we find that MultKAN layers suffer from a few key drawbacks including limited applicability in output layers, bulky parameterizations with extraneous activations, and the inclusion of complex hyperparameters. To address these issues, we propose LeanKANs, a direct and modular replacement for MultKAN and traditional AddKAN layers. LeanKANs address these three drawbacks of MultKAN through general applicability as output layers, significantly reduced parameter counts for a given network structure, and a smaller set of hyperparameters. As a one-to-one layer replacement for standard AddKAN and MultKAN layers, LeanKAN is able to provide these benefits to traditional KAN learning problems as well as augmented KAN structures in which it serves as the backbone, such as KAN Ordinary Differential Equations (KAN-ODEs) or Deep Operator KANs (DeepOKAN). We demonstrate LeanKAN's simplicity and efficiency in a series of demonstrations carried out across a standard KAN toy problem as well as ordinary and partial differential equations learned via KAN-ODEs, where we find that its sparser parameterization and compact structure serve to increase its expressivity and learning capability, leading it to outperform similar and even much larger MultKANs in various tasks.