# Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN

arXiv: https://arxiv.org/abs/2407.18373 · HTML: https://mlsanity.com/paper/2407.18373/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 46 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 14 of 14)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (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)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [ExSpliNet: An interpretable and expressive spline-based neural network](https://mlsanity.com/paper/2205.01510/) (2022)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspective](https://mlsanity.com/paper/2401.08667/) (2024)

## Built on by (most cited first, 25 of 25)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Deep Learning Alternatives of the Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2410.01990/) (2024)
- [SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2411.06286/) (2024)
- [LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior](https://mlsanity.com/paper/2502.17844/) (2025)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Physics-informed time series analysis with Kolmogorov-Arnold Networks under Ehrenfest constraints](https://mlsanity.com/paper/2509.18483/) (2025)
- [ChemKANs for Combustion Chemistry Modeling and Acceleration](https://mlsanity.com/paper/2504.12580/) (2025)
- [Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement](https://mlsanity.com/paper/2505.18131/) (2025)
- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [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)
- [Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning](https://mlsanity.com/paper/2504.15240/) (2025)
- [KANITE: Kolmogorov-Arnold Networks for ITE estimation](https://mlsanity.com/paper/2503.13912/) (2025)
- [Uncertainty Quantification with Bayesian Higher Order ReLU KANs](https://mlsanity.com/paper/2410.01687/) (2024)
- [Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.22409/) (2026)
- [Multilevel Training for Kolmogorov Arnold Networks](https://mlsanity.com/paper/2603.04827/) (2026)
- [Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.03806/) (2025)
- [Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning](https://mlsanity.com/paper/2505.11682/) (2025)
- [Kolmogorov Arnold Networks (KANs) for Imbalanced Data -- An Empirical Perspective](https://mlsanity.com/paper/2507.14121/) (2025)
- [Performance-Carbon Trade-Offs across Architectural Biases in Shear Flow Forecasting](https://mlsanity.com/paper/2509.24517/) (2025)
- [Worst-Case Distance-Aware Error Bounds for Neural Networks](https://mlsanity.com/paper/2510.22021/) (2025)
- [Synergizing Kolmogorov-Arnold Networks with Dynamic Adaptive Weighting for High-Frequency and Multi-Scale PDE Solutions](https://mlsanity.com/paper/2512.22283/) (2025)
- [The Enhanced Physics-Informed Kolmogorov-Arnold Networks: Applications of Newton's Laws in Financial Deep Reinforcement Learning (RL) Algorithms](https://mlsanity.com/paper/2602.01388/) (2026)
- [Geometric Kolmogorov--Arnold Network (GeoKAN)](https://mlsanity.com/paper/2605.06740/) (2026)
- [K-U-KAN: Koopman-Enhanced U-KAN for 3D Dental Reconstruction from a Single Panoramic X-ray Radiograph](https://mlsanity.com/paper/2605.25163/) (2026)
- [LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems](https://mlsanity.com/paper/2607.03682/) (2026)

## Abstract (arXiv)

Physics-informed neural networks have proven to be a powerful tool for solving differential equations, leveraging the principles of physics to inform the learning process. However, traditional deep neural networks often face challenges in achieving high accuracy without incurring significant computational costs. In this work, we implement the Physics-Informed Kolmogorov-Arnold Neural Networks (PIKAN) through efficient-KAN and WAV-KAN, which utilize the Kolmogorov-Arnold representation theorem. PIKAN demonstrates superior performance compared to conventional deep neural networks, achieving the same level of accuracy with fewer layers and reduced computational overhead. We explore both B-spline and wavelet-based implementations of PIKAN and benchmark their performance across various ordinary and partial differential equations using unsupervised (data-free) and supervised (data-driven) techniques. For certain differential equations, the data-free approach suffices to find accurate solutions, while in more complex scenarios, the data-driven method enhances the PIKAN's ability to converge to the correct solution. We validate our results against numerical solutions and achieve 99 % accuracy in most scenarios.