# Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks

arXiv: https://arxiv.org/abs/2407.17611 · HTML: https://mlsanity.com/paper/2407.17611/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 64 references that are in the map.
- First posted on arXiv 24 Jul 2024.

## Builds on (most cited first, 30 of 37)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [U-KAN Makes Strong Backbone for Medical Image Segmentation and Generation](https://mlsanity.com/paper/2406.02918/) (2024)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [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)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [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 Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [TKAN: Temporal Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.07344/) (2024)
- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [A Temporal Kolmogorov-Arnold Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.02486/) (2024)
- [Kolmogorov-Arnold Network for Satellite Image Classification in Remote Sensing](https://mlsanity.com/paper/2406.00600/) (2024)
- [KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Search](https://mlsanity.com/paper/2406.17630/) (2024)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)

## Built on by (most cited first, 30 of 38)

- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [PointNet with KAN versus PointNet with MLP for 3D Classification and Segmentation of Point Sets](https://mlsanity.com/paper/2410.10084/) (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)
- [Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.08026/) (2024)
- [Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs](https://mlsanity.com/paper/2505.03595/) (2025)
- [Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study](https://mlsanity.com/paper/2509.03417/) (2025)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [High precision PINNs in unbounded domains: application to singularity formulation in PDEs](https://mlsanity.com/paper/2506.19243/) (2025)
- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamics](https://mlsanity.com/paper/2508.03965/) (2025)
- [KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics example](https://mlsanity.com/paper/2408.02743/) (2024)
- [Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement](https://mlsanity.com/paper/2505.18131/) (2025)
- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (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)
- [Automatic Grid Updates for Kolmogorov-Arnold Networks using Layer Histograms](https://mlsanity.com/paper/2511.08570/) (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)
- [Explainable fault and severity classification for rolling element bearings using Kolmogorov-Arnold networks](https://mlsanity.com/paper/2412.01322/) (2024)
- [Training Deep Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.23501/) (2025)
- [Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities](https://mlsanity.com/paper/2410.04096/) (2024)
- [KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks](https://mlsanity.com/paper/2501.13456/) (2025)
- [Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2502.06153/) (2025)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have emerged as a robust framework for solving Partial Differential Equations (PDEs) by approximating their solutions via neural networks and imposing physics-based constraints on the loss function. Traditionally, Multilayer Perceptrons (MLPs) have been the neural network of choice, with significant progress made in optimizing their training. Recently, Kolmogorov-Arnold Networks (KANs) were introduced as a viable alternative, with the potential of offering better interpretability and efficiency while requiring fewer parameters. In this paper, we present a fast JAX-based implementation of grid-dependent Physics-Informed Kolmogorov-Arnold Networks (PIKANs) for solving PDEs, achieving up to 84 times faster training times than the original KAN implementation. We propose an adaptive training scheme for PIKANs, introducing an adaptive state transition technique to avoid loss function peaks between grid extensions, and a methodology for designing PIKANs with alternative basis functions. Through comparative experiments, we demonstrate that the adaptive features significantly enhance solution accuracy, decreasing the L^2 error relative to the reference solution by up to 43.02%. For the studied PDEs, our methodology approaches or surpasses the results obtained from architectures that utilize up to 8.5 times more parameters, highlighting the potential of adaptive, grid-dependent PIKANs as a superior alternative in scientific and engineering applications.