Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs

• URL: http://arxiv.org/abs/2506.07958v1
• Date: Mon, 09 Jun 2025 17:30:13 GMT
• Title: Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs
• Authors: Salah A. Faroughi, Farinaz Mostajeran,
• Abstract summary: We aim to advance the theoretical understanding of cPIKANs by analyzing them using Neural Tangent Kernel (NTK) theory.<n>We first derive the NTK of standard cKANs in a supervised setting, and then extend the analysis to the physics-informed context.<n>Results indicate a tractable behavior for NTK in the context of cPIKANs, which exposes learning dynamics that standard physics-informed neural networks (PINNs) cannot capture.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-informed Kolmogorov-Arnold Networks (PIKANs), and in particular their Chebyshev-based variants (cPIKANs), have recently emerged as promising models for solving partial differential equations (PDEs). However, their training dynamics and convergence behavior remain largely unexplored both theoretically and numerically. In this work, we aim to advance the theoretical understanding of cPIKANs by analyzing them using Neural Tangent Kernel (NTK) theory. Our objective is to discern the evolution of kernel structure throughout gradient-based training and its subsequent impact on learning efficiency. We first derive the NTK of standard cKANs in a supervised setting, and then extend the analysis to the physics-informed context. We analyze the spectral properties of NTK matrices, specifically their eigenvalue distributions and spectral bias, for four representative PDEs: the steady-state Helmholtz equation, transient diffusion and Allen-Cahn equations, and forced vibrations governed by the Euler-Bernoulli beam equation. We also conduct an investigation into the impact of various optimization strategies, e.g., first-order, second-order, and hybrid approaches, on the evolution of the NTK and the resulting learning dynamics. Results indicate a tractable behavior for NTK in the context of cPIKANs, which exposes learning dynamics that standard physics-informed neural networks (PINNs) cannot capture. Spectral trends also reveal when domain decomposition improves training, directly linking kernel behavior to convergence rates under different setups. To the best of our knowledge, this is the first systematic NTK study of cPIKANs, providing theoretical insight that clarifies and predicts their empirical performance.

Related papers

• Generalization Bound of Gradient Flow through Training Trajectory and Data-dependent Kernel [55.82768375605861]
  We establish a generalization bound for gradient flow that aligns with the classical Rademacher complexity for kernel methods.<n>Unlike static kernels such as NTK, the LPK captures the entire training trajectory, adapting to both data and optimization dynamics.
  arXiv  Detail & Related papers  (2025-06-12T23:17:09Z)
• Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ? [3.591122855617648]
  We study the neural tangent kernel (NTK) for general partial differential equations based on physics-informed neural networks (PINNs)<n>As we all know, the training of an artificial neural network can be converted to the evolution of NTK.
  arXiv  Detail & Related papers  (2024-12-09T02:41:39Z)
• The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks [0.0]
  We show how the NTK perspective falls short in the nonlinear scenario. We explore the convergence guarantees of such methods in both linear and nonlinear cases.
  arXiv  Detail & Related papers  (2024-02-06T10:24:36Z)
• Architectural Strategies for the optimization of Physics-Informed Neural Networks [30.92757082348805]
  Physics-informed neural networks (PINNs) offer a promising avenue for tackling both forward and inverse problems in partial differential equations (PDEs) Despite their remarkable empirical success, PINNs have garnered a reputation for their notorious training challenges across a spectrum of PDEs.
  arXiv  Detail & Related papers  (2024-02-05T04:15:31Z)
• Connecting NTK and NNGP: A Unified Theoretical Framework for Wide Neural Network Learning Dynamics [6.349503549199403]
  We provide a comprehensive framework for the learning process of deep wide neural networks.<n>By characterizing the diffusive phase, our work sheds light on representational drift in the brain.
  arXiv  Detail & Related papers  (2023-09-08T18:00:01Z)
• Neural tangent kernel analysis of PINN for advection-diffusion equation [0.0]
  Physics-informed neural networks (PINNs) numerically approximate the solution of a partial differential equation (PDE) PINNs are known to struggle even in simple cases where the closed-form analytical solution is available. This work focuses on a systematic analysis of PINNs for the linear advection-diffusion equation (LAD) using the Neural Tangent Kernel (NTK) theory.
  arXiv  Detail & Related papers  (2022-11-21T18:35:14Z)
• Extrapolation and Spectral Bias of Neural Nets with Hadamard Product: a Polynomial Net Study [55.12108376616355]
  The study on NTK has been devoted to typical neural network architectures, but is incomplete for neural networks with Hadamard products (NNs-Hp) In this work, we derive the finite-width-K formulation for a special class of NNs-Hp, i.e., neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK.
  arXiv  Detail & Related papers  (2022-09-16T06:36:06Z)
• Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks [72.09574528342732]
  Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs) They often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under gradient descent with momentum (SGDM)
  arXiv  Detail & Related papers  (2022-06-29T19:03:10Z)
• The Spectral Bias of Polynomial Neural Networks [63.27903166253743]
  Polynomial neural networks (PNNs) have been shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a $textitspectral bias$ towards low-frequency functions, which yields faster learning of low-frequency components during training. Inspired by such studies, we conduct a spectral analysis of the Tangent Kernel (NTK) of PNNs. We find that the $Pi$-Net family, i.e., a recently proposed parametrization of PNNs, speeds up the
  arXiv  Detail & Related papers  (2022-02-27T23:12:43Z)
• Convex Analysis of the Mean Field Langevin Dynamics [49.66486092259375]
  convergence rate analysis of the mean field Langevin dynamics is presented. $p_q$ associated with the dynamics allows us to develop a convergence theory parallel to classical results in convex optimization.
  arXiv  Detail & Related papers  (2022-01-25T17:13:56Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)
• Weighted Neural Tangent Kernel: A Generalized and Improved Network-Induced Kernel [20.84988773171639]
  The Neural Tangent Kernel (NTK) has recently attracted intense study, as it describes the evolution of an over- parameterized Neural Network (NN) trained by gradient descent. We introduce the Weighted Neural Tangent Kernel (WNTK), a generalized and improved tool, which can capture an over- parameterized NN's training dynamics under different gradients. With the proposed weight update algorithm, both empirical and analytical WNTKs outperform the corresponding NTKs in numerical experiments.
  arXiv  Detail & Related papers  (2021-03-22T03:16:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.