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Synergizing Kolmogorov-Arnold Networks with Dynamic Adaptive Weighting for High-Frequency and Multi-Scale PDE Solutions

arXiv 2512.22283 ↗First posted 25 Dec 2025cs.LG cs.AIcited by 0

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Abstract (arXiv)

PINNs enhance scientific computing by incorporating physical laws into neural network structures, leading to significant advancements in scientific computing. However, PINNs struggle with multi-scale and high-frequency problems due to pathological gradient flow and spectral bias, which severely limit their predictive power. By combining an enhanced network architecture with a dynamically adaptive weighting mechanism featuring upper-bound constraints, we propose the Dynamic Balancing Adaptive Weighting Physics-Informed Kolmogorov-Arnold Network (DBAW-PIKAN). The proposed method effectively mitigates gradient-related failure modes and overcomes bottlenecks in function representation. Compared to baseline models, the proposed method accelerates the convergence process and improves solution accuracy by at least an order of magnitude without introducing additional computational complexity. Numerical results on the Klein-Gordon, Burgers, and Helmholtz equations demonstrate that DBAW-PIKAN achieves superior accuracy and generalization performance.

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References · 15

  1. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  2. 2018On the Spectral Bias of Neural Networkscited by 1,321
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  4. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  5. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  6. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  7. 2024U-KAN Makes Strong Backbone for Medical Image Segmentation and Generationcited by 176
  8. 2019Towards Understanding the Spectral Bias of Deep Learningcited by 173
  9. 2018Deep Learning of Vortex Induced Vibrationscited by 74
  10. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 64
  11. 2024Unveiling the Power of Wavelets: A Wavelet-based Kolmogorov-Arnold Network for Hyperspectral Image Classificationcited by 33
  12. 2024Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KANcited by 25
3 more
  1. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  2. 2024Physics-Informed Kolmogorov-Arnold Networks for Power System Dynamicscited by 14
  3. 2025Self-adaptive weighting and sampling for physics-informed neural networkscited by 6

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