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

TL;DR

This paper introduces DBAW-PIKAN, a novel neural network architecture that combines adaptive weighting and enhanced design to effectively solve high-frequency and multi-scale PDEs, significantly improving accuracy and convergence.

Contribution

The paper presents DBAW-PIKAN, a new physics-informed neural network that mitigates gradient issues and spectral bias, outperforming baseline models in solving complex PDEs.

Findings

Achieves at least an order of magnitude improvement in accuracy.

Accelerates convergence compared to baseline models.

Demonstrates superior performance on Klein-Gordon, Burgers, and Helmholtz equations.

Abstract

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.