DBAW-PIKAN: Dynamic Balance Adaptive Weight Kolmogorov-Arnold Neural Network for Solving Partial Differential Equations

Abstract

Physics-informed neural networks (PINNs) have led to significant advancements in scientific computing by integrating fundamental physical principles with advanced data-driven techniques. However, when dealing with problems characterized by multi-scale or high-frequency features, PINNs encounter persistent and severe challenges related to stiffness in gradient flow and spectral bias, which significantly limit their predictive capabilities. To address these issues, this paper proposes a Dynamic Balancing Adaptive Weighting Physics-Informed Kolmogorov-Arnold Network (DBAW-PIKAN), designed to mitigate such gradient-related failure modes and overcome the bottlenecks in function representation. The core of DBAW-PIKAN combines the Kolmogorov-Arnold network architecture, based on learnable B-splines, with an adaptive weighting strategy that incorporates a dynamic decay upper bound. 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. A series of numerical benchmarks, including the Klein-Gordon, Burgers, and Helmholtz equations, demonstrate the significant advantages of DBAW-PIKAN in enhancing both accuracy and generalization performance.