解决高频率多尺度偏微分方程求解难题，提升物理信息神经网络精度与收敛速度。
Synergizing Kolmogorov-Arnold Networks with Dynamic Adaptive Weighting for High-Frequency and Multi-Scale PDE Solutions
- 融合Kolmogorov-Arnold网络与动态自适应加权机制，缓解梯度失效问题。
- 在Klein-Gordon、Burgers和Helmholtz方程上实现精度提升一个数量级。
- 无需额外计算开销，适合需要高保真科学模拟的工程与科研场景。
PINNs通过将物理规律嵌入神经网络结构，在科学计算中取得显著进展。然而，由于病态梯度流和谱偏差，传统PINNs在多尺度和高频问题上表现不佳，严重限制了其预测能力。本文提出动态平衡自适应加权物理信息Kolmogorov-Arnold网络（DBAW-PIKAN），结合改进网络架构与带上限约束的动态自适应权重机制，有效缓解梯度相关失败模式并突破函数表示瓶颈。相比基线模型，该方法加速收敛过程，且在不增加额外计算复杂度的前提下，解决方案精度至少提升一个数量级。在Klein-Gordon、Burgers和Helmholtz方程上的数值实验表明，DBAW-PIKAN具有更优的精度与泛化性能。
原文摘要 · Abstract (English)
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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