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.

翻译：物理信息神经网络通过将物理定律融入神经网络结构，显著推动了科学计算的发展。然而，由于病态梯度流与谱偏差的存在，物理信息神经网络在处理多尺度与高频问题时面临挑战，这严重限制了其预测能力。通过将增强的网络架构与具有上界约束的动态自适应加权机制相结合，本文提出了动态平衡自适应加权物理信息Kolmogorov-Arnold网络。该方法有效缓解了与梯度相关的失效模式，并突破了函数表示的瓶颈。与基线模型相比，所提方法在不引入额外计算复杂度的前提下，加速了收敛过程，并将求解精度提升至少一个数量级。在Klein-Gordon方程、Burgers方程和Helmholtz方程上的数值实验表明，DBAW-PIKAN实现了卓越的精度与泛化性能。