融合动态自适应加权的Kolmogorov-Arnold网络用于高频和多尺度PDE求解
Synergizing Kolmogorov-Arnold Networks with Dynamic Adaptive Weighting for High-Frequency and Multi-Scale PDE Solutions
AI总结：
DBAW-PIKAN通过融合动态自适应加权机制与改进网络架构，有效解决高频和多尺度PDE求解中的梯度流动和频谱偏差问题，提升求解精度和收敛速度。
摘要：
PINNs通过将物理定律融入神经网络结构，显著提升了科学计算的水平。然而，PINNs在处理多尺度和高频问题时面临病态梯度流动和频谱偏差的挑战，严重限制了其预测能力。通过结合改进的网络架构与具有上界约束的动态自适应加权机制，我们提出动态平衡自适应加权物理信息Kolmogorov-Arnold网络（DBAW-PIKAN）。所提方法有效缓解了与梯度相关的失败模式，并克服了函数表示的瓶颈。与基线模型相比，所提方法通过至少一个数量级的加速收敛过程和提高求解精度，而无需引入额外的计算复杂性。在Klein-Gordon、Burgers和Helmholtz方程上的数值结果表明，DBAW-PIKAN实现了更优越的精度和泛化性能。
英文摘要：
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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