发表机构
Worcester Polytechnic Institute; Sandia National Laboratories（伍斯特理工学院; 桑迪亚国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出Norm-PCGrad梯度手术方法，解决PINN和PIKAN在域分解中的冲突梯度问题，实现最低相对L2误差，并引入SPINN降低计算成本，验证了方法的通用性。
AI 中文摘要
科学机器学习方法，如物理信息神经网络（PINNs），在求解复杂几何上的偏微分方程（PDEs）时，越来越依赖域分解以获得更好的可扩展性，然而由此产生的由残差项、边界项和界面项组成的复合损失极易受到冲突梯度的影响，从而降低训练效果。本研究将域分解与基于投影的梯度手术相结合，以系统性地缓解二维和三维设置中的此类冲突。我们评估了两种现有的基于投影的算法PCGrad和ConFIG，并发现了它们在特定场景下的性能下降，例如具有多个重叠界面的三维域。为了解决这一局限性，我们提出了Norm-PCGrad，一种归一化变体，在一系列二维和三维域分解问题中达到了最先进的精度。在所有考虑的基准测试中，与不使用梯度手术的训练以及现有算法如PCGrad和ConFIG相比，Norm-PCGrad始终实现了最低的相对$L_2$误差，同时仅产生可忽略不计的额外计算开销。为了提高域分解框架（如扩展PINN（XPINN））的计算效率，我们提出在选定的子域中用可分离架构（如可分离PINN（SPINN））替换普通PINN，将计算成本从二次（或三次）降低到线性。我们还证明了梯度手术可扩展到物理信息Kolmogorov-Arnold网络（PIKANs），为三维域分解带来了显著的精度提升，并确认了所提出方法在不同网络架构上的通用性。
英文摘要
Scientific machine learning methods such as physics-informed neural networks (PINNs) increasingly rely on domain decomposition for better scalability while solving partial differential equations (PDEs) over complex geometries, yet the resulting composite loss comprising residual, boundary, and interface terms is highly susceptible to conflicting gradients that degrade training. This work bridges domain decomposition with projection-based gradient surgery to systematically mitigate such conflicts in 2D and 3D settings. We evaluate two existing projection-based algorithms, PCGrad and ConFIG, and identify their performance degradation in specific scenarios such as 3D domains with multiple overlapping interfaces. To address this limitation, we propose Norm-PCGrad, a normalized variant that achieves state-of-the-art accuracy across a range of 2D and 3D domain decomposition problems. Across the benchmarks considered, Norm-PCGrad consistently achieves the lowest relative $L_2$ error compared to training without gradient surgery as well as to existing algorithms such as PCGrad and ConFIG, while incurring negligible additional computational overhead. To improve computational efficiency of domain decomposition frameworks such as Extended PINN (XPINN), we propose replacing vanilla PINNs in selected subdomains with separable architectures such as Separable PINN (SPINN), reducing the computational cost from quadratic (or cubic) to linear. We additionally demonstrate that gradient surgery extends to physics-informed Kolmogorov-Arnold Networks (PIKANs), yielding substantial accuracy improvements for 3D domain decomposition and confirming the generality of the proposed approach across network architectures.
Comments46 pages, 31 figures
