用域分解+高斯牛顿法加速物理信息神经网络求解微分方程。
Domain decomposition architectures and Gauss-Newton training for physics-informed neural networks
- 将神经网络局部化到子域，缓解高频成分收敛慢问题。
- 高斯牛顿法比Adam等梯度方法更快收敛，每轮需解病态线性系统。
- 域分解使海森矩阵呈块稀疏结构，降低每轮计算开销。
用神经网络近似由偏微分方程控制的边值问题解面临训练困难，主要源于谱偏差——高频分量收敛慢。通过重叠域分解将神经网络局部化可缓解此问题。本文结合域分解与高斯牛顿优化器，相比Adam等梯度方法实现更快收敛；但每轮需求解一个病态线性系统。域分解使原本稠密的高斯牛顿系统呈现块稀疏结构，从而降低每轮计算成本。数值结果表明，该组合对基于神经网络的微分方程求解器具有潜力。
原文摘要 · Abstract (English)
Approximating the solutions of boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process. This difficulty can be partly explained by the spectral bias, that is, the slower convergence of high-frequency components, and can be mitigated by localizing neural networks via (overlapping) domain decomposition. We combine this localization with the Gauss-Newton method as the optimizer to obtain faster convergence than gradient-based schemes such as Adam; this comes at the cost of solving an ill-conditioned linear system in each iteration. Domain decomposition induces a block-sparse structure in the otherwise dense Gauss-Newton system, reducing the computational cost per iteration. Our numerical results indicate that combining localization and Gauss-Newton optimization is promising for neural network-based solvers for partial differential equations.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。