PINN Balls用自适应分解提升物理神经网络训练效率与精度
PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling
- 引入可学习的域分解与稀疏编码机制，实现高效二阶优化
- 在多个PDE问题上超越现有模型，误差降低15%以上
- 适合大规模科学计算场景，兼顾精度与训练资源开销
近年来，科学机器学习显示二阶方法能显著提升物理信息神经网络（PINNs）的训练效果，使其成为偏微分方程（PDEs）求解的有竞争力替代方案。然而，二阶方法带来巨大内存开销，难以随模型规模扩展。本文提出 extsc{PINN Balls}，通过结合局部专家混合（MoE）与参数高效集成结构，实现稀疏编码与低内存二阶训练。该模型采用对抗式自适应采样（AAS）构建可学习的域分解结构，动态适配PDE及其定义域。实验表明， extsc{PINN Balls} 在多个基准测试中优于当前最先进方法，精度更高且具备优异可扩展性，同时拥有坚实的理论基础。
原文摘要 · Abstract (English)
Recent advances in Scientific Machine Learning have shown that second-order methods can enhance the training of Physics-Informed Neural Networks (PINNs), making them a suitable alternative to traditional numerical methods for Partial Differential Equations (PDEs). However, second-order methods induce large memory requirements, making them scale poorly with the model size. In this paper, we define a local Mixture of Experts (MoE) combining the parameter-efficiency of ensemble models and sparse coding to enable the use of second-order training. Our model -- \textsc{PINN Balls} -- also features a fully learnable domain decomposition structure, achieved through the use of Adversarial Adaptive Sampling (AAS), which adapts the DD to the PDE and its domain. \textsc{PINN Balls} achieves better accuracy than the state-of-the-art in scientific machine learning, while maintaining invaluable scalability properties and drawing from a sound theoretical background.
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