AI 中文总结
研究针对物理信息神经网络训练成本高的问题，提出基于分层配置点细化的MPU-PINNs策略，先粗后细训练并结合缩放技术，经实验验证该策略能大幅减少训练时间，提升计算效率与可扩展性。
AI 中文摘要
物理信息神经网络（PINNs）为求解偏微分方程（PDEs）提供了灵活框架，但大量配置点会使训练计算成本高昂。为缓解此成本，引入基于多重网格的参数更新PINNs（MPU-PINNs），这是一种从粗到细的训练策略，在学习过程中逐步增加训练点数量。该方法先在一组粗配置点上训练神经网络，再将学习到的参数转移到更精细层级。为进一步提高高频问题性能，还纳入缩放技术以减轻训练期间的频谱偏差影响。在多个基准PDEs上评估MPU-PINNs，数值实验表明其大大减少训练时间，同时达到与传统PINNs及其他代表性变体相当的精度。结果还表明，所提出的从粗到细学习策略大幅减少了精细层级所需的优化工作量。总体而言，MPU-PINNs提供了一个高效的单网络训练框架，增强了PINNs对广泛PDE问题的计算效率和可扩展性。
英文摘要
Physics-informed neural networks (PINNs) offer a flexible framework for solving partial differential equations (PDEs), but training can become computationally expensive when a large number of collocation points are required to accurately enforce the governing equations. To alleviate this cost, we introduce multigrid-based parameter-updated PINNs (MPU-PINNs), a coarse-to-fine training strategy that progressively increases the number of training points throughout the learning process. The proposed approach begins by training a neural network on a coarse set of collocation points and then transfers the learned parameters to successively finer levels. This initialization strategy enables the network to capture the global features of the solution at a relatively low computational cost before refining local details with additional training points. To further improve performance for high-frequency problems, we incorporate a scaling technique that mitigates the effects of spectral bias during training. We evaluate MPU-PINNs on several benchmark PDEs, including two- and three-dimensional Poisson equations, a convection-diffusion-reaction equation, and the Helmholtz equation. Numerical experiments indicate that MPU-PINNs greatly reduce training time while achieving accuracy comparable to that of conventional PINNs and other representative variants such as SA-PINNs and XPINNs. The results further suggest that the proposed coarse-to-fine learning strategy substantially decreases the optimization effort required at finer levels. Overall, MPU-PINNs provide an efficient single-network training framework that enhances the computational efficiency and scalability of PINNs for a broad range of PDE problems.
