PINN在高维噪声热扩散中表现远超传统方法，适合高维强噪声场景。
Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion
- 用物理信息神经网络建模，融合方程约束与数据噪声鲁棒性
- 3D下20%边界噪声时精度达91%，传统有限差分法仅36%
- 高维场景下节点更少、精度更高，适合真实噪声环境
在存在不可避免物理噪声的高维瞬态热扩散问题中，经典数值方法精度会灾难性下降。本文提出一种物理信息神经网络（PINN）框架，适用于一、二、三空间维度，明确界定了解算器选择的新操作区间。在3D下20%边界噪声条件下，PINN保持约91%精度，而有限差分法（FDM）降至36%，优势显著。在真实铜热系统实验中，PINN在实际噪声下将边界重构误差降低3.3倍。该方法兼具噪声鲁棒性与维度效率：在3D中所需时空节点少于FDM且精度更高，揭示了经典离散化在高维下的真实成本。研究重新定义了解算器选择标准——关键不再是单一精度，而是噪声暴露与维度的联合影响。当噪声和维度均高时，传统解法不足；本工作为将PINN确立为该类场景的操作标准提供了依据。
原文摘要 · Abstract (English)
High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems. Under 20% boundary noise in 3D, PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage. This is further confirmed in a physical copper thermal system, where PINN reduces boundary reconstruction error by 3.3 times under realistic noise conditions. This noise resilience is accompanied by a dimensionality-driven efficiency crossover: PINN requires fewer spacetime nodes than FDM in 3D while achieving superior accuracy, exposing the true cost of classical discretization at scale. These findings reframe solver selection: the decisive axis is not accuracy alone, but noise exposure and dimensionality jointly. When noise and dimensionality are both high, the classical solver paradigm is insufficient; this work provides the foundation to justify PINN as the operational standard in such regimes.
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。