从统计学习视角解析物理信息神经网络的训练机制与性能表现
Statistical Learning Analysis of Physics-Informed Neural Networks
- 将PINN参数估计重构为统计学习问题，揭示物理约束本质是无限间接数据源
- 证明PINN学习属于奇异学习问题，用局部学习系数分析热方程求解结果
- 为预测不确定性量化和外推能力提供理论支持，适合研究模型可解释性者
我们从统计学习角度研究了物理信息神经网络（PINNs）在初值与边值问题（IBVP）上的训练与性能。针对硬性初始与边界条件约束的参数化形式，将PINN参数估计重构为统计学习问题。从该视角看，物理残差惩罚项并非正则化项，而是无限的间接数据来源；学习过程等价于通过最小化真实分布δ(0)q(x,t)与PINN残差分布p(y|x,t,w)q(x,t)之间的KL散度来拟合。进一步分析表明，基于随机优化的PINN学习是奇异学习问题，我们采用局部学习系数（Lau et al., 2025）工具对热方程IBVP的参数估计进行分析。最后讨论了该分析对预测不确定性量化及外推能力的启示。
原文摘要 · Abstract (English)
We study the training and performance of physics-informed learning for initial and boundary value problems (IBVP) with physics-informed neural networks (PINNs) from a statistical learning perspective. Specifically, we restrict ourselves to parameterizations with hard initial and boundary condition constraints and reformulate the problem of estimating PINN parameters as a statistical learning problem. From this perspective, the physics penalty on the IBVP residuals can be better understood not as a regularizing term bus as an infinite source of indirect data, and the learning process as fitting the PINN distribution of residuals $p(y \mid x, t, w) q(x, t) $ to the true data-generating distribution $δ(0) q(x, t)$ by minimizing the Kullback-Leibler divergence between the true and PINN distributions. Furthermore, this analysis show that physics-informed learning with PINNs is a singular learning problem, and we employ singular learning theory tools, namely the so-called Local Learning Coefficient (Lau et al., 2025) to analyze the estimates of PINN parameters obtained via stochastic optimization for a heat equation IBVP. Finally, we discuss implications of this analysis on the quantification of predictive uncertainty of PINNs and the extrapolation capacity of PINNs.
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。