S$^{2}$-PINN：随机可分离物理信息神经网络

中文摘要 AI 辅助

随机偏微分方程（PDE）的不确定性量化（UQ）在计算科学与工程中无处不在。然而，针对此类问题的经典谱求解器面临维数灾难，而现有的神经求解器往往忽略随机结构，而这种结构使得矩和校准变得易于处理。我们引入了一种随机可分离物理信息神经网络，称为S$^{2}$-PINN，它通过可学习的Gaussian空间字典、Fourier时间特征和广义多项式混沌（gPC）随机基来表示随机PDE的解$u(t,\mathbf{x},\mathbf{Z})$，并由低秩Canonical Polyadic（CP）张量分解核心耦合。该方法使用混合强形式和gPC投影残差损失进行训练。我们的理论分析表明，在温和条件下，可分离类在$L^2$中是稠密的，并且投影残差恰好对应于随机Galerkin约束。此外，我们证明了小批量投影系数对gPC模式数量的依赖是对数级的，并且正交性惩罚控制了学习到的空间字典的条件数。使用四个构造的随机PDE基准，我们展示了S$^{2}$-PINN在均值和方差精度以及校准方面优于九个基线，同时使用的参数显著更少。对非构造的Poisson和Darcy问题、随机Navier-Stokes问题、更高随机维度的扩散缩放研究以及两个随机逆问题的进一步评估揭示了所提出结构的泛化能力。这些结果共同支持随机可分离性作为物理信息神经UQ的有效设计原则。实验代码可在https URL中找到。

英文摘要

Uncertainty quantification (UQ) for random partial differential equations (PDEs) is ubiquitous in computational science and engineering. However, classical spectral solvers for this class of problems face the curse of dimensionality, and existing neural solvers often ignore the stochastic structure that makes moments and calibration tractable. We introduce a stochastic separable physics-informed neural network, dubbed S$^{2}$-PINN, that represents the solution $u(t,\mathbf{x},\mathbf{Z})$ of a random PDE with a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos (gPC) stochastic basis, coupled by a low-rank Canonical Polyadic (CP) tensor decomposition core. The method is trained with a hybrid strong-form and gPC-projected residual loss. Our theoretical analysis establishes that the separable class is dense in $L^2$ under mild conditions, and the projected residual corresponds exactly to a stochastic Galerkin constraint. Furthermore, we show that mini-batch projection coefficients are logarithmically dependent on the number of gPC modes, and that the orthogonality penalty controls the conditioning of the learned spatial dictionary. Using four manufactured random PDE benchmarks, we show that S$^{2}$-PINN outperforms nine baselines in terms of mean and variance accuracy, as well as calibration, while using significantly fewer parameters. Further evaluations on non-manufactured Poisson and Darcy problems, a stochastic Navier--Stokes problem, a diffusion scaling study of higher random dimensions, and two stochastic inverse problems reveal the generalization capabilities of the proposed structure. Together, these results support stochastic separability as an effective design principle for physics-informed neural UQ. The code for the experiments can be found in https://github.com/DMax1314/s2pinn

发表机构

• Lehigh University（理海大学）

机构由 AI 辅助整理，请以论文原文为准。