S²-PINN：面向随机偏微分方程不确定性量化的物理信息神经网络
S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks
解决随机 PDE 不确定性量化的新网络结构，比九个基线更准还更省参数，做科学计算的可以看看开源代码。
arXiv 论文提出 S²-PINN，用可学习高斯空间字典、傅里叶时间特征与 gPC 随机基，并通过低秩 CP 张量分解耦合，来求解随机偏微分方程。理论部分证明了可分离类在 L² 中稠密，投影残差等价于随机 Galerkin 约束。在四个构造随机 PDE 基准上，S²-PINN 在均值、方差精度和校准上超过九个基线，参数量更少。在 Poisson、Darcy、随机 Navier-Stokes 及两个随机逆问题上的进一步评测验证了泛化能力，代码已在 GitHub 开源。
S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks
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