S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks

2026-10-02

AI Summary

This study addresses the curse of dimensionality in traditional uncertainty quantification for stochastic partial differential equations and the neglect of stochastic structure by existing neural solvers. To this end, we propose S²-PINN, which introduces a stochastic separability design principle. By leveraging CP tensor decomposition, it couples Gaussian spatial dictionaries, Fourier temporal features, and generalized polynomial chaos bases, trained via a hybrid residual loss. Theoretically, we establish its density and demonstrate that its projection residual is equivalent to a stochastic Galerkin constraint, significantly reducing computational complexity. Empirically, S²-PINN outperforms nine baselines across multiple benchmarks, achieving superior accuracy in mean and variance estimation as well as better calibration with fewer parameters, while exhibiting strong generalization capabilities.

Abstract

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

Problem

• Uncertainty quantification
• Random partial differential equations
• Curse of dimensionality
• Physics-informed neural networks

Innovation

• Physics-Informed Neural Networks
• Uncertainty Quantification
• Generalized Polynomial Chaos
• Tensor Decomposition
• Random Partial Differential Equations