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

TL;DR

S²-PINN is a physics-informed neural network that models stochastic PDE solutions using a separable structure combining spatial, temporal, and stochastic bases via low-rank tensor decomposition. It enables accurate uncertainty quantification with fewer parameters by leveraging gPC expansion, Fourier features, and a learnable spatial dictionary, trained with a hybrid residual loss. The method demonstrates superior accuracy and calibration across diverse benchmarks.

Why It Matters

It addresses the scalability and accuracy challenges of uncertainty quantification in high-dimensional stochastic PDEs, offering a parameter-efficient alternative to traditional spectral methods and existing neural solvers. This enables reliable simulation and inference in complex engineering and scientific applications with uncertain inputs.

Key Findings

• The separable architecture is theoretically dense in L², ensuring expressive power for stochastic PDE solutions.
• The projected residual loss enforces stochastic Galerkin constraints, improving solution consistency and calibration.
• The model achieves higher accuracy in mean and variance with fewer parameters than nine baselines across multiple problem types.