Physics-informed neural networks improve uncertainty estimates for random pdes
S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks
Machine Learning
Summary
Solving equations that depend on random factors is a common challenge in science and engineering, but existing neural network methods often miss important patterns in the randomness. The authors created a new approach called S²-PINN that breaks the problem into parts based on space, time, and randomness, making it easier to learn accurate solutions with fewer parameters. Their method combines special mathematical tools with neural networks to give better estimates of average behaviors and uncertainties. Tests on several example problems show it can outperform existing methods while being more efficient.
What this means in practice
- •For engineering simulation teams: Generate accurate simulations of physical systems influenced by uncertain conditions with improved efficiency and uncertainty estimates.
- •For environmental modeling groups: Model complex environmental processes affected by random inputs using fewer computational resources and better calibrated uncertainty.
Authors
Zhendong Li, Akwum Onwunta
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