environmental modeling groups

Abstract: Physics-informed Gaussian process (GP) methods provide a probabilistic framework for PDE solving. Existing approaches incorporate PDE information in different ways: pointwise conditioning uses the governing equation only at finitely many selected locations, globally constrained GPs restrict the admissible functions to satisfy the PDE exactly, and residual-penalized methods control the PDE mismatch over the whole domain through an additional penalty weight. What is still missing is a GP formulation that directly incorporates global PDE-residual information of the entire domain into its covariance structure, while retaining model flexibility and standard GP conditioning without a separate residual-weight parameter. To solve this issue, this study proposes PDE-residual-informed Kriging (PRIK) based on a PDE-residual kernel. The PDE-residual kernel is constructed by equipping a kernel-section space with an inner product defined by the $L^{2}$ inner product of PDE residuals. This construction makes PRIK yield the posterior mean with minimum integrated squared PDE residual. It also provides direct statistical interpretations: the variance-scale estimate equals the normalized PDE-residual energy (i.e., the integrated squared PDE residual of the posterior mean divided by the number of labeled observations), and the profile likelihood balances this residual energy against model complexity. Under the stated regularity and stability conditions, we establish $H^{2}$-quasi-optimality and convergence rates for the posterior mean, together with a nonasymptotic bound for its PDE-residual norm. We further show that the normalized posterior kernel converges to a variational Green-type kernel and derive a uniform nonasymptotic contraction bound for the posterior variance. Numerical studies provide empirical evidence consistent with the theory and demonstrate the PDE-solving capability of PRIK.

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