hybrid spde-gp inference

To address scalability bottlenecks in large-scale probabilistic PDE solving caused by dense covariance matrices, this paper proposes a scalable physics-informed Bayesian solver. Methodologically, it introduces the first coupling of Gaussian Markov random fields (GMRFs) with stochastic PDE (SPDE) priors, yielding an SPDE-GMRF joint prior that inherits Markovian sparsity; combined with a variational inference framework leveraging sparse linear algebra and embedded physical constraints, it explicitly models discretization error, parameter uncertainty, and observational noise. Compared to conventional dense Gaussian process approaches, the method achieves order-of-magnitude reductions in memory and runtime overhead for nonlinear PDE inverse problems, while significantly accelerating convergence. The core contribution is a unified modeling paradigm that overcomes expressivity limitations of standard covariance functions—simultaneously ensuring physical interpretability, computational scalability, and rigorous uncertainty quantification.

This paper addresses the adaptivity of posterior contraction rates in approximate Gaussian process (GP) regression. It identifies a fundamental suboptimality of the stationary SPDE–finite element method with fixed smoothness when the true function is smoother than assumed, whereas lattice-based GP methods with inverse-Gamma bandwidth priors achieve adaptive optimal rates. To unify these paradigms, we propose a general GP approximation framework centered on linear combinations of compactly supported basis functions. We establish, for the first time, that lattice GPs achieve optimal posterior contraction over all unknown smoothness classes under a prior on the number of basis functions, while the SPDE approach fails to adapt due to its fixed smoothness assumption. We develop a unified theory of posterior concentration rates and design computationally efficient inference strategies. Numerical experiments corroborate the theoretical guarantees and demonstrate the practical advantages of our framework.

This work addresses the computational inefficiency of spatiotemporal Gaussian process (GP) inference, which suffers from cubic time complexity and high memory costs due to misalignment between observed and prediction locations. The authors propose the Vanilla-SPDE Exchange method, which establishes—for the first time—a direct equivalence between standard GPs and stochastic partial differential equation (SPDE)-based state-space models, thereby enabling a hybrid inference framework. By integrating sparse approximations with SPDE representations, the approach substantially reduces both computational and memory complexity while preserving predictive accuracy. Theoretical analysis and empirical evaluations demonstrate that the method achieves superior joint spatiotemporal inference efficiency, particularly on dense grids and under non-aligned observation settings.

This work addresses the challenge of modeling spatial heterogeneity in Gaussian process (GP) posterior covariance fields. We systematically characterize the joint influence of kernel bandwidth and observation distribution on the structure of the posterior covariance matrix. First, we establish a geometric analytical framework for posterior covariance, enabling principled interpretation of its spatial variability. Building on this, we propose a theoretically grounded estimator for the absolute value of the covariance field, integrating concepts from adaptive finite element error estimation. The estimator supports efficient low-rank and sparse approximations as well as preconditioning. Crucially, it accurately identifies high- and low-covariance regions, significantly accelerating matrix compression and linear solves in large-scale GP inference. Experiments demonstrate superior trade-offs between accuracy and computational cost compared to existing methods. Our approach provides a novel, scalable tool for uncertainty quantification in high-dimensional GPs.

Gaussian processes (GPs) face prohibitive computational complexity and poor scalability when modeling large-scale spatial data. To address this, we propose the Full-Scale Approximation (FSA) framework, which unifies global and local covariance structures within a single coherent model. We introduce a novel adaptive preconditioner that substantially accelerates convergence and enhances robustness of the conjugate gradient method for solving linear systems arising from GP inference. Additionally, we design an efficient prediction variance estimator based on stochastic trace approximation, circumventing costly matrix inversions. Empirical evaluations on both synthetic and real-world datasets demonstrate that FSA achieves accuracy comparable to exact inference via Cholesky decomposition while reducing computation time by orders of magnitude. The core implementation is provided as an open-source C++ library, with Python and R interfaces for broad accessibility.

This study addresses the challenge of modeling spatiotemporal convection–diffusion processes defined on two-dimensional manifolds, such as the sphere, by proposing a Gaussian random field framework grounded in convection–diffusion stochastic partial differential equations (SPDEs). The method constructs a covariance structure via Galerkin discretization on smooth, compact Riemannian manifolds and enables scalable spatiotemporal prediction through Bayesian inference. To the best of our knowledge, this work is the first to systematically apply the convection–diffusion SPDE framework to domains with complex geometry, thereby integrating physical interpretability with statistical flexibility. Empirical evaluations demonstrate that the model achieves both high predictive accuracy and computational efficiency on simulated spherical data as well as real-world global aerosol optical thickness observations.

This work investigates the non-asymptotic characterization of posterior contraction rates and finite-sample Bernstein–von Mises (BvM) behavior in nonparametric Bayesian models. By interpreting the posterior distribution as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, the study extends diffusion-based analytical frameworks to infinite-dimensional settings for the first time. This approach enables precise control over posterior moments and yields sharp non-asymptotic concentration rates in Hilbert norm, along with a quantitative Laplace approximation of the posterior. In the context of nonparametric linear Gaussian inverse problems, the theory elucidates how likelihood curvature and prior regularity jointly govern posterior contraction rates and finite-sample BvM phenomena.

Existing deterministic neural surrogate models struggle to accurately capture the long-term statistical properties and predictive uncertainty of stochastic partial differential equation (SPDE) systems subject to random perturbations. This work proposes the TRIE evaluation framework—the first comprehensive benchmark tailored for surrogate models of stochastic PDEs—focusing on invariant measure recovery, uncertainty calibration, and efficient probabilistic generation. The authors evaluate multiple approaches, including generative models, Monte Carlo Dropout, and heteroscedastic Gaussian likelihoods, across two chaotic stochastic field SPDEs under various parameter settings. Experimental results demonstrate that the proposed latent generative model consistently achieves the lowest Continuous Ranked Probability Score (CRPS) across all configurations, accurately reproducing the underlying statistical structure while accelerating inference by approximately 12× without compromising long-term statistical fidelity.

This work addresses the challenge of applying Gaussian processes (GPs) to high-dimensional inputs, where they are prone to the curse of dimensionality, and existing two-stage dimensionality reduction approaches often compromise either predictive accuracy or reliable uncertainty quantification. To overcome this limitation, the authors propose an end-to-end Bayesian joint modeling framework that seamlessly integrates input dimensionality reduction within the GP formulation. By placing a prior on the Stiefel manifold to enforce orthogonality of the projection matrix and employing Riemannian Hamiltonian Monte Carlo for posterior inference along geodesics, the method achieves, for the first time, a unified Bayesian treatment of GP regression and dimensionality reduction. The framework is further extended to deep Gaussian processes to enhance representational capacity. Experimental results demonstrate superior performance over conventional two-stage methods in both prediction accuracy and uncertainty calibration, albeit at increased computational cost.

This work addresses the challenge of data-driven modeling of parametric spatiotemporal fields over fixed or parameter-dependent domains, where conventional Gaussian processes struggle to scale to large grids and lack efficient uncertainty quantification. The authors propose a scalable Gaussian process framework based on deep product kernels, leveraging Kronecker matrix algebra to achieve near-linear training complexity. The method enables continuous predictions at arbitrary spatiotemporal coordinates and efficient computation of posterior variances at the same cost as the posterior mean, on both structured and unstructured grids. This significantly overcomes the scalability limitations of traditional Gaussian processes. Experiments demonstrate that the approach matches the accuracy of Fourier Neural Operators and Deep Operator Networks, and outperforms projection-based reduced-order models on the one-dimensional unsteady Burgers equation, confirming its effectiveness as an efficient surrogate model with built-in uncertainty quantification.