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Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions…

Gaussian Conditional Random Fields (GCRF), as a structured regression model, is designed to achieve higher regression accuracy than unstructured predictors at the expense of execution time, taking into account the objects similarities and…

Data in many applications follows systems of Ordinary Differential Equations (ODEs). This paper presents a novel algorithmic and symbolic construction for covariance functions of Gaussian Processes (GPs) with realizations strictly following…

Gaussian Markov random fields (GMRFs) are useful in a broad range of applications. In this paper we tackle the problem of learning a sparse GMRF in a high-dimensional space. Our approach uses the l1-norm as a regularization on the inverse…

A challenge in multivariate problems with discrete structures is the inclusion of prior information that may differ in each separate structure. A particular example of this is seismic amplitude versus angle (AVA) inversion to elastic…

Series expansions of isotropic Gaussian random fields on $\mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localised structure provide an alternative to…

This work is motivated by constructing a weather simulator for precipitation. Temperature and humidity are two of the most important driving forces of precipitation, and the strategy is to have a stochastic model for temperature and…

We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of stochastic area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation…

In large-scale regression problems, random Fourier features (RFFs) have significantly enhanced the computational scalability and flexibility of Gaussian processes (GPs) by defining kernels through their spectral density, from which a finite…

We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the…

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning…

We study the application of graph random features (GRFs) - a recently introduced stochastic estimator of graph node kernels - to scalable Gaussian processes on discrete input spaces. We prove that (under mild assumptions) Bayesian inference…

This paper tackles the challenge of parameter calibration in stochastic models, particularly in scenarios where the likelihood function is unavailable in an analytical form. We introduce a gradient-based simulated parameter estimation…

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and…

Gaussian processes (GPs) are frequently used in machine learning and statistics to construct powerful models. However, when employing GPs in practice, important considerations must be made, regarding the high computational burden,…

Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on…

In this paper, we present a new statistical approach to the problem of incorporating experimental observations into a mathematical model described by linear partial differential equations (PDEs) to improve the prediction of the state of a…

Variational approximation methods have proven to be useful for scaling Bayesian computations to large data sets and highly parametrized models. Applying variational methods involves solving an optimization problem, and recent research in…

Gaussian Markov random fields (GMRFs) are probabilistic graphical models widely used in spatial statistics and related fields to model dependencies over spatial structures. We establish a formal connection between GMRFs and convolutional…

Key challenges in the analysis of highly multivariate large-scale spatial stochastic processes, where both the number of components (p) and spatial locations (n) can be large, include achieving maximal sparsity in the joint precision…