Related papers: Probabilistic Solutions To Ordinary Differential E…

Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional…

Several machine learning problems arising in natural language processing can be modeled as a sequence labeling problem. We provide Gaussian process models based on pseudo-likelihood approximation to perform sequence labeling. Gaussian…

Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed…

Sequential Bayesian Filtering aims to estimate the current state distribution of a Hidden Markov Model, given the past observations. The problem is well-known to be intractable for most application domains, except in notable cases such as…

This paper is concerned with a state-space approach to deep Gaussian process (DGP) regression. We construct the DGP by hierarchically putting transformed Gaussian process (GP) priors on the length scales and magnitudes of the next level of…

In this work, a Gaussian process regression(GPR) model incorporated with given physical information in partial differential equations(PDEs) is developed: physics-assisted Gaussian processes(PAGP). The targets of this model can be divided…

Ordinary differential equation (ODE) models are widely used to describe systems in many areas of science. To ensure these models provide accurate and interpretable representations of real-world dynamics, it is often necessary to infer…

Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain unknown parameters, say $\theta$ of physical significance which…

We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…

All discretized numerical models contain modelling errors - this reality is amplified when reduced-order models are used. The ability to accurately approximate modelling errors informs statistics on model confidence and improves…

Operator learning focuses on approximating mappings $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\mathcal{V}$ between infinite-dimensional spaces of functions, such as $u: \Omega_u\rightarrow\mathbb{R}$ and $v:…

In many real-world applications we are interested in approximating costly functions that are analytically unknown, e.g. complex computer codes. An emulator provides a fast approximation of such functions relying on a limited number of…

Gaussian processes (GPs) provide a framework for Bayesian inference that can offer principled uncertainty estimates for a large range of problems. For example, if we consider regression problems with Gaussian likelihoods, a GP model enjoys…

Bayesian models based on Gaussian processes (GPs) offer a flexible framework to predict spatially distributed variables with uncertainty. But the use of nonstationary priors, often necessary for capturing complex spatial patterns, makes…

We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This…

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of…

Bayesian estimation strategies represent the most fundamental formulation of the state estimation problem available, and apply readily to nonlinear systems with non-Gaussian uncertainties. The present paper introduces a novel method for…

Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric…

This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an explicit solution in the simpler case of linear Gaussian state space models. The key…

Gaussian processes (GPs) are distributions over functions, which provide a Bayesian nonparametric approach to regression and classification. In spite of their success, GPs have limited use in some applications, for example, in some cases a…