Related papers: Gaussian Process Priors for Boundary Value Problem…

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant…

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary…

Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high-fidelity numerical simulations across parameter spaces remain computationally prohibitive.…

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…

We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss--Markov prior and…

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…

Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications,…

Gaussian processes (GPs) are widely used as surrogate models for emulating computer code, which simulate complex physical phenomena. In many problems, additional boundary information (i.e., the behavior of the phenomena along input…

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of…

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is…

Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of…

Gaussian processes (GPs) are ubiquitous tools for modeling and predicting continuous processes in physical and engineering sciences. This is partly due to the fact that one may employ a Gaussian process as an interpolator while facilitating…

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost for big data. In this paper, we propose a new Bayesian approach, EigenGP, that learns both…

This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors…

Differential equations are important mechanistic models that are integral to many scientific and engineering applications. With the abundance of available data there has been a growing interest in data-driven physics-informed models.…

Automatic forecasting is the task of receiving a time series and returning a forecast for the next time steps without any human intervention. Gaussian Processes (GPs) are a powerful tool for modeling time series, but so far there are no…

We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian…

Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical…

This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Mat\'ern-type Gaussian field priors that enable flexible modeling…