Related papers: Multifidelity Gaussian process regression for solv…
We propose a data fusion method based on multi-fidelity Gaussian process regression (GPR) framework. This method combines available data of the quantity of interest (QoI) and its gradients with different fidelity levels, namely, it is a…
This paper explores the application of kernel learning methods for parameter prediction and evaluation in the Algebraic Multigrid Method (AMG), focusing on several Partial Differential Equation (PDE) problems. AMG is an efficient iterative…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges, primarily due to the curse of dimensionality and the inherent limitations of traditional numerical…
We develop in this paper a multi-grade deep learning method for solving nonlinear partial differential equations (PDEs). Deep neural networks (DNNs) have received super performance in solving PDEs in addition to their outstanding success in…
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…
We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper,…
Gaussian processes (GPs) are powerful probabilistic models that define flexible priors over functions, offering strong interpretability and uncertainty quantification. However, GP models often rely on simple, stationary kernels which can…
In supervised learning, the output variable to be predicted is often represented as a function, such as a spectrum or probability distribution. Despite its importance, functional output regression remains relatively unexplored. In this…
The Gaussian process (GP) is a popular statistical technique for stochastic function approximation and uncertainty quantification from data. GPs have been adopted into the realm of machine learning in the last two decades because of their…
Structural kernels are a flexible learning paradigm that has been widely used in Natural Language Processing. However, the problem of model selection in kernel-based methods is usually overlooked. Previous approaches mostly rely on setting…
Combining Gaussian processes with the expressive power of deep neural networks is commonly done nowadays through deep kernel learning (DKL). Unfortunately, due to the kernel optimization process, this often results in losing their Bayesian…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
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…
Gaussian processes provide a powerful probabilistic kernel learning framework, which allows learning high quality nonparametric regression models via methods such as Gaussian process regression. Nevertheless, the learning phase of Gaussian…
Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
Large-scale Gaussian process inference has long faced practical challenges due to time and space complexity that is superlinear in dataset size. While sparse variational Gaussian process models are capable of learning from large-scale data,…