Related papers: Kernel Methods for Bayesian Elliptic Inverse Probl…

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The…

In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in…

In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop…

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on…

Inverse problems for Partial Differential Equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In…

Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for…

Gaussian Process (GP) regression is a powerful nonparametric Bayesian framework, but its performance depends critically on the choice of covariance kernel. Selecting an appropriate kernel is therefore central to model quality, yet remains…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are considered, such as…

There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…

In this article, we propose a non-parametric Bayesian level-set method for simultaneous reconstruction of two different piecewise constant coefficients in an elliptic partial differential equation. We show that the Bayesian formulation of…

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations…

Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle…

Bayesian approach, as a useful tool for quantifying uncertainties, has been widely used for solving inverse problems of partial differential equations (PDEs). One of the key difficulties for employing Bayesian approach for the issue is how…

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such…

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions…

We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous…

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian…

These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental…

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying…