Related papers: Physics-Informed Gaussian Process Regression Gener…
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work,…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Gaussian processes provide a compact representation for modeling and estimating an unknown function, that can be updated as new measurements of the function are obtained. This paper extends this powerful framework to the case where the…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an…
We consider identifiability of partially linear additive structural equation models with Gaussian noise (PLSEMs) and estimation of distributionally equivalent models to a given PLSEM. Thereby, we also include robustness results for errors…
In this paper, we investigate a class of approximate Gaussian processes (GP) obtained by taking a linear combination of compactly supported basis functions with the basis coefficients endowed with a dependent Gaussian prior distribution.…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
Recently, Gaussian processes have been used to model the vector field of continuous dynamical systems, referred to as GPODEs, which are characterized by a probabilistic ODE equation. Bayesian inference for these models has been extensively…
This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In scientific and engineering applications of regression analysis, the underlying regression functions are piecewise continuous in…
Gaussian processes (GPs) provide a probabilistic nonparametric representation of functions in regression, classification, and other problems. Unfortunately, exact learning with GPs is intractable for large datasets. A variety of approximate…
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…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are either intended for unbounded domains…
Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs.…
We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining…