Related papers: Physics-Informed Gaussian Process Regression Gener…

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for…

The key task of physical simulation is to solve partial differential equations (PDEs) on discretized domains, which is known to be costly. In particular, high-fidelity solutions are much more expensive than low-fidelity ones. To reduce the…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

Rolling bearings are subject to various faults due to its long-time operation under harsh environment, which will lead to unexpected breakdown of machinery system and cause severe accidents. Deep learning methods recently have gained…

We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…

Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then…

This paper proposes a new way to learn Physics-Informed Neural Network loss functions using Generalized Additive Models. We apply our method by meta-learning parametric partial differential equations, PDEs, on Burger's and 2D Heat…

Gaussian processes (GPs) with derivatives are useful in many applications, including Bayesian optimization, implicit surface reconstruction, and terrain reconstruction. Fitting a GP to function values and derivatives at $n$ points in $d$…

Gaussian processes (GPs) are widely used in nonparametric regression, classification and spatio-temporal modeling, motivated in part by a rich literature on theoretical properties. However, a well known drawback of GPs that limits their use…

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…

Gaussian processes (GPs) are Bayesian nonparametric models for function approximation with principled predictive uncertainty estimates. Deep Gaussian processes (DGPs) are multilayer generalizations of GPs that can represent complex marginal…

Deep Learning Gaussian Processes (DL-GP) are proposed as a methodology for analyzing (approximating) computer models that produce heteroskedastic and high-dimensional output. Computer simulation models have many areas of applications,…

This work is concerned with the quantification of the epistemic uncertainties induced the discretization of partial differential equations. Following the paradigm of probabilistic numerics, we quantify this uncertainty probabilistically.…

We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a…

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is…

Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times,…