Related papers: Error Estimates for Deep Learning Methods in Fluid…
Neural networks (NNs) can achieved high performance in various fields such as computer vision, and natural language processing. However, deploying NNs in resource-constrained safety-critical systems has challenges due to uncertainty in the…
The developments over the last five decades concerning numerical discretisations of the incompressible Navier--Stokes equations have lead to reliable tools for their approximation: those include stable methods to properly address the…
The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools…
We investigate uncertainty estimation and multimodality via the non-deterministic predictions of Bayesian neural networks (BNNs) in fluid simulations. To this end, we deploy BNNs in three challenging experimental test-cases of increasing…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
A machine-learning strategy for investigating the stability of fluid flow problems is proposed herein. The goal is to provide a simple yet robust methodology to find a nonlinear mapping from the parametric space to an indicator representing…
Extracting information on fluid motion directly from images is challenging. Fluid flow represents a complex dynamic system governed by the Navier-Stokes equations. General optical flow methods are typically designed for rigid body motion,…
This article focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier-Stokes equations. For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the…
We consider non-spherical rigid body particles in an incompressible fluid in the regime where the particles are too large to assume that they are simply transported with the fluid without back-coupling and where the particles are also too…
Recently, physics-driven deep learning methods have shown particular promise for the prediction of physical fields, especially to reduce the dependency on large amounts of pre-computed training data. In this work, we target the…
Turbulent problems in industrial applications are predominantly solved using Reynolds Averaged Navier Stokes (RANS) turbulence models. The accuracy of the RANS models is limited due to closure assumptions that induce uncertainty into the…
We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in…
Rigorous assessment of uncertainty is crucial to the utility of DNS results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier-Stokes equations. Due to the…
In recent years, Physics-Informed Neural Networks (PINNs) have emerged as a powerful and robust framework for solving nonlinear differential equations across a wide range of scientific and engineering disciplines, including biology,…
Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the…
In this paper, we present an error estimate of a second-order linearized finite element (FE) method for the 2D Navier-Stokes equations with variable density. In order to get error estimates, we first introduce an equivalent form of the…
In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using…
We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase…
This paper focuses on deriving optimal-order full moment error estimates in strong norms for both velocity and pressure approximations in the Euler-Maruyama time discretization of the stochastic Navier-Stokes equations with multiplicative…
Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks…