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We propose a fourth order Navier-Stokes solver based on the immersed interface method (IIM), for flow problems with stationary and one-way coupled moving boundaries and interfaces. Our algorithm employs a Runge-Kutta-based projection method…

We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem…

Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics and plasma physics. Fluids are well described by the Navier-Stokes equations, but solving these equations at…

We provide spatial discretizations of nonlinear incompressible Navier-Stokes equations with inputs and outputs in the form of matrices ready to use in any numerical linear algebra package. We discuss the assembling of the system operators…

In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of…

We design the helicity-conservative physics-informed neural network model for the Navier-Stokes equation in the ideal case. The key is to provide an appropriate PDE model as loss function so that its neural network solutions produce…

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…

A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical…

Finding the distribution of the velocities and pressures of a fluid by solving the Navier-Stokes equations is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of…

Advancements in computational fluid mechanics have largely relied on Newtonian frameworks, particularly through the direct simulation of Navier-Stokes equations. In this work, we propose an alternative computational framework that employs…

In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the existing…

The control of nonlinear large-scale dynamical models such as the incompressible Navier-Stokes equations is a challenging task. The computational challenges in the controller design come from both the possibly large state space and the…

In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood…

Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we…

Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in…

Conventional mathematical models for simulating incompressible fluid flow problems are based on the Navier-Stokes equations expressed in terms of pressure and velocity. In this context, pressure-velocity coupling is a key issue, and…

ADCME is a novel computational framework to solve inverse problems involving physical simulations and deep neural networks (DNNs). This paper benchmarks its capability to learn spatially-varying physical fields using DNNs. We demonstrate…

The unsteady Micropolar Navier-Stokes Equations (MNSE) are a system of parabolic partial differential equations coupling linear velocity and pressure with angular velocity: material particles have both translational and rotational degrees…

This proposed work introduces a data-assimilation-assisted approach to train neural networks, aimed at effectively reducing epistemic uncertainty in state estimates of separated flows. This method, referred to as model-consistent training,…

We develop a new numerical technique for approximating solutions of the Navier-Stokes equations on moving domains. The method aims at simulating an incompressible fluid past an object whose motion is assigned a priori using a level-set…