相关论文: An optimal control deep learning method to design …
We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. We use polynomials of degree $k+1$ and $k \ge 0$…
This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the…
Neural network optimization remains one of the most consequential yet poorly understood challenges in modern AI research, where improvements in training algorithms can lead to enhanced feature learning in foundation models,…
We propose a neural network approach to model general interaction dynamics and an adjoint based stochastic gradient descent algorithm to calibrate its parameters. The parameter calibration problem is considered as optimal control problem…
In this paper, online convex optimization is applied to the problem of controlling linear dynamical systems. An algorithm similar to online gradient descent, which can handle time-varying and unknown cost functions, is proposed. Then,…
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we…
We devise and analyze a class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems. Discrete duality relations are derived that lead to optimal error estimates in the case of total-variation…
Kinetic schemes for compressible flow of gases are constructed by exploiting the connection between Boltzmann equation and the Navier-Stokes equations. This connection allows us to construct a flux splitting for the Navier-Stokes equations…
We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned…
We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem and we propose multigrid methods to solve the discretized system. We prove that the $W$-cycle algorithm is uniformly convergent in the energy…
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…
In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced…
In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order…
The discontinuous Galerkin (DG) method is widely being used to solve hyperbolic partial differential equations (PDEs) due to its ability to provide high-order accurate solutions in complex geometries, capture discontinuities, and exhibit…
We present a multilevel stochastic gradient descent method for the optimal control of systems governed by partial differential equations under uncertain input data. The gradient descent method used to find the optimal control leverages a…
The training of machine learning models is typically carried out using some form of gradient descent, often with great success. However, non-asymptotic analyses of first-order optimization algorithms typically employ a gradient smoothness…
In recent years, high-order finite element methods on high-order meshes have attracted considerable attention. This work investigates the isoparametric upwind discontinuous Galerkin method for the radiation transport equation on a bounded…
We first propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a \emph{convection dominated} Dirichlet boundary control problem. Dirichlet boundary control problems and convection dominated problems are…
In this paper, a uniformly high-order discontinuous Galerkin gas kinetic scheme (DG-HGKS) is proposed to solve the Euler equations of compressible flows. The new scheme is an extension of the one-stage compact and efficient high-order GKS…
Natural gradient descent (NGD) is a powerful optimization technique for machine learning, but the computational complexity of the inverse Fisher information matrix limits its application in training deep neural networks. To overcome this…