In recent work it has been established that deep neural networks (DNNs) are capable of
approximating solutions to a large class of parabolic partial differential equations without …

Deep neural networks with rectified linear units (ReLU) are getting more and more popular
due to their universal representation power and successful applications. Some theoretical …

Sparse-grid methods have recently gained interest in reducing the computational cost of
solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid …

In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport
equations and apply it to kinetic simulations. The method uses the weak formulations of …

This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG) method for
computing high dimensional partial differential equations (PDEs) and its software …

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann
equation with multi-dimensional random inputs. Themethod uses locally supported …

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme
for time-dependent transport equations in multidimensions. The method is constructed using …

Multiwavelets (MW) enable the compression, analysis and assembly of model data on a
multiresolution grid within Godunov-type solvers based on second-order discontinuous …

In this paper, a discrete unified gas kinetic scheme (DUGKS) with a sparse grid method
applied in velocity space (DUGKS-SG) is proposed for simulating rarefied gas flows. The …

In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-
Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics …