Related papers: Tensor Network Space-Time Spectral Collocation Met…
We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor…
The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov--Galerkin method…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
Quasi-periodic trajectories with two or more incommensurate frequencies are ubiquitous in nonlinear dynamics, yet the classical Fourier-based time-spectral method is tied to strictly periodic responses. We introduce a torus time-spectral…
We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms…
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The…
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a…
Solving large tensor linear systems poses significant challenges due to the high volume of data stored, and it only becomes more challenging when some of the data is missing. Recently, Ma et al. showed that this problem can be tackled using…
Transformer is a successful deep neural network (DNN) architecture that has shown its versatility not only in natural language processing but also in music information retrieval (MIR). In this paper, we present a novel Transformer-based…
We present a novel method to approximate optimal feedback laws for nonlinear optimal control based on low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variational principle with the modification that the…
We are concerned with the computation of the mean-time-to-absorption (MTTA) for a large system of loosely interconnected components, modeled as continuous time Markov chains. In particular, we show that splitting the local and…
Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we…
Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization…
We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated through time-space…
We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved…
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the,…
We study low-rank tensor methods for the numerical solution of Schr\"odinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum…
In intelligent transportation systems, traffic data imputation, estimating the missing value from partially observed data is an inevitable and challenging task. Previous studies have not fully considered traffic data's multidimensionality…
We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or non-conservative binary interactions. The method is…
In this paper, we consider a fast and second-order implicit difference method for approximation of a class of time-space fractional variable coefficients advection-diffusion equation. To begin with, we construct an implicit difference…