Related papers: Combining Monte Carlo and Tensor-network Methods f…
We first propose the regular sketch-and-project method for solving tensor equations with respect to the popular t-product. Then, three adaptive sampling strategies and three corresponding adaptive sketch-and-project methods are derived. We…
Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may…
We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, a good trial wavefunction to guide the random…
In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train.…
For tensor linear systems with respect to the popular t-product, we first present the sketch-and-project method and its adaptive variants. Their Fourier domain versions are also investigated. Then, considering that the existing sketching…
Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling…
In this paper, we propose a tensor type of discretization and optimization process for solving high dimensional partial differential equations. First, we design the tensor type of trial function for the high dimensional partial differential…
Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing…
We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely,…
In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In…
We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear…
We use tensor network techniques to obtain high order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all…
Tensor network contraction is a fundamental mathematical operation that generalizes the dot product and matrix multiplication. It finds applications in numerous domains, such as database systems, graph theory, machine learning, probability…
Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the…
Low-rank approximation of tensors has been widely used in high-dimensional data analysis. It usually involves singular value decomposition (SVD) of large-scale matrices with high computational complexity. Sketching is an effective data…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
We introduce forward-backward stochastic differential equations, highlighting the connection between solutions of these and solutions of partial differential equations, related by the Feynman-Kac theorem. We review the technique of…
The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination…
Tensor network states are powerful variational ans\"atze for many-body ground states of quantum lattice models. The use of Monte Carlo sampling techniques in tensor network approaches significantly reduces the cost of tensor contractions,…
Stochastic differential equations play an important role in various applications when modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a…