Related papers: Tensor neural networks for high-dimensional Fokker…
Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture…
Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor neural network based machine learning method. This means…
Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of…
Physics-Informed Neural Networks (PINNs) have shown continuous and increasing promise in approximating partial differential equations (PDEs), although they remain constrained by the curse of dimensionality. In this paper, we propose a…
The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by our early study in \cite{li2018data}, we…
Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the…
Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper…
Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised…
Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Although there has…
This paper studies the prediction task of tensor-on-tensor regression in which both covariates and responses are multi-dimensional arrays (a.k.a., tensors) across time with arbitrary tensor order and data dimension. Existing methods either…
Modern sensing and metrology systems now stream terabytes of heterogeneous, high-dimensional (HD) data profiles, images, and dense point clouds, whose natural representation is multi-way tensors. Understanding such data requires regression…
Dynamic multilayer networks are frequently used to describe the structure and temporal evolution of multiple relationships among common entities, with applications in fields such as sociology, economics, and neuroscience. However,…
Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost…
The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…
In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the…
Tensor networks are factorizations of high-dimensional tensors into networks of smaller tensors. They have applications in physics and mathematics, and recently have been proposed as promising machine learning architectures. To ease the…
The groundbreaking performance of deep neural networks (NNs) promoted a surge of interest in providing a mathematical basis to deep learning theory. Low-rank tensor decompositions are specially befitting for this task due to their close…
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…
Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum…
Learning high-dimensional distributions is an important yet challenging problem in machine learning with applications in various domains. In this paper, we introduce new techniques to formulate the problem as solving Fokker-Planck equation…