Related papers: Tensor Numerical Methods for High-dimensional PDEs…
The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that…
Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-$N$ tensor from exponential…
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However,…
Numerous complex real-world systems, such as those in biological, ecological, and social networks, exhibit higher-order interactions that are often modeled using polynomial dynamical systems or homogeneous polynomial dynamical systems…
In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…
Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and…
Tensor decomposition of high-dimensional data often struggles to capture semantically or physically meaningful structures, particularly when relying on reconstruction objectives and fixed-rank constraints. We introduce a no-rank tensor…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
Unlike 2D raster images, there is no single dominant representation for 3D visual data processing. Different formats like point clouds, meshes, or implicit functions each have their strengths and weaknesses. Still, grid representations such…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
Most currently used tensor regression models for high-dimensional data are based on Tucker decomposition, which has good properties but loses its efficiency in compressing tensors very quickly as the order of tensors increases, say greater…
We introduce two nonlinear sufficient dimension reduction methods for regressions with tensor-valued predictors. Our goal is two-fold: the first is to preserve the tensor structure when performing dimension reduction, particularly 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…
Modern technological advances have enabled an unprecedented amount of structured data with complex temporal dependence, urging the need for new methods to efficiently model and forecast high-dimensional tensor-valued time series. This paper…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
This work presents a non-intrusive surrogate modeling scheme based on machine learning technology for predictive modeling of complex systems, described by parametrized time-dependent PDEs. For these problems, typical finite element…
We consider the representation of operators in terms of tensor networks and their application to ground-state approximation and time evolution of systems with long-range interactions. We provide an explicit construction to represent an…