Related papers: Neural-HSS: Hierarchical Semi-Separable Neural PDE…
In this work we introduce a new multiscale artificial neural network based on the structure of $\mathcal{H}$-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial…
The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved…
Partition of unity networks (POU-Nets) have been shown capable of realizing algebraic convergence rates for regression and solution of PDEs, but require empirical tuning of training parameters. We enrich POU-Nets with a Gaussian noise model…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
Hyperspectral images are of crucial importance in order to better understand features of different materials. To reach this goal, they leverage on a high number of spectral bands. However, this interesting characteristic is often paid by a…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
We describe a fast, direct solver for elliptic partial differential equations on a two-dimensional hierarchy of adaptively refined, Cartesian meshes. Our solver, inspired by the Hierarchical Poincar\'e-Steklov (HPS) method introduced by…
Hyperspectral images involve abundant spectral and spatial information, playing an irreplaceable role in land-cover classification. Recently, based on deep learning technologies, an increasing number of HSI classification approaches have…
When simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales, stochastic noise and high-dimensionality can make simulations prohibitively expensive. The computational cost is dictated by…
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
In this paper, we introduces a Pseudo-Symplectic Neural Network (PSNN) for learning general Hamiltonian systems (both separable and non-separable) from data. To address the limitations of existing structure-preserving methods (e.g.,…
Deep neural networks have been extremely successful at various image, speech, video recognition tasks because of their ability to model deep structures within the data. However, they are still prohibitively expensive to train and apply for…
The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential…
Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
Retrieval, the initial stage of a recommendation system, is tasked with down-selecting items from a pool of tens of millions of candidates to a few thousands. Embedding Based Retrieval (EBR) has been a typical choice for this problem,…
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with…
A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…