Related papers: From Monte Carlo to neural networks approximations…

The resolution of the Poisson equation is usually one of the most computationally intensive steps for incompressible fluid solvers. Lately, Deep Learning, and especially Convolutional Neural Networks (CNN), has been introduced to solve this…

There is a growing interest in subject-specific predictions using deep neural networks (DNNs) because real-world data often exhibit correlations, which has been typically overlooked in traditional DNN frameworks. In this paper, we propose a…

A Monte Carlo algorithm is said to be adaptive if it automatically calibrates its current proposal distribution using past simulations. The choice of the parametric family that defines the set of proposal distributions is critical for good…

Recently, a class of efficient spectral Monte-Carlo methods was developed in \cite{Feng2025ExponentiallyAS} for solving fractional Poisson equations. These methods fully consider the low regularity of the solution near boundaries and…

Probabilistic prediction of sequences from images and other high-dimensional data is a key challenge, particularly in risk-sensitive applications. In these settings, it is often desirable to quantify the uncertainty associated with the…

We introduce a novel combination of Bayesian Models (BMs) and Neural Networks (NNs) for making predictions with a minimum expected risk. Our approach combines the best of both worlds, the data efficiency and interpretability of a BM with…

Solving partial differential equations in high dimensions by deep neural network has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo…

Monte Carlo simulations of physics processes at particle colliders like the Large Hadron Collider at CERN take up a major fraction of the computational budget. For some simulations, a single data point takes seconds, minutes, or even hours…

In the following article we provide an exposition of exact computational methods to perform parameter inference from partially observed network models. In particular, we consider the duplication attachment (DA) model which has a likelihood…

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the…

As the deep neural networks are being applied to complex tasks, the size of the networks and architecture increases and their topology becomes more complicated too. At the same time, training becomes slow and at some instances inefficient.…

The Poisson Generalized Linear Model (GLM) is a foundational tool for analyzing neural spike train data. However, standard implementations rely on discretizing spike times into binned count data, limiting temporal resolution and…

Discrete time stochastic optimal control problems and Markov decision processes (MDPs) are fundamental models for sequential decision-making under uncertainty and as such provide the mathematical framework underlying reinforcement learning…

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale…

Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations…

The reliable and accurate numerical approximation of the $p$-Laplacian is particularly challenging in the extreme regimes $p \to 1^{+}$ and $p \gg 1$, where the operator becomes either highly singular or strongly degenerate, often causing…

The approximation of a high-dimensional vector by a small combination of column vectors selected from a fixed matrix has been actively debated in several different disciplines. In this paper, a sampling approach based on the Monte Carlo…

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…

In supervised learning using kernel methods, we often encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space (RKHS). Large-scale finite-sum problems can be solved using efficient variants of Newton method,…

In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…