Related papers: Numerical Simulations in Two-Dimensional Neural Fi…
This article describes methods for the deterministic simulation of the collisional Boltzmann equation. It presumes that the transport and collision parts of the equation are to be simulated separately in the time domain. Time stepping…
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system…
To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network…
We analyze the convergence of gradient-based optimization algorithms that base their updates on delayed stochastic gradient information. The main application of our results is to the development of gradient-based distributed optimization…
This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully…
This paper describes the generation of initial conditions for numerical simulations in cosmology with multiple levels of resolution, or multiscale simulations. We present the theory of adaptive mesh refinement of Gaussian random fields…
Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their…
Multi-component polymer mixtures are ubiquitous in biological self-organization but are notoriously difficult to study computationally. Plagued by both slow single molecule relaxation times and slow equilibration within dense mixtures,…
We present an algorithm to approximate the real trilogarithm for a real argument with IEEE 754-1985 double precision accuracy. The approximation is structured such that it can make use of instruction-level parallelism when executed on…
This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is…
Deep neural networks have been successfully applied in many different fields like computational imaging, medical healthcare, signal processing, or autonomous driving. In a proof-of-principle study, we demonstrate that computational optical…
We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to…
Recent advances in deep learning have allowed neural networks (NNs) to successfully replace traditional numerical solvers in many applications, thus enabling impressive computing gains. One such application is time domain simulation, which…
As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…
Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…
This article presents an empirical validation of the functional multidimensional scaling model, a novel approach that improves the smoothness of time-varying dissimilarities in a low-dimensional space, embedding a modified Adam stochastic…
Neural Fields have emerged as a transformative approach for 3D scene representation in computer vision and robotics, enabling accurate inference of geometry, 3D semantics, and dynamics from posed 2D data. Leveraging differentiable…
Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques…
We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…