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The numerical integration of particle trajectories in curved spacetimes is fundamental for obtaining realistic models of the particle dynamics around massive compact objects such as black holes and neutron stars. Generalized algorithms…
Despite recent progress in time-series foundation models, challenges persist in improving representation learning and adapting to diverse downstream tasks. We introduce a General Time-series Model (GTM), which advances representation…
In this paper, we analyze and provide numerical illustrations for a moving finite element method applied to convection-dominated, time-dependent partial differential equations. We follow a method of lines approach and utilize an underlying…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in…
A symplectic pseudospectral time-domain (SPSTD) scheme is developed to solve Schrodinger equation. Instead of spatial finite differences in conventional finite-difference time-domain (FDTD) method, the fast Fourier transform is used to…
This paper presents a spatial two-grid (STG) compact difference scheme for a two-dimensional (2D) nonlinear diffusion-wave equation with variable exponent, which describes, e.g., the propagation of mechanical diffusive waves in viscoelastic…
In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(\frac{1}{2},1)$. A sharp regularity estimate of the mild solution and the numerical…
We introduce PRISM, a method for real-time filtering in a probabilistic generative model of agent motion and visual perception. Previous approaches either lack uncertainty estimates for the map and agent state, do not run in real-time, do…
We present a high-order space-time discretization equipped with fully-discrete entropy stability properties for general choices of volume and surface quadrature rules. The formulation uses flux reconstruction (FR) in the spatial dimension…
Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a…
We consider fully discrete numerical schemes for a downscaling data assimilation algorithm aimed at approximating the velocity field of the 2D Navier-Stokes equations corresponding to given coarse mesh observational measurements. The time…
In this work, we present an efficient approach to solve nonlinear high-contrast multiscale diffusion problems. We incorporate the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and…
Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…
In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$. The basic idea is to add and…
Stable diffusion is an outstanding image generation model for text-to-image, but its time-consuming generation process remains a challenge due to the quadratic complexity of attention operations. Recent token merging methods improve…
This work considers to numerically solve a subdiffusion equation involving constant time delay $\tau$ and Riemann-Liouville fractional derivative. First, a fully discrete finite element scheme is developed for the considered problem under…
The diffusion approximation of stochastic gradient descent (SGD) in current literature is only valid on a finite time interval. In this paper, we establish the uniform-in-time diffusion approximation of SGD, by only assuming that the…
In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Gr\"unwald formulae is…
This article proposes a new class of general linear method with $p=q$ and $r=s=p+1$. The construction of the present method is carried out using order conditions and error minimization subject to $A$- stability constraints. The proposed…