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Diffusion models play a pivotal role in contemporary generative modeling, claiming state-of-the-art performance across various domains. Despite their superior sample quality, mainstream diffusion-based stochastic samplers like DDPM often…
This study presents a numerical analysis of the Field-Noyes reaction-diffusion model with nonsmooth initial data, employing a linear Galerkin finite element method for spatial discretization and a second-order exponential Runge-Kutta scheme…
Simulation-based techniques such as variants of stochastic Runge-Kutta are the de facto approach for inference with stochastic differential equations (SDEs) in machine learning. These methods are general-purpose and used with parametric and…
This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carath\'eodory…
Neural networks have proven to be efficient surrogate models for tackling partial differential equations (PDEs). However, their applicability is often confined to specific PDEs under certain constraints, in contrast to classical PDE solvers…
The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the…
In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for…
This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank…
Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently,…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
The application of Runge-Kutta schemes designed to enjoy a large region of absolute stability can significantly increase the efficiency of numerical methods for PDEs based on a method of lines approach. In this work we investigate the…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be…
In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for…
In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more…
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…
In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain…
In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation…