Related papers: Extensions of the MapDE algorithm for mappings rel…
This paper is the first of a series in which we develop exact and approximate algorithms for mappings of systems of differential equations. Here we introduce the MapDE algorithm and its implementation in Maple, for mappings relating…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…
A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential…
A diffusion probabilistic model (DPM) is a generative model renowned for its ability to produce high-quality outputs in tasks such as image and audio generation. However, training DPMs on large, high-dimensional datasets such as…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a…
An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists.…
A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without…
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes…
We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations…
We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference…
We investigate the problem of recovering coefficients in scalar nonlinear ordinary differential equations that can be exactly linearized. This contribution builds upon prior work by Lyakhov, Gerdt, and Michels, which focused on obtaining a…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a…
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
The key assumption underlying linear Markov Decision Processes (MDPs) is that the learner has access to a known feature map $\phi(x, a)$ that maps state-action pairs to $d$-dimensional vectors, and that the rewards and transitions are…
In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second order nonlinear ODEs. We show that besides the conventional point, Sundman and generalized…
In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high…
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…