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A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose…
Generative models based on flow matching have attracted significant attention for their simplicity and superior performance in high-resolution image synthesis. By leveraging the instantaneous change-of-variables formula, one can directly…
A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as…
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that…
Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems…
We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the…
A Koopman decomposition of a complex system leads to a representation in which nonlinear dynamics appear to be linear. The existence of a linear framework with which to analyse nonlinear dynamical systems brings new strategies for…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
In this work, we present a scalable Linear Matrix Inequality (LMI) based framework to verify the stability of a set of linear Partial Differential Equations (PDEs) in one spatial dimension coupled with a set of Ordinary Differential…
Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary…
Lie symmetry transformations that leave a differential equation invariant play a fundamental role in science and mathematics. Such Lie symmetry groups uniquely determine their Lie symmetry algebras. Exact differential elimination algorithms…
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
A new integrable class of Davey--Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev--Petviashvili equation by means of an asymptotically exact nonlinear…
In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural…
The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$) and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$ ($p\in (1, \infty)$)…
We propose a new multistep deep learning-based algorithm for the resolution of moderate to high dimensional nonlinear backward stochastic differential equations (BSDEs) and their corresponding parabolic partial differential equations (PDE).…
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of…