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We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and…
We consider a discrete equation, defined on the two-dimensional square lattice, which is linearizable, namely, of the Burgers type and depends on a parameter $\alpha$. For any natural number $N$ we choose $\alpha$ so that the equation…
Inverse problems for Partial Differential Equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In…
A linearizable version of multidimensional system of $n$-wave type nonlinear PDEs is proposed. This system is derived using the spectral representation of its solution via the procedure similar to the dressing method for the ISTM-integrable…
After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux…
Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization…
When numerical solution of elliptic and parabolic partial differential equations is required to be highly accurate in space, the discrete problem usually takes the form of large-scale and sparse linear systems. In this work, as an…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…
In this paper, approximate analytical solutions of nonlinear Emden-Fowler type equations are obtained by the differential transform method (DTM). The DTM is a numerical as well as analytical method for solving integral equations, ordinary…
In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order \(\geq 2\)) partial differential equations (PDEs). Our approach centers on a novel loss function…
This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. Method allows one to construct the systems of multidimensional Partial Differential…
We extend the Eruguin result exposed in the paper "Construction of the whole set of ordinary differential equations with a given integral curve" published in 1952 and construct a differential system in $\Bbb{R}^N$ which admits a given set…
We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the…
We propose a new reduced order modeling strategy for tackling parametrized Partial Differential Equations (PDEs) with linear constraints, in particular Darcy flow systems in which the constraint is given by mass conservation. Our approach…