Related papers: Long-time Integration of Nonlinear Wave Equations …
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…
Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…
Recent advances in learning dynamical systems from data have shown significant promise. However, many existing methods assume access to the full state of the system -- an assumption that is rarely satisfied in practice, where systems are…
Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning-based components to support model reuse, inference, and integration across large computational processes. Despite the…
It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff…
In this paper, numerical and solitonic solutions of Korteweg de Vries(KdV) and Korteweg de Vries-Burger's (KdVB) equations with initial and boundary conditions are calculated by sinc-collocation method. The basis of method is sinc…
This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We consider the approximation of a class of dynamic partial differential equations (PDE) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the…
In this paper, we propose a novel Physics-Informed Neural Network (PINN) framework based on the Cord\`{e}s condition for solving both linear and fully nonlinear partial differential equations (PDEs) in non-divergence form, together with…
Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every…
We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory.…
We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the…
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to…
Reduction operators, i.e. the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities is investigated within the framework of singular reduction operator. A…
Neural operators have been validated as promising deep surrogate models for solving partial differential equations (PDEs). Despite the critical role of boundary conditions in PDEs, however, only a limited number of neural operators robustly…
Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein-Gordon (NLKG), the Korteweg-deVries (KdV) and water wave equations, it is shown that the leading…
The Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) methods, introduced in [42], were originally designed for solving linear partial differential equations. In this paper, we extend the LRNN-DG methods to solve…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…