相关论文: Multiplicity of Solutions for Linear Partial Diffe…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
This work focuses on the study of partial differential equation (PDE) based basis function for Discontinuous Galerkin methods to solve numerically wave-related boundary value problems with variable coefficients. To tackle problems with…
In this study, we concern the multidimensional viscosity solutions theory of a kind of semi-linear partial differential equations (PDEs). A new definition of viscosity solution for this multidimensional semi-linear PDEs which is related to…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned…
We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincar\'e differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular…
A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…
We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…
This work introduces the families of generalized energy operators $([[.]^p]_k^+)_{k\in\mathbb{Z}}$ and $([[.]^p]_k^-)_{k\in\mathbb{Z}}$ ($p$ in $\mathbb{Z}^+$). One shows that with Lemma 1, the successive derivatives of $\big…
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…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space…
We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific geometries; a second one is to provide new…
Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…
We introduce a synthetic approach to global pluripotential theory, covering in particular the case of a compact K\"ahler manifold and that of a projective Berkovich space over a non-Archimedean field. We define and study the space of…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…